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We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L 2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori , each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field. • Semi-implicit FV/FE method for incompressible viscous and resistive MHD equations. • Well-balanced and exactly divergence-free on general unstructured mixed-element grids. • Constrained L2 projection for an exactly divergence-free reconstruction. • Thorough study of the lid-driven MHD cavity problem (reference solution is provided). • Stable long-time simulation of Grad-Shafranov equilibria in 3D tokamak geometries. [ABSTRACT FROM AUTHOR]

This paper presents a model order reduction method to accelerate broadband topology optimization of structural-acoustic interaction systems by coupling Finite Element Methods and Boundary Element Methods. The finite element method is used for simulating thin-shell vibration and the boundary element method for exterior acoustic fields. Moreover, the finite element and boundary element methods are implemented in the context of isogeometric analysis, whereby the geometric accuracy and high order continuity of Kirchhoff-Love shells can be guaranteed and meantime no meshing is necessary. The topology optimization method takes continuous material interpolation functions in the density and bulk modulus, and adopts adjoint variable methods for sensitivity analysis. The reduced order model is constructed based on second-order Arnoldi algorithm combined with Taylor's expansions which eliminate the frequency dependence of the system matrices. Numerical results show that the proposed algorithm can significantly improve the efficiency of broadband topology optimization analysis. • Broadband topology optimization of 3D structural-acoustic interaction with reduced order isogeometric FEM/BEM. • Model order reduction is conducted for topology optimization of structural-acoustic interaction by coupling FEM/BEM. • The isogeometric analysis is applied to the topology optimization framework with reduced order FEM/BEM. [ABSTRACT FROM AUTHOR]

In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite aero-structures is presented. The crucial novelty lies in the introduction of A-harmonicity in the local approximation spaces, which in contrast to Babuška and Lipton (2011) [30] is enforced more efficiently via a constraint in the local eigenproblems. This significant modification leads to excellent approximation properties, which turn out to be essential to capture accurately material strains and stresses with a low dimensional approximation space, hence maximizing model order reduction. The implementation of the framework in the Distributed and Unified Numerics Environment (DUNE) software package, as well as a detailed description of all components of the method are presented and exemplified on a composite laminated beam under compressive loading. The excellent parallel scalability of the method, as well as its superior performance compared to the related, previously introduced GenEO method are demonstrated on two realistic application cases, including a C-shaped wing spar with complex geometry. Further, by allowing low-cost approximate solves for closely related models or geometries this efficient, novel technology provides the basis for future applications in optimization or uncertainty quantification on challenging problems in composite aero-structures. [ABSTRACT FROM AUTHOR]

In this paper, we develop a class of high-order conservative methods for simulating non-equilibrium radiation diffusion problems. Numerically, this system poses significant challenges due to strong nonlinearity within the stiff source terms and the degeneracy of nonlinear diffusion terms. Explicit methods require impractically small time steps, while implicit methods, which offer stability, come with the challenge to guarantee the convergence of nonlinear iterative solvers. To overcome these challenges, we propose a predictor-corrector approach and design proper implicit-explicit time discretizations. In the predictor step, the system is reformulated into a nonconservative form and linear diffusion terms are introduced as a penalization to mitigate strong nonlinearities. We then employ a Picard iteration to secure convergence in handling the nonlinear aspects. The corrector step guarantees the conservation of total energy, which is vital for accurately simulating the speeds of propagating sharp fronts in this system. For spatial approximations, we utilize local discontinuous Galerkin finite element methods, coupled with positive-preserving and TVB limiters. We validate the orders of accuracy, conservation properties, and suitability of using large time steps for our proposed methods, through numerical experiments conducted on one- and two-dimensional spatial problems. In both homogeneous and heterogeneous non-equilibrium radiation diffusion problems, we attain a time stability condition comparable to that of a fully implicit time discretization. Such an approach is also applicable to many other reaction-diffusion systems. • High order conservative LDG-IMEX methods for non-equilibrium radiation diffusion problems are developed. • Degeneracy, strong nonlinearity and stiff source are all taken into account. • A predictor-corrector procedure based on an IMEX scheme is proposed. • Only mildly nonlinear system needs to be solved and is robust for convergence. • The scheme is of high order, conservative, positivity-preserving and allow large time steps. [ABSTRACT FROM AUTHOR]

This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial discretization of the model problem using constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). This approach consists of two stages. First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. The multiscale basis functions are obtained in the second stage using the auxiliary space by solving local energy minimization problems over the oversampling domains. The basis functions have exponential decay outside the corresponding local oversampling regions. We shall consider the first and second-order explicit exponential Runge-Kutta approach for temporal discretization and to build a fully discrete numerical solution. The exponential integration strategy for the time variable allows us to take full advantage of the CEM-GMsFEM as it enables larger time steps due to its stability properties. We derive the error estimates in the energy norm under the regularity assumption. Finally, we will provide some numerical experiments to sustain the efficiency of the proposed method. • Multiscale reduction methods are efficient and accurate to solve problems with high-contrast media. • The presence of high-contrast coefficients reduces the stability region of time discretization. • Implicit methods must solve nonlinear equations, which can be a bottleneck computation. • Exponential integrators are alternative techniques and use robust time-stepping. [ABSTRACT FROM AUTHOR]

Stochastic Klein–Gordon–Schrödinger (KGS) equations are important mathematical models and describe the interaction between scalar nucleons and neutral scalar mesons in the stochastic environment. In this paper, we propose novel structure-preserving schemes to numerically solve stochastic KGS equations with additive noise, which preserve averaged charge evolution law, averaged energy evolution law, symplecticity, and multi-symplecticity. By applying central difference, sine pseudo-spectral method, or finite element method in space and modifying finite difference in time, we present some charge and energy preserved fully-discrete schemes for the original system. In addition, combining the symplectic Runge-Kutta method in time and finite difference in space, we propose a class of multi-symplectic discretizations preserving the geometric structure of the stochastic KGS equation. Finally, numerical experiments confirm theoretical findings. [ABSTRACT FROM AUTHOR]

The coupling effects in multiphysics processes are often neglected in designing multiscale methods. The coupling may be described by a non-positive definite operator, which in turn brings significant challenges in multiscale simulations. In the paper, we develop a regularized coupling multiscale method based on the generalized multiscale finite element method (GMsFEM) to solve coupled thermomechanical problems, and it is referred to as the coupling generalized multiscale finite element method (CGMsFEM). The method consists of defining the coupling multiscale basis functions through local regularized coupling spectral problems in each coarse-grid block, which can be implemented by a novel design of two relaxation parameters. Compared to the standard GMsFEM, the proposed method can not only accurately capture the multiscale coupling correlation effects of multiphysics problems but also greatly improve computational efficiency with fewer multiscale basis functions. In addition, the convergence analysis is also established, and the optimal error estimates are derived, where the upper bound of errors is independent of the magnitude of the relaxation coefficient. Several numerical examples for periodic, random microstructure, and random material coefficients are presented to validate the theoretical analysis. The numerical results show that the CGMsFEM shows better robustness and efficiency than uncoupled GMsFEM. [ABSTRACT FROM AUTHOR]

