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In this paper, we develop a finite element method (FEM) to solve the Drude metamaterial perfectly matched layer (PML) model by edge elements. Both the stability analysis and error estimate for the scheme are established. To our best knowledge, this is the first paper on developing and analyzing a FEM for this Drude PML model. Numerical results are presented to justify the theoretical error estimate and demonstrate the effectiveness of this PML in absorbing outgoing waves in the Drude metamaterial. [ABSTRACT FROM AUTHOR]

Partial data is a common challenge in many applied problems, where we can measure only the part of the solution. This paper considers the poroelasticity problem in fractured media, where only partial measurements of fracture pressures are available. The poroelasticity models have applications in many engineering and scientific fields, such as geomechanics, hydrology, and reservoir simulation. This paper aims to develop a Bayesian decision making approach using partial data and consider its application to the poroelasticity problem in fractured media. The methodology of the proposed approach combines Proper Orthogonal Decomposition, Discrete Empirical Interpolation Method, Bayesian Inversion Approach, and Bayesian Decision Theory. It allows us to compute posterior probabilities and decide according to the optimal decision rule based on the partial data. Moreover, we apply partially explicit temporal discretization to solve forward problems efficiently. For testing the proposed approach, we consider a two-dimensional model problem. We simulate the cases when we need to make a single decision about the reservoir or decisions about the elements of the set of reservoirs. The results show that the proposed approach can efficiently compute the posterior probabilities and make a decision in the presence of only partial data for both cases. [ABSTRACT FROM AUTHOR]

This paper aims to present in a systematic form the stability and convergence analysis of a numerical method defined in nonuniform grids for nonlinear elliptic and parabolic convection–diffusion–reaction equations with Neumann boundary conditions. The method proposed can be seen simultaneously as a finite difference scheme and as a fully discrete piecewise linear finite element method. We establish second convergence order with respect to a discrete H 1 -norm which shows that the method is simultaneously supraconvergent and superconvergent. Numerical results to illustrate the theoretical results are included. [ABSTRACT FROM AUTHOR]

Numerical simulation of oil-water miscible displacement is discussed in this paper, and a compressible problem of energy mathematics is solved potentially. The mathematical model defined by a nonlinear system includes mainly two partial differential equations (PDEs): a parabolic equation for the pressure and a convection-diffusion equation for the saturation. The pressure is obtained by a conservative mixed finite volume element method (MFVE). The computational accuracy is improved for Darcy velocity. A conservative upwind mixed finite volume element method (UMFVE) is applied to compute the saturation on changing meshes. The diffusion is discretized by a mixed finite volume element method, and the convection is computed by upwind differences. The upwind method can solve convection-dominated diffusion equations accurately and avoids numerical dispersion and nonphysical oscillation. The saturation and the adjoint vector function are obtained simultaneously. An optimal order error estimates is obtained. Finally, numerical examples are provided to show the accuracy, efficiency and possible applications. [ABSTRACT FROM AUTHOR]

Randomized Neural Networks (RNNs) are a variety of neural networks in which the hidden-layer parameters are fixed to randomly assigned values, and the output-layer parameters are obtained by solving a linear system through least squares. This improves the efficiency without degrading the accuracy of the neural network. In this paper, we combine the idea of the Local RNN (LRNN) and the Discontinuous Galerkin (DG) approach for solving partial differential equations. RNNs are used to approximate the solution on the subdomains, and the DG formulation is used to glue them together. Taking the Poisson problem as a model, we propose three numerical schemes and provide convergence analysis. Then we extend the ideas to time-dependent problems. Taking the heat equation as a model, three space–time LRNN with DG formulations are proposed. Finally, we present numerical tests to demonstrate the performance of the methods developed herein. We evaluate the performance of the proposed methods by comparing them with the finite element method and the conventional DG method. The LRNN-DG methods can achieve higher accuracy with the same degrees of freedom, and can solve time-dependent problems more precisely and efficiently. This indicates that this new approach has great potential for solving partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new combined multiscale finite element method to solve the elliptic problem which has both a highly oscillating diffusion coefficient and a very complicated boundary. To design a numerical method for this kind of problems is difficult because that the microscopically geometrical information of the boundary and oscillation information of the coefficient should be taken into account simultaneously, which usually requires a very fine mesh. The main idea of the proposed method is to adopt unfitted fine scale meshes on the complicated boundary by using the cut finite element, and adopt the coarse scale meshes for the other part by using the recently developed local orthogonal decomposition (LOD) technique. Through this way, the introduced method can deal with arbitrarily complex boundaries and cut down some unnecessary computational effort. To ensure that the discrete systems are well-posed, we utilize the penalty technique on the artificial interface between coarse and fine meshes as well as on the boundary. The ghost penalty technique is used on the boundary to avoid the ill-condition of the system matrix. The error estimate and the condition number of the system matrix for the proposed method are derived without any periodic assumptions on the coefficient and boundary shape. Especially, the result is independent of the way that the boundary cut the mesh. Numerical experiments further confirm the theoretical results and demonstrate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, We propose the residual-based a posteriori error estimator of the weak Galerkin finite element method with the backward Euler time discretization for the linear parabolic partial differential equation. For the a posteriori error estimator, we introduce the Helmholtz decomposition technique to prove its reliability. We mainly study WG element (P j (K) , P ℓ (∂ K) , V (K , r) = R T j (K)). Numerical experiments based on the lowest order case, i.e. (P 0 (K) , P 0 (∂ K) , R T 0 (K)) , are provided to verify the theoretical research. [ABSTRACT FROM AUTHOR]