In this paper, the Petrov-Galerkin immersed finite element method is developed for simulating flow in fractured porous media. We consider the hybrid-dimensional fracture model as an elliptic interface problem with multiple physical coupling interface conditions, and embed the information of low-dimensional fractures into the local functions of interface elements. The immersed finite element space is constructed by explicitly representing the local solution functions in the interface element. This approach addresses the limitation of conforming meshes. More importantly, it can capture both the continuity and discontinuity of pressure, making it effective for both conductive and blocking fractures. We provide a detailed illustration of how to construct local functions on triangular and rectangular meshes. Finally, numerical examples and benchmarks are presented to validate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a selective discontinuous Galerkin (SDG) method and a polynomial preserving recovery (PPR) method are developed and integrated with the immersed-finite-element particle-in-cell (IFE-PIC) method, in order to carry out the plasma-material interaction simulation on adaptive Cartesian meshes (ACM). To significantly save the computational cost in practice, the PIC simulation often needs to use Cartesian meshes for the whole domain and the ACM for some focused regions in the plasma-material interaction problems. Therefore, we utilize the SDG method with immersed finite elements for the field solver of the IFE-PIC method, as the key technique to allow the local Cartesian mesh refinement which will generate hanging nodes. To minimize the degree of freedom of the SDG on the ACM and reduce the computational cost, a new selective technique of the discontinuous global basis functions is proposed for the SDG. Various types of nodes in the ACM are discussed for the implementation of this selective technique, including the hanging nodes. Meanwhile, the gathering and scattering steps of the PIC simulation also need to be appropriately performed on the ACM. The PPR method is a generic gradient recovery technique to be incorporated into the IFE-PIC method for more accurate force deposition on the ACM. In addition, two other algorithmic structures of the traditional IFE-PIC method have been modified to accommodate the ACM in the simulation, including a new mapping array structure for the particle positioning and a new charge deposition with some special particle positions in the ACM. As a result, the integrated immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method can efficiently perform the plasma-material interactions simulation on the Cartesian meshes with local mesh refinement independent of the interface. Several numerical experiments are performed to demonstrate the convergence, efficiency, and applicability of the proposed ISDG-PIC method. • Immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method. • Polynomial preserving recovery (PPR) method. • A new mapping array structure for the particle positioning. • A new charge deposition with some special particle positions in the adaptive Cartesian meshes. • A series of numerical examples for validation. [ABSTRACT FROM AUTHOR]

• Construction of transparent boundary conditions (TBCs) for exural wave cavity scattering in two dimensions, ensuring the satisfaction of the sommerfeld radiation conditions. • Deduction of a decomposed problem for the biharmonic plate wave equation using two auxiliary functions and establishment of the unique-ness of the solution for the decomposed problem. • Proposal of the linear _nite element method with interior penalty term (IP-FEM) and boundary penalty term (BP-FEM) to solve the coupled boundary helmholtz and modi_ed helmholtz equations, leading to stable numerical solutions. • Introduction of an analytical solution for exural wave scattering by a circular cavity with a clamped boundary, facilitating comparative analysis. Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution on the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a high-order accurate finite element method for incompressible variable density flow. The method uses high-order Taylor-Hood velocity-pressure elements in space and backward differentiation formula (BDF) time stepping in time. This way of discretization leads to two main issues: (i) a saddle point system that needs to be solved at each time step; (i i) a stability issue when the viscosity of the flow goes to zero or if the density profile has a discontinuity. We address the first issue by using Schur complement preconditioning and artificial compressibility approaches. We observed similar performance between these two approaches. To address the second issue, we introduce a modified artificial Guermond-Popov viscous flux where the viscosity coefficients are constructed using a newly developed residual-based shock-capturing method. Numerical validations confirm high-order accuracy for smooth problems and accurately resolved discontinuities for problems in 2D and 3D with varying density ratios. • A robust and highly accurate finite element method for incompressible variable density flow is presented. • A modified Schur complement preconditioning is proposed that has comparable performance to artificial compressibility methods. • The proposed viscous regularization is kinetic energy stable. • A newly developed residual-based shock-capturing method is proposed. • High-order accuracy for smooth problems and accurately resolved discontinuities for problems with varying density ratios. [ABSTRACT FROM AUTHOR]

This paper proposes a strategy to solve the problems of the conventional s-version of finite element method (SFEM) fundamentally. Because SFEM can reasonably model an analytical domain by superimposing meshes with different spatial resolutions, it has intrinsic advantages of local high accuracy, low computation time, and simple meshing procedure. However, it has disadvantages such as accuracy of numerical integration and matrix singularity. Although several additional techniques have been proposed to mitigate these limitations, they are computationally expensive or ad-hoc, and detract from the method's strengths. To solve these issues, we propose a novel strategy called B-spline based SFEM. To improve the accuracy of numerical integration, we employed cubic B-spline basis functions with C 2 -continuity across element boundaries as the global basis functions. To avoid matrix singularity, we applied different basis functions to different meshes. Specifically, we employed the Lagrange basis functions as local basis functions. The numerical results indicate that using the proposed method, numerical integration can be calculated with sufficient accuracy without any additional techniques used in conventional SFEM. Furthermore, the proposed method avoids matrix singularity and is superior to conventional methods in terms of convergence for solving linear equations. Therefore, the proposed method has the potential to reduce computation time while maintaining a comparable accuracy to conventional SFEM. [ABSTRACT FROM AUTHOR]

In this paper, we propose a finite element method for solving the coupled model of classical drift-diffusion equations and Schrödinger-Poisson equations in simulating a resonant tunneling diode (RTD). The coupling coefficient Θ and the quantum electron density n qu are defined by weighted integrals of | ψ p | 2 where ψ p is the wave function of Schrödinger equation for each momentum p ∈ R. We first prove a growth rate of | ψ p | as | p | → + ∞. This enables us to truncate the momentum space R to a bounded interval [ − P , P ] , which corresponds to an energy interval [ 0 , E P ]. We propose an energy-adaptive algorithm based on a posteriori error estimate for computing Θ and n qu approximately on [ 0 , E P ]. The algorithm reduces the number of nodes greatly by capturing resonance peaks accurately and adjusting the partition of [ 0 , E P ] adaptively. Numerical experiment for an RTD shows that, based on energy-adaptive grids, the Gummel method for solving the coupled model is robust and can converge in several iterative steps. • We first prove a growth rate of | ψ p | as | p | → + ∞. Based on this, the momentum space can be truncated to a bounded interval. • We propose an adaptive algorithm which adjusts the partition of the momentum domain based on a posteriori error estimate. • We propose a modified Gummel algorithm (MGA) for solving the coupled SPDD model based on the adaptive algorithm. [ABSTRACT FROM AUTHOR]

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective boundary conditions. The fixed point method to solve the system is shown to be monotone. The discretization is done with a P 1 Finite Element Method. The convolution integrals are precomputed at every vertex of the mesh and stored in compressed hierarchical matrices, using Partially Pivoted Adaptive Cross-Approximation. Then the fixed point iterations involve only matrix vector products. The method is O (N N 3 ln ⁡ N) , with respect to the number of vertices, when everything is smooth. A numerical implementation is proposed and tested on two examples. As there are some analogies with ray tracing the programming is complex. • We present a method and a an algorithm using an H-matrix compression scheme to compute the temperature of a gas under electromagnetic radiations. • The method is now capable of handling reflective boundaries. • The paper improves on previously existing results and the numerical methods use state of the art computational tools. [ABSTRACT FROM AUTHOR]

As the number of processors on supercomputers has increased dramatically, there is a growing interest in developing scalable algorithms with a high degree of parallelism for extreme-scale reservoir simulation. However, traditional simulators and algorithms for such nonlinear problems are usually based on the family of time-marching methods, where parallelization is restricted to the spatial dimension only. In this paper, we propose a family of parallel-in-time (PinT) reservoir algorithms for solving multiphase flow problems in porous media to fully exploit the parallelism of supercomputers, in which the design of efficient nonlinear and linear space-time algorithms plays an essential role. More precisely, we first introduce a space-time mixed finite element method for the fully-implicit discretization. The nonlinear system arising at each parallel-in-time step is solved by a variant of bound-preserving Newton methods with a nonlinear elimination preconditioner, where the corresponding linear system is solved by the space-time restricted additive Schwarz (stRAS) method. Since the straightforward extension of one-level stRAS to multilevel does not work because of the pollution effects from the temporal direction, we accordingly present a nonstandard V-cycle multilevel stRAS method with a pollution-removing strategy to accelerate the convergence and improve the robustness. Numerical experiments are presented to demonstrate that the aforementioned parallel-in-time algorithm can not only achieve a high degree of parallelism in accurately resolving reservoir transport features in a heterogeneous medium, but also successfully circumvent the convergence and scalability issues associated with the constraint of large time steps and the boundedness requirement of the solution. • Space-time mixed finite element scheme for parallel in-time discretization. • Active-set reduced-space method with nonlinear elimination for bound-preserving solution. • Space-time restricted additive Schwarz algorithm for linear preconditioning. • Nonlinear parallel-in-time simulations with up to 13,872 processor cores. [ABSTRACT FROM AUTHOR]

The optimal control of thermally convective flows is usually modeled by an optimization problem with constraints of Boussinesq equations that consist of the Navier-Stokes equation and an advection-diffusion equation. This optimal control problem is challenging from both theoretical analysis and algorithmic design perspectives. For example, the nonlinearity and coupling of fluid flows and energy transports prevent direct applications of gradient type algorithms in practice. In this paper, we propose an efficient numerical method to solve this problem based on the operator splitting and optimization techniques. In particular, we employ the Marchuk-Yanenko method leveraged by the L 2 -projection for the time discretization of the Boussinesq equations so that the Boussinesq equations are decomposed into some easier linear equations without any difficulty in deriving the corresponding adjoint system. Consequently, at each iteration, four easy linear advection-diffusion equations and two degenerated Stokes equations at each time step are needed to be solved for computing a gradient. Then, we apply the Bercovier-Pironneau finite element method for space discretization, and design a BFGS type algorithm for solving the fully discretized optimal control problem. We look into the structure of the problem, and design a meticulous strategy to seek step sizes for the BFGS efficiently. Efficiency of the numerical approach is promisingly validated by the results of some preliminary numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed numerical scheme for a single-phase compressible flow without diffusion of a multi-component mixture in porous media with the higher-order approximation in both space and time. The mathematical model consists of Darcy velocity, transport equations for each component of a mixture, pressure equation and associated relations for physical quantities such as viscosity or equation of state. The discrete problem is obtained using a combination of the discontinuous Galerkin method for the transport equations and the mixed-hybrid finite element method for the Darcy velocity and the pressure equation. In both methods the higher-order approximation is used. The resulting nonlinear problem for concentrations is solved with the fully mass-conservative iterative IMPEC method. Experimental order of convergence analysis (EOC) and some numerical experiments of a 2D flow are carried out. • A new numerical scheme is presented for simulation of single-phase flow in a porous medium. • The second-order approximation is attained in both space and time. • The second-order approximation in both time and space is verified using the experimental order of convergence. • Several computational examples illustrate advantageous features of our approach. [ABSTRACT FROM AUTHOR]

We present an open-source adaptive front-tracking solver for biological capsules in viscous flows. The membrane elastic and bending forces are solved on a Lagrangian triangulation using a linear Finite Element Method and a paraboloid fitting method. The fluid flow is solved on an octree adaptive grid using the open-source platform Basilisk. The Lagrangian and Eulerian grids communicate using an Immersed Boundary Method by means of Peskin-like regularized Dirac delta functions. We demonstrate the accuracy of our solver with extensive validations: in Stokes conditions against the Boundary Integral Method, and in the presence of inertia against similar (but not adaptive) front-tracking solvers. Excellent qualitative and quantitative agreements are shown. We then demonstrate the robustness of the present solver in challenging cases featuring extreme membrane deformations, very large computational domains and high volume fractions. Moreover, we illustrate the capability of the solver to simulate inertial capsule-laden flows in complex STL-defined geometries, opening the door for bioengineering applications featuring large three-dimensional channel structures. The source code and all the test cases presented in this paper are freely available. • Front Tracking/Immersed Boundary method to simulate a 3D flow laden with deformable capsules in a complex geometry. • Extension of the method to adaptive mesh refinement on Cartesian octree grids. • Implementation in the open-source software Basilisk. • Complex geometries are considered via STL files. • Comprehensive validations to verify the robustness and accuracy of the method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a deep learning based reduced order modeling method for stochastic underground flow problems in highly heterogeneous media. We aim to utilize supervised learning to build a reduced surrogate mapping from the stochastic parameter space that characterizes the possible highly heterogeneous media to the solution space of a stochastic flow problem. Offline preparation includes collecting a set of full-order solutions to form a global snapshot space and extracting dominating POD modes using a particular well-designed spectral problem to represent full-order solutions. Due to the small dimension of reduced-order solutions, the complexity of neural network is significantly lightened, which effectively relieves the burden of training. We adopt the generalized multiscale finite element method (GMsFEM), where a set of local multiscale basis functions that can capture the heterogeneity of the media and source information are constructed to efficiently generate globally defined snapshot space. In an online stage, one can rapidly obtain the reduced-order pressure and then corresponding full-order solution is recovered using a linear transformation. Due to the decoupling of offline and online procedures, the proposed reduced order method serves as a non-intrusive tool that can achieve fast simulations independent of input parameters. Rigorous theoretical analyses are provided and extensive numerical experiments for linear and nonlinear stochastic flows are provided to verify the superior performance of the proposed method. • Develop a reduced order neural network for highly heterogeneous stochastic flow. • Use K-fold cross validation to partition data into training and test parts. • Compare the deep learning approach with a previous reduced order method, whose error bound is rigorously proved in the paper. [ABSTRACT FROM AUTHOR]

• We have designed a generalized multiscale model for heterogeneous Cosserat media without scale separation. • We have designed multiscale spaces with and without coupling of processes and investigated them. • Numerical results are presented that show the proposed methods provide a good accuracy with a few degrees of freedom. In this paper, we develop a multiscale computational approach for heterogeneous Cosserat media without scale separation and high contrast. Cosserat medium has been used in many applications to describe materials whose particles possess rotational degrees of freedom. Many previous findings investigate Cosserat medium in the presence of heterogeneities. Approaches that include homogenization and numerical homogenization have been developed for problems with scale separation, such as periodicity. In these approaches, macroscopic models are derived that show the resulting equations can be Cauchy medium or Cosserat medium depending on the Cosserat intrinsic length scale. In our paper, we consider computational macroscopic models for Cosserat medium without scale separation. We use Generalized Multiscale Finite Element Method (GMsFEM) to derive a macroscopic model on a coarse computational grid, that does not resolve the small scales and the contrast. In this approach, multiple macroscopic basis functions are used to describe the small scales. The resulting macroscopic model is similar to multicontinua model as it contains multiple macroscopic parameters at each macroscopic point (or grid). We present numerical results for different heterogeneous media types. Our numerical results show that using a few multiscale basis functions (i.e., macroscopic parameters), we can achieve a good accuracy. [ABSTRACT FROM AUTHOR]

Partial discharges and the electrical treeing represent one of key aspects for the ageing of the insulating parts of electrical components. To predict the end of life of dielectric systems several different modelling approaches have been developed. One of the most promising is the modelling using first principles and a set of partial differential equations. In this field, one of the toughest challenges is to cope with the very different scales of the treeing (each branch is a few micrometers wide) and of electrical components (raging from a few centimetres to a few meters). In this paper we propose an approach, based on the finite element and finite volume methods, which uses two completely independent grids to describe internal defects and electrical components. Some interpolation algorithms have been developed to couple the solution on the two grids. We have also tested the accuracy of our approach by comparing it with previously published methods that use only one grid. • Partial discharges simulation using plasma physics for challenging geometries. • Efficient treatment of tiny internal defects in electric equipments. • Grid decoupling between a gaseous defect and a dielectric bulk. [ABSTRACT FROM AUTHOR]

In this paper, the Half-Space Matching (HSM) method, first introduced for scalar problems, is extended to elastodynamics, to solve time-harmonic 2D scattering problems, in locally perturbed infinite anisotropic homogeneous media. The HSM formulation couples a variational formulation around the perturbations with Fourier integral representations of the outgoing solution in four overlapping half-spaces. These integral representations involve outgoing plane waves, selected according to their group velocity, and evanescent waves. Numerically, the HSM method consists in a finite element discretization of the HSM formulation, together with an approximation of the Fourier integrals. Numerical results, validating the method, are presented for different materials, isotropic and anisotropic. Comparisons with the Perfectly Matched Layers (PML) method are performed for several anisotropic materials. These results highlight the robustness of the HSM method compared to the sensitivity of the PML method with respect to its parameters. [ABSTRACT FROM AUTHOR]

In this paper, the multivariate well-conditioned asymptotic waveform evaluation (MWCAWE) algorithm is presented. It is based on a univariate version of the algorithm, previously introduced for fast frequency sweeps. Taking advantage of the robustness of this univariate algorithm, the MWCAWE enables to effectively generate multi-parameter reduced-order models. Additionally, a residue-based contour following approach is introduced for a two-stage generation of a projection basis, here applied on bi-variate acoustic problems where frequency and a material parameter are in focus. The effectiveness of the proposed method is demonstrated on two poro-acoustic problems, highlighting both the good convergence property of the algorithm and the potential of the multi-patch strategy for parametric applications. • A parametric model order reduction is demonstrated on vibro-acoustic problems. • A multivariate moment-matching algorithm is proposed based on the WCAWE algorithm. • A residue-based, contour-following sampling is introduced for multi-interval strategies. • The effectiveness of the method is demonstrated on 3D bivariate FE problems. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of unsteady Stokes equations in patient-specific arterial-like domains in 3D. A Stokes-like solver is a necessary component in a more sophisticated nonlinear Navier-Stokes method, for which several multilevel domain decomposition methods have been introduced recently. Because of the complex geometry, the construction and the solve of the coarse problem usually take a large percentage of the total compute time. In this paper, we introduce a parameterized one-dimensional Stokes solver defined along the centerline of the artery and use its stabilized finite element discretization to construct a coarse preconditioner. With suitable 3D-to-1D restriction and 1D-to-3D extension operators on fully unstructured meshes, a two-level additive Schwarz preconditioner can be constructed. Some numerical experiments for flows in realistic arteries are presented to show the efficiency and robustness of the new coarse preconditioner whose computational cost is considerably lower than the existing three-dimensional coarse preconditioners. • A two-level DDM is proposed for unsteady Stokes flows in patient-specific arteries. • The coarse preconditioner is obtained by a 1D Stokes model on the centerline. • Multiscale 1D-3D restriction and extension matrices are introduced by interpolations. • Numerical results show high efficiencies of the proposed method for complex arteries. [ABSTRACT FROM AUTHOR]

In this paper, the immersed finite element (IFE) coupled with the conservative semi-Lagrangian (CSL) kinetic scheme is developed for plasma-material interactions. The proposed method (IFE-CSL) enjoys respective advantages of the IFE and CSL, i.e., the flexibility and efficiency for treating complex boundary conditions, mass conservation and being free from the Courant-Friedrichs-Lewy (CFL) condition. In the current IFE-CSL, the IFE method based on structured interface-independent meshes is developed for the spatial discretization of the field equation, which provides an accurate approach with convenient implementations to solve the field problems with irregular interface. The CSL scheme combined with an immersed method termed half-way ghost-cell is proposed for the spatial discretization of the Vlasov equation, which enables the proposed method to exactly conserve mass and conveniently handle curved boundaries. Then the IFE and CSL method are coupled via the charge density depending on full kinetic or hybrid kinetic model, where the appropriate IFE solver is developed for the linear or nonlinear Poisson equation, respectively. In the current IFE-CSL, different geometries can be treated automatically for both IFE and CSL through the specific geometric information. As a result, the proposed CSL-IFE can be conveniently and efficiently applied to simulate the plasma-material interactions with structured interface-independent meshes. Finally, several numerical experiments are performed to demonstrate the capabilities of the proposed method. • The immersed finite element method is coupled to the conservative semi- Lagrangian scheme (IFE-CSL) for the first time. • This IFE-CSL can be used for plasma simulations involving complex interfaces. • The inject and charge accumulation boundary conditions are implemented. • The nonlinear IFE solver is developed for hybrid kinetic simulation. • This CSL-IFE is accurate and enjoys mass conservation, positivity preservation as well extra-large time stepping sizes. [ABSTRACT FROM AUTHOR]

Conventionally, a finite element framework for coupled hydro-mechanical-chemical processes in porous media adopts simplified chemical reactions, which consider superficial chemical effects of complicated multiphysics mechanisms. This paper presents a novel computational framework for geochemistry of reactive multi-component systems to explicitly capture the mineral dissolution/precipitation of saturated porous media. The resultant of this framework can be applied in oil recovery engineering and used to simulate the production scenarios. The model is also fundamental for studying on creep behaviors from the view of the coupling effects between hydro-geochemistry. In particular, a complete chemistry system of ▪ is considered, including intra-aqueous chemical reactions and kinetic rate-limiting chemical reactions. Besides, an algorithm is proposed such that the complete chemistry system is scalable for various chemical reactions and is easily incorporated into existing finite element frameworks for hydro-mechanical processes of porous media. This complete chemistry system is implemented to establish a local chemical equilibrium. The geochemical reactive transport equations are then sequentially coupled with the balance laws such that coupling among fluid advection-diffusion, solid deformation, chemical concentration advection-diffusion, intra-aqueous chemical reaction, and kinetic rate-limiting chemical reaction can be simulated. Phenomenological relations for porosity and permeability changes with mineral saturation in solutions are employed. We first verify the proposed framework to capture the advection-diffusion of different chemical species under steady-state and transient conditions. Then a consolidation problem under the chemical reactions is presented to further demonstrate the capabilities of the proposed computational framework. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel interface preserving and residual-based adaptivity procedure for the phase-field modeling of a fully Eulerian fluid-structure interaction. The proposed adaptive fully Eulerian procedure involves a fixed background unstructured finite element mesh on which the fluid-structure boundary is treated implicitly via a diffused interface description. We employ the stabilized finite element formulation and a partitioned iterative procedure to solve the coupled system of the Allen-Cahn phase-field equation with the unified momentum equation comprising solid and fluid dynamics. To evaluate the solid stresses, the left Cauchy-Green deformation tensor is convected at each time step to trace the evolution of the solid strain in the Eulerian reference frame. We utilize a residual based error indicator and the newest vertex bisection algorithm for the adaptive refinement/coarsening of the unstructured mesh. Through the adaptive finite element method, we aim to preserve the profile of the diffused interface in the presence of convective distortion. We propose an early stopping criterion for the adaptivity of the interface dynamics by combining the effects of the phase-field interface thickness parameter and the mesh resolution. In addition, we control the size of the refined/coarsened elements by introducing a characteristic length scale governed by a weighted harmonic mean of the desired bulk and the interface mesh resolutions. The proposed stopping criterion and the characteristic length scale provide significant reductions in the computational cost while simultaneously ensuring convergence properties of the coupled fluid-structure system. We perform a detailed convergence and accuracy analysis via two benchmark problems namely, the pure solid system and a coupled fluid-solid system with an interface in a rectangular domain. We further provide a comparative study between feature- and residual based adaptive strategies. We systematically assess the performance of the adaptive procedure in terms of conservation properties and computational efficiency. Finally, we demonstrate our fully Eulerian adaptive solver to simulate the contact and bouncing phenomenon between an elastic solid and a rigid wall. • Novel interface preserving adaptivity on unstructured grid for fully Eulerian FSI. • Early stopping criterion based on interface thickness and mesh resolution. • New characteristic length scale by weighted harmonic mean of bulk and interface resolutions. • Significant improvements in conservation properties with proposed adaptivity. • Demonstration of FSI contact dynamics via bouncing elastic ball problem. [ABSTRACT FROM AUTHOR]

This paper presents a novel Generalized Finite Element Method with global-local enrichment (GFEMgl) to solve time-dependent parabolic and hyperbolic problems with subscale features in both space and time. Compared with currently available GFEMgl, the proposed method requires a solution of local problems to solve transient PDEs, possibly with a different time integrator from the global problem or other local problems. The fine-scale solution can be easily retrieved. The proposed method is tested for the solution of the heat, advection, and advection-diffusion equations whose accuracy, stability, scalability, and convergence are thoroughly analyzed. Compared to direct analysis, numerical results show that the proposed method has the following properties 1) the accuracy closely matches direct analysis result with a fine mesh; 2) critical time step size is loosened; 3) optimal convergence rate is achieved. Moreover, the proposed method allows local problems to be turned off if their solutions are not helpful for the current global time step, which can further improve its efficiency. • Subscale features of parabolic and hyperbolic problems are captured on coarse meshes. • Method is optimally convergent on uniform meshes. • No a-priori knowledge about the solution is required. • Critical time step is larger than for methods that provide similar accuracy. • Adaptive enrichment reduces problem size. [ABSTRACT FROM AUTHOR]

In this paper, we consider the three-dimensional (3D) nonlinear Maxwell's equations in the optical media characterized by linear Lorentz dispersion, nonlinear Kerr effect and delayed Raman scattering. Using Crank-Nicolson time discretization and special treatment of the nonlinearity, we propose a second order accurate (in time) finite element method for the nonlinear problem. Fully discrete energy stability analysis is established to show that the scheme is unconditionally stable. The nonlinear system is solved by Newton-Krylov method. In order to efficiently solve the linearized algebraic system in each Newton sub-iteration, we further develop an effective preconditioner by using suitable approximations to the nonlinearity and the auxiliary space preconditioning technique. Numerical experiments are provided to examine the accuracy, the energy stability, the parallel scalability and the robustness in preconditioning of our proposed scheme. We then apply the scheme to simulate the spatial soliton propagation in 3D nonlinear glass. We also demonstrate the performance of the scheme in handling complex geometry through simulating the wave scattering by a spherical hole in the nonlinear optical glass. • Unconditionally energy-stable FE method is developed for the nonlinear Maxwell equations. • Fully discrete energy stability analysis is established for the proposed scheme. • Robust and scalable Newton-Krylov iterative method is proposed for the discrete problem. • Extensive numerical examples are provided to show the robustness and efficiency of the solver. [ABSTRACT FROM AUTHOR]

In this paper, we propose a recycling preconditioning method with auxiliary tip subspace for solving a sequence of highly ill-conditioned linear systems of equations of different sizes arising from elastic crack propagation problems discretized by the extended finite element method. To construct a Schwarz type preconditioner, the finite element mesh is decomposed into crack tip subdomains, which contain all the degrees of freedom (DOFs) of the branch enrichment functions, and regular subdomains, which contain the standard DOFs and the DOFs of the Heaviside and the Junction enrichment functions. As cracks propagate these subdomains are modified accordingly, and the subdomain matrices are constructed as the restriction of the global matrix to the subdomains. In the overlapping Schwarz preconditioners, the crack tip subproblems are solved exactly and the regular subproblems are solved by some inexact solvers, such as ILU. We consider problems with and without crack intersections and develop a simple scheme to update, instead of re-computing, the subdomain problems as cracks propagate, in which only crack tip subdomains are updated around the new crack tips and all the regular subdomains remain unchanged. Therefore, no extra search is required, and the sizes of crack tip subproblems do not increase as cracks propagate, which greatly saves the computational cost. Moreover, starting from the second system, the Krylov subspace method uses a nontrivial initial guess constructed using the solution of the previous system with a modification around the new crack tips. The strategy accelerates the convergence remarkably. Numerical experiments demonstrate the efficiency of the proposed algorithms applied to problems with several types of cracks. [ABSTRACT FROM AUTHOR]

In this paper, we consider flow and transport problems in thin domains. Modeling problems in thin domains occur in many applications, where thin domains lead to some type of reduced models. A typical example is one dimensional reduced-order model for flows in pipe-like geometries (e.g., blood vessels). In many reduced-order models, the model equations are described apriori by some analytical approaches. In this paper, we propose the use of multiscale methods, which are alternative to reduced-order models and can represent reduced-dimension modeling by using fewer basis functions (e.g., the use of one basis function corresponds to one dimensional approximation). The mathematical model considered in the paper is described by a system of equations for velocity, pressure, and concentration, where the flow is described by the Stokes equations, and the transport is described by an unsteady convection-diffusion equation with non-homogeneous boundary conditions on walls (reactive boundaries). We start with the finite element approximation of the problem on unstructured grids and use it as a reference solution for two and three-dimensional model problems. Fine grid approximation resolves complex geometries on the grid level and leads to a large discrete system of equations that is computationally expensive to solve. To reduce the size of the discrete systems, we develop a multiscale model reduction technique, where we construct local multiscale basis functions to generate a lower-dimensional model on a coarse grid. The proposed multiscale model reduction is based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsGEM). In DG-GMsFEM for flow problems, we start with constructing the snapshot space for each interface between coarse grid cells to capture possible flows. For the reduction of the snapshot space size, we perform a dimension reduction via a solution of the local spectral problem and use eigenvectors corresponding to the smallest eigenvalues as multiscale basis functions for the approximation on the coarse grid. For the transport problem, we construct multiscale basis functions for each interface between coarse grid cells and present additional basis functions to capture non-homogeneous boundary conditions on walls. Finally, we will present some numerical simulations for three test geometries for two and three-dimensional problems to demonstrate the method's performance. • Design of multiscale methods and concepts for thin domain problems. • Dimension reduction using multiscale methods. • Multiscale methods for problems with irregular boundaries. • Coupled flow and transport problems in thin domains. [ABSTRACT FROM AUTHOR]

• Analysis of multiscale methods for quasi-gas dynamics. • The multiscale setup of quasi-gas dynamics. • Constraint energy minimizing basis construction for quasi-gas dynamics. In this paper, we consider the quasi-gas-dynamic (QGD) model in a multiscale environment. The model equations can be regarded as a hyperbolic regularization and are derived from kinetic equations. So far, the research on QGD models has been focused on problems with constant coefficients. In this paper, we investigate the QGD model in multiscale media, which can be used in porous media applications. This multiscale problem is interesting from a multiscale methodology point of view as the model problem has a hyperbolic multiscale term, and designing multiscale methods for hyperbolic equations is challenging. In the paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) combined with the central difference scheme in time to solve this problem. The CEM-GMsFEM provides a flexible and systematical framework to construct crucial multiscale basis functions for approximating the solution to the problem with reduced computational cost. With this approach of spatial discretization, we establish the stability of the fully discretized scheme, based on the coarse grid, under a coarse-scale CFL condition. Complete convergence analysis of the proposed method is presented. Numerical results are provided to illustrate and verify the theoretical findings. [ABSTRACT FROM AUTHOR]

A single ion channel is a membrane protein with an ion selectivity filter that allows only a single species of ions (such as potassium ions) to pass through in the "open" state. Its selectivity filter also naturally separates a solvent domain into an intracellular domain and an extracellular domain. Such biological and geometrical characteristics of a single ion channel are novelly adopted in the construction of a new kind of dielectric continuum ion channel model, called the Poisson-Nernst-Planck single ion channel (PNPSIC) model, in this paper. An effective PNPSIC finite element solver is then developed and implemented as a software package workable for a single ion channel with a three-dimensional X-ray crystallographic molecular structure and a mixture of multiple ionic species. Numerical results for a potassium channel confirm the convergence and efficiency of the PNPSIC finite element solver and demonstrate the high performance of the software package. Moreover, the PNPSIC model is applied to the calculation of electric current and validated by biophysical experimental data. • Construct a PNP single ion channel (PNPSIC) model by single ion channel selectivity property. • Solve the PNPSIC model by an effective finite element iterative algorithm. • Develop a PNPSIC software for crystallographic molecular structures and multiple ionic species. • Demonstrate the high performance of the PNPSIC software for a potassium channel. • Validate the PNPSIC model by comparing predicted I-V curves with biophysical experimental data. [ABSTRACT FROM AUTHOR]

The development of stable numerical schemes for viscoelastic multiphase flows has implications for many areas of engineering applications. The principal original contribution of this paper is the implementation of a conservative level-set method to define implicitly the interface between fluid phases, fully integrated into the mathematical framework of viscoelastic flow. The governing equations are discretized using the finite element method and stabilisation of the constitutive equation is achieved using either the discontinuous Galerkin (DG) or streamline upwinding (SU) method. The discrete elastic viscous stress splitting gradient (DEVSS-G) formulation is also employed in the Navier-Stokes equations to balance the hyperbolic characteristics of the polymeric stress tensor. The numerical scheme is validated with reference to several benchmark problems and excellent quantitative agreement with published data is found for Newtonian and viscoelastic fluids, for both single and multiphase flows. The motion of a gas bubble rising in a viscoelastic fluid is studied in detail. The influence of polymer concentration, surface tension, fluid elasticity and shear-thinning behaviour, on flow features such as the development of filaments and cusps and the generation of negative wakes is explored. • Conservative level-set method applied to viscoelastic multiphase flows. • Discontinuous Galerkin, streamline upwinding and DEVSS-G stabilisation techniques applied to multiphase viscoelastic flow. • FEM discretisation and algorithm provided for conservative level-set, Navier-Stokes and viscoelastic constitutive equations. • Single- and two-phase flow benchmarks considered from the literature. • Large polymer concentrations considered for the viscoelastic rising bubble case. [ABSTRACT FROM AUTHOR]

In fluid flow simulation, the multi-continuum model is a useful strategy. When the heterogeneity and contrast of coefficients are high, the system becomes multiscale, and some kinds of reduced order methods are demanded. Combining these techniques with nonlinearity, we will consider in this paper a dual-continuum model which is generalized as a multi-continuum model for a coupled system of nonlinear Richards equations as unsaturated flows, in complex heterogeneous fractured porous media; and we will solve it by a novel multiscale approach utilizing the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). In particular, such a nonlinear system will be discretized in time and then linearized by Picard iteration (whose global convergence is proved theoretically). Subsequently, we tackle the resulting linearized equations by the CEM-GMsFEM and obtain proper offline multiscale basis functions to span the multiscale space (which contains the pressure solution). More specifically, we first introduce two new sources of samples, and the GMsFEM is used over each coarse block to build local auxiliary multiscale basis functions via solving local spectral problems, that are crucial for detecting high-contrast channels. Second, per oversampled coarse region, local multiscale basis functions are created through the CEM as constrainedly minimizing an energy functional. Various numerical tests for our approach reveal that the error converges with the coarse-grid size and that only few oversampling layers as well as basis functions are needed. • We consider the multi-continuum Richards equations, which form a coupled system of nonlinear partial differential equations. • The concepts of constraint energy minimization and spectral decomposition are used to construct multiscale basis functions. • The nonlinearity is tackled by Picard iteration on a low-dimensional multiscale space. • The global convergence of Picard iterative procedure is theoretically proven. • The error converges with the coarse-grid size, according to our numerical results, which are also very accurate. [ABSTRACT FROM AUTHOR]

Within finite element models of fluids, vector-valued fields such as velocity or momentum variables are commonly discretised using the Raviart-Thomas elements. However, when using the lowest-order quadrilateral Raviart-Thomas elements, standard finite element discretisations of the vector transport equation typically have a low order of spatial accuracy. This paper describes two schemes that improve the accuracy of transporting such vector-valued fields on two-dimensional curved manifolds. The first scheme that is presented reconstructs the transported field in a higher-order function space, where the transport equation is then solved. The second scheme applies a mixed finite element formulation to the vector transport equation, simultaneously solving for the transported field and its vorticity. An approach to stabilising this mixed vector-vorticity formulation is presented that uses a Streamline Upwind Petrov-Galerkin (SUPG) method. These schemes are then demonstrated, along with their accuracy properties, through some numerical tests. Two new test cases are used to assess the transport of vector-valued fields on curved manifolds, solving the vector transport equation in isolation. The improvement of the schemes is also shown through two standard test cases for rotating shallow-water models. • Standard finite element discretisations of the vector transport equation have poor accuracy with the lowest-order Raviart-Thomas elements. • Two new approaches have been explored to improve the accuracy. • One approach recovers the transported field in a higher-order function space, where the transport is performed. • The other approach solves a mixed finite element problem for the transported vector field and its vorticity. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new multiscale approach with a meshfree coarse scale. A coarse scale is constructed on the basis of an already existing computational grid on a fine scale, depending on the heterogeneous parameters of the problem. This approach is based on the Generalized Multiscale Finite Element Method (GMsFEM), where the heterogeneous parameters of the problem are taken into account on a coarse scale using multiscale basis functions. These multiscale basis functions are constructed at an offline stage using local spectral problems. To represent the fractures on a fine grid, the Discrete Fracture Model (DFM) is used. The results of a numerical solution for two-dimensional and three-dimensional problems are presented. • We have designed the GMsFEM, where the coarse scale is constructed using the meshfree method. • A point cloud is used instead of a structured coarse grid; The multiscale space is constructed on a fine grid depending on heterogeneous parameters. • Numerical results show that the proposed method can provide good accuracy. [ABSTRACT FROM AUTHOR]

Based on the numerical method proposed in Hu et al. (2018) [22] for Kohn-Sham equation, further improvement on the efficiency is obtained in this paper by i). designing a numerical method with the strategy of separately handling the nonlinear Hartree potential and exchange-correlation potential, and ii). parallelizing the algorithm in an eigenpairwise approach. The feasibility of two approaches is analyzed in detail, and the new algorithm is described completely. Compared with previous results, a significant improvement of numerical efficiency can be observed from plenty of numerical experiments, which make the new method more suitable for the practical problems. • An accelerating multilevel correction AFEM is designed to solve Kohn-Sham equation. • Solving large-scale nonlinear eigenvalue problem is avoided. • A small-scale Kohn-Sham equation and large-scale boundary value problem are required. • Hartree potential and exchange-correlation potential are handling in a nested scheme. • The novel algorithm can be performed in an eigenpairwise approach. [ABSTRACT FROM AUTHOR]

This paper develops a numerical method for predicting the nonlinear structural-acoustic interactions of a hyperelastic solid immersed in an unbounded compressible viscous fluid. The nonlinear acoustic wave model with perfectly matched layers adopted is proposed in an arbitrary Lagrangian-Eulerian frame to absorb acoustic waves at truncated fluid boundaries, whereas the geometric and material nonlinearities of the hyperelastic solid are characterized in a total Lagrangian form. Both the fluid and the solid are discretized by the finite element method, and both models are coupled together by the momentum equilibrium and kinematic compatibility on the common interface. The convergence and accuracy of the proposed method are assessed by comparing the computed results with existing solutions. A hyperelastic ring in a viscous fluid is analyzed to provide physical insights into the nonlinear dynamics behaviors of the coupled hyperelastic solid and acoustic fluid system. The hyperelastic ring in acoustic fluid exhibits deformation instability under a uniform distributed excitation load, producing a series of subharmonics, ultrasubharmonics, and superharmonics in the structural and acoustic responses. Nonlinear internal resonance and bifurcation phenomena in the vibration of the ring and the acoustic waves of the fluid are also discussed. • A numerical method for nonlinear structural-acoustic interaction of a hyperelastic solid immersed in an unbounded fluid is proposed. • Arbitrary Lagrangian-Eulerian technique is adopted for the nonlinear acoustic wave model in a time-dependent acoustic domain. • A loosely coupled serially staggered procedure for nonlinear structural-acoustic interaction is presented. • Internal resonance and bifurcation phenomena of a hyperelastic ring and the acoustic waves are discussed. [ABSTRACT FROM AUTHOR]

• Present a nonuniform size modified Poisson-Boltzmann ion channel model (nuSMPBIC). • Partition nuSMPBIC into three-submodels to overcome singularity difficulties. • Propose efficient iterative methods for solving a nuSMPBIC finite element system. • Develop a nuSMPBIC finite element package for a 3D ion channel molecular structure. • Report numerical tests for voltage-dependent anion channel and four ionic species. In this paper, a nonuniform size modified Poisson-Boltzmann ion channel (nuSMPBIC) model is presented as a nonlinear system of an electrostatic potential and multiple ionic concentrations. It mixes nonlinear algebraic equations with a Poisson boundary value problem involving Dirichlet-Neumann mixed boundary value conditions and a membrane surface charge density to reflect the effects of ion sizes and membrane charges on electrostatics and ionic concentrations. To overcome the difficulties of strong singularities and exponential nonlinearities, it is split into three submodels with a solution of Model 1 collecting all the singular points and Models 2 and 3 much easier to solve numerically than the original nuSMPBIC model. A damped two-block iterative method is then presented to solve Model 3, along with a novel modified Newton iterative scheme for solving each related nonlinear algebraic system. To this end, an effective nuSMPBIC finite element solver is derived and then implemented as a program package that works for an ion channel protein with a three-dimensional molecular structure and a mixture of multiple ionic species. Numerical results for a voltage-dependent anion channel (VDAC) in a mixture of four ionic species demonstrate a fast convergence rate of the damped two-block iterative method, the high performance of the software package, and the importance of considering nonuniform ion sizes. Moreover, the nuSMPBIC model is validated by the anion selectivity property of VDAC. [ABSTRACT FROM AUTHOR]

In this paper, we construct a combined multiscale finite element method (MsFEM) using the Local Orthogonal Decomposition (LOD) technique to solve the multiscale problems which may have singularities in some special portions of the computational domain. For example, in the simulation of steady flow transporting through highly heterogeneous porous media driven by extraction wells, the singularities lie in the near-well regions. The basic idea of the combined method is to utilize the traditional finite element method (FEM) directly on a fine mesh of the problematic part of the domain and using the LOD-based MsFEM on a coarse mesh of the other part. The key point is how to define local correctors for the basis functions of the elements near the coarse and fine mesh interface, which require meticulous treatment. The proposed method takes advantages of the traditional FEM and the LOD-based MsFEM, which uses much less DOFs than the standard FEM and may be more accurate than the LOD-based MsFEM for problems with singularities. The error analysis is carried out for highly varying coefficients, without any assumptions on scale separation or periodicity. Numerical examples with periodic and random highly oscillating coefficients, as well as the multiscale problems on the L-shaped domain, and multiscale problems with high-contrast channels or well-singularities are presented to demonstrate the efficiency and accuracy of the proposed method. • A combined MsFEM is proposed to solve the multiscale problem with singularities. • A special definition of the local correctors for the interface-elements is given. • The error analysis is carried out for arbitrary varying coefficients. • Error estimates are independent of the ratio of the coarse-fine mesh sizes. • The method is very efficient for multiscale problems with well-singularities. [ABSTRACT FROM AUTHOR]

In this paper, Particle-in-Cell algorithms for the Vlasov–Poisson system are presented based on its Poisson bracket structure. The Poisson equation is solved by finite element methods, in which the appropriate finite element spaces are taken to guarantee that the semi-discretized system possesses a well defined discrete Poisson bracket structure. Then, splitting methods are applied to the semi-discretized system by decomposing the Hamiltonian function. The resulting discretizations are proved to be Poisson bracket preserving. Moreover, the conservative quantities of the system are also well preserved. In numerical experiments, we use the presented numerical methods to simulate various physical phenomena. Due to the huge computational effort of the practical computations, we employ the strategy of parallel computing. The numerical results verify the efficiency of the new derived numerical discretizations. [ABSTRACT FROM AUTHOR]

This paper proposes a new coupled multiphysics model and decoupled finite element method for the realistic naturally fractured reservoir consisting of the triple-porosity medium combined with a free flow region. More specifically, the triple-porosity medium consists of three transmittable, interacting contiguous porous mediums with more-permeable macrofractures, less-permeable microfractures, and stagnant-matrix region, respectively, which is described by the dual-fracture-matrix equations. Moreover, the free fluid flow region is governed by the evolutionary Stokes equation. To couple the triple-porosity medium with the conduit region, five physically valid coupling conditions, are utilized to capture the interfacial phenomena efficiently. The variational formulation and the well-posedness of the model are reported. To lay a solid ground, a fully-implicit algorithm is proposed in a traditional format. On the other hand, based on the idea of partitioned time-stepping, the five interface conditions, and the mass exchange terms in the triple-porosity model, the second algorithm is decoupled into four levels. The decoupling technique allows a non-iterative splitting of the coupled problem into four parallel subproblems. The unconditional stability and optimal convergence analysis are derived for the partitioned time-stepping scheme. For the petroleum engineering applications, we design two numerical experiments and perform parameter sensitivity analysis, which shows the applicability and complicated flow characteristics of the triple-porosity-Stokes interface system. • A new coupled multiphysics model is proposed to study the realistic naturally fractured reservoirs. • The triple-porosity-Stokes system consists of conduit region and three contiguous porous medium. • Five physically valid interface conditions are utilized to couple the free flow and porous media flow. • Stability and optimal convergence analysis are derived for the partitioned time-stepping scheme. • Two numerical experiments are designed and illustrate the parameter sensitivity for the petroleum engineering applications. [ABSTRACT FROM AUTHOR]

Magneto-hydrodynamics is one of the foremost models in plasma physics with applications in inertial confinement fusion, astrophysics and elsewhere. Advanced numerical methods are needed to get an insight into the complex physical phenomena. The classical Lagrangian methods are typically limited to the low orders of convergence and suffer from violation of the divergence-free condition for magnetic field or conservation of the invariants. This paper is the first part of a new series about high-order non-ideal magneto-hydrodynamics, where a multi-dimensional conservative Lagrangian method based on curvilinear finite elements is presented. The condition on zero divergence of magnetic field and conservation of mass, momentum, magnetic flux and the total energy are satisfied exactly. The curvilinear elements prevent entangling of the computational mesh and its imprinting into the solution. A high-order conservative time integration is applied, where an arbitrary order of convergence is attained for problems of ideal magneto-hydrodynamics. The resistive magnetic field diffusion is solved by an implicit scheme. Description of the method is given and multiple test problems demonstrating properties of the scheme are performed. The construction of the method and possible future directions of development are discussed. • Multi-dimensional numerical scheme based on high-order curvilinear finite elements. • Exact conservation of mass, momentum, magnetic flux and the total energy. • Divergence-free magnetic field maintained on moving Lagrangian meshes. • High-order symplectic time integration for ideal MHD. • Conservative Joule heating and (semi-)implicit magnetic field diffusion. [ABSTRACT FROM AUTHOR]

• Novel FEM method: G-continuous Bezier elements. • Non-linear MHD simulations of fusion plasmas in realistic geometry. • Numerical methods for non-linear simulations. The international tokamak ITER is progressing towards assembly completion and first-plasma operation, which will be a physics and engineering challenge for the fusion community. In the preparation for ITER experimental scenarios, non-linear MHD simulations are playing an essential role to actively understand and predict the behaviour and stability of tokamak plasmas in future fusion power plant. The development of MHD codes like JOREK is a key aspect of this research effort, and provides invaluable insight into the plasma stability and the control of global and localised plasma events, like Edge-Localised-Mode and disruptions. In this paper, we present an operational implementation of a new, generalised formulation of Bezier finite-elements applied to the JOREK code, a significant advancement from the previously G1-continuous bi-cubic Bezier elements. This new mathematical method enables any polynomial order of Bezier elements, with a guarantee of G-continuity at the level of (n − 1) / 2 , for any odd n , where n is the order of the Bezier polynomials. The generalised method is defined, and a rigorous mathematical proof is provided for the G-continuity requirement. Key details on the code implementation are mentioned, together with a suite of tests to demonstrate the mathematical reliability of the finite-element method, as well as the practical usability for typical non-linear tokamak MHD simulations. A demonstration for a state-of-the-art simulation of an Edge-Localised-Mode instability in the future ITER tokamak, with realistic grid geometry, finalises the study. [ABSTRACT FROM AUTHOR]

Anisotropic elliptic interface problems with non-homogeneous jump conditions are of great importance but hard to solve numerically. In this work, we develop an easy-to-implement exact-interface-fitted mesh generation (EIFMG) algorithm, and then propose a linearity-preserving finite volume (FV) scheme for solving such problems. Firstly, by curving the cell edges to align with the material interface, the EIFMG algorithm produces a structured exact-interface-fitted curved quadrilateral mesh without changing the total number of cells and associated topology. Secondly, by employing the so-called linearity preserving criterion, our derived FV scheme can be exact on curved quadrilateral meshes for constant coefficient problems with a linear solution even with extremely coarse mesh, which is generally not easy for finite element or finite difference methods to achieve. Moreover, a new method for interpolating vertex unknowns with cell-centered ones is also developed for non-homogeneous jump conditions of solutions and normal fluxes. Finally, numerical results are presented to verify the second-order accuracy and linearity-preserving ability of the proposed FV scheme. As far as we know, the algorithm based on FV method in this paper seems to be the best way to handle anisotropic elliptic interface problems with nonhomogeneous jump conditions, judging from the computational performance and efficiency of many typical numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we address the compressible Navier–Stokes equations written in the so-called conservative formulation. In particular, we focus on the possibility of uncoupling the computation of the problem unknowns, namely, density, linear momentum and total energy, a technique usually labeled as fractional step method, which allows to reduce the associated computational cost. The proposed methodology is a finite-element solver supplemented with a stabilization technique within the Variational Multi-Scale framework. In this regard, we consider orthogonal and dynamic definitions for the subscales. This discretization in space shows an adequate stability, permitting in particular the use of equal interpolation for all variables in play. However, we complement it with a shock-capturing operator in order to solve problems involving shocks. Several representative benchmark flow simulations are performed, which demonstrate the suitability of the proposed algorithm for a vast range of regimes. • Novel fractional step method for compressible flows in conservative variables. • Scheme (of first and second order) designed at the algebraic level. • A stabilized finite element method is employed. • A shock capturing technique is introduced to deal with the appearance of shocks. • Numerical experiments show the excellent performance of the formulations proposed. [ABSTRACT FROM AUTHOR]

• Analysis of the algebraic structure of the global interface problem. • Generalization of sweeping preconditioners for Helmholtz problems. • Application to time-harmonic scattering of acoustic waves. • Illustration for the acoustic case using the high-order finite element method. This paper explores a family of generalized sweeping preconditioners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (e.g. horizontal and diagonal). Several directions can be combined thanks to the flexible version of GMRES, allowing for the rapid transfer of information in the different zones of the computational domain, then accelerating the convergence of the final iterative solution procedure. Several two-dimensional finite element results are proposed to study and to compare the sweeping preconditioners, and to illustrate the performance on cases of increasing complexity. [ABSTRACT FROM AUTHOR]

• Energy evolution law of stochastic wave equation with multiplicative noise. • Energy-preserving spatial semi-discretizations via CFD and DG finite element method. • A new methodology of constructing energy-preserving fully-discrete schemes. In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the interior penalty discontinuous Galerkin finite element method to discretize space variable and present two semi-discrete schemes, respectively. Then we make use of the discrete gradient method and the Padé approximation to propose efficient fully-discrete schemes. These semi-discrete and fully-discrete schemes are proved to preserve the discrete averaged energy evolution law. In particular, we also prove that the proposed fully-discrete schemes exactly inherit the energy evolution law almost surely if the considered model is driven by additive noise. Numerical experiments are given to confirm theoretical findings. [ABSTRACT FROM AUTHOR]

While many numerical methods for the linear transport equation are available in the literature in 1D or on Cartesian meshes, fewer works are dedicated to the resolution of this model on unstructured meshes. In the context of radiative hydrodynamics, we need a method capable to handle a wide range of radiation regimes going from free-streaming to diffusion and to be coupled with a Lagrangian hydrodynamics solver. In this paper we design a method based on the micro-macro paradigm and to the Discrete Ordinates (S N) angular discretization, which fulfills these requirements. It allows to choose the limit transport scheme and the limit diffusion scheme. It is compared on challenging test problems to a Discontinuous Finite Element (DFE) method. • Arbitrary grid method to solve a linear transport equation based on a micro-macro decomposition. • Conservation of radiative energy. • Handle all the regimes from free-streaming to diffusion (Asymptotic Preserving). • Positivity preserving limit diffusion scheme. • Comparison with a Discontinuous Finite Element method. [ABSTRACT FROM AUTHOR]