In this paper, a strain gradient thermoelastic problem is studied from the numerical point of view. The heat conduction is modeled by using the type II thermal law and the second gradient of the thermal displacement is also included in the set of independent constitutive variables. An existence and uniqueness result is recalled. Then, the fully discrete approximations are introduced by using the implicit Euler scheme and the finite element method. A discrete stability property and a main a priori error estimates results are proved. Then, some numerical simulations are performed, including some issues as the numerical convergence of the approximations, the effect of two possible dissipative terms (second- and fourth-order) or a comparison with the type II strain gradient thermoelasticity. [ABSTRACT FROM AUTHOR]

This paper develops a lowest-order Petrov–Galerkin immersed Q 1 − Q 0 method for solving Stokes interface problem. We make use of a uniform, interface-unfitted Cartesian mesh. An immersed Petrov–Galerkin formulation is presented, where the test spaces are conventional finite element spaces and the solution spaces satisfying the jump conditions. For the Stokes and Navier–Stokes interface problem, simple stabilized items are introduced. The nonlinear convective term is treated using Picard's iteration. Extensive numerical experiments validate the feasibility and optimal convergence order for the Petrov–Galerkin immersed Q 1 − Q 0 scheme both with homogeneous and non-homogeneous jumps. [ABSTRACT FROM AUTHOR]

Generalized isogeometric analysis (GIGA) enables accurate numerical solutions for interface problem with optimal convergence rates. However, the approach is not stable in terms of the spline functions' degree and is not robust with the position of the interface. In this paper, we introduce a stable GIGA (SGIGA) through a specific modification of the enrichment space. We prove that SGIGA can retain optimal convergence rates, and provide necessary well-conditioning for arbitrary degree and arbitrary location of the interface. We validate the theoretical results in various illustrations and compare the method with the stable generalized finite element method and GIGA. [ABSTRACT FROM AUTHOR]

This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete H 2 norm and the usual L 2 norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension. [ABSTRACT FROM AUTHOR]

In this paper, we consider language competition in a heterogeneous medium. The mathematical model considers bilingualism, population growth, and social pressure to use a specific language (diglossia). The transition from one language group to another is determined by probabilities that depend on status, group size, and spatial location. The model considers not only diffusive spread but also convective spread, which models seasonal migrations of speaker populations. We develop multiscale approaches for coarse grid modeling while maintaining reasonable accuracy. We present offline and online approaches based on the Generalized Multiscale Finite Element Method. We consider a model problem in a heterogeneous medium to test these approaches. The results show that both approaches can significantly reduce the discrete problem's size. However, the online approach achieves better accuracy. • We consider a language competition process between two languages in a heterogeneous medium. • The mathematical model considers diffusive-convective spatial spread, bilingualism, logistic growth, and diglossia. • For efficient modeling, we present offline and online multiscale approaches based on the GMsFEM and the Residual-Driven Online GMsFEM, respectively. • Numerical results show that both multiscale approaches have high accuracy, but the online approach provides better results than the offline one. • The online adaptive strategy provides accuracy similar to the uniform one but with fewer degrees of freedom. [ABSTRACT FROM AUTHOR]

This paper develops a multiscale solver for the problem of two languages competing in a heterogeneous medium. The mathematical model contains terms for language group switching, diglossia, population growth, and diffusive-convective spatial spread. The fine grid approximation is based on the finite element method. We use the finite difference method with an implicit scheme for time approximation. We use the Picard method to linearize the nonlinear terms. We use the Generalized Multiscale Finite Element Method for approximation on a coarse grid. Numerical results are presented for a model two-language interaction problem. The results show that the proposed multiscale approach can significantly reduce the computational cost of the problem while maintaining high accuracy. [ABSTRACT FROM AUTHOR]

This paper presents a new fully computable a posteriori error estimates for the primal hybrid finite element methods based on equilibrated flux and potential reconstructions. The reconstructed potential is obtained from a local L 2 orthogonal projection of the numerical solution on a continuous function space over the mesh skeleton. The equilibrated flux is the solution of a local mixed problem with a Neumann boundary condition given by the Lagrange multipliers of the primal hybrid finite element solution. For that, a divergence-consistent finite element pair is used. The upper and lower bounds of the error estimator are proved, and numerical results illustrate the efficiency of the error indicators. [ABSTRACT FROM AUTHOR]

This paper discusses the multiscale asymptotic expansion for combustion system with composite materials or porous media in thermal protection system. The combustion system consists of two nonlinear equations with rapidly oscillating coefficients, which are posed on a time-varying domain and computationally expensive to solve with the finite element method. To efficiently solve this system, we propose the standard/novel multiscale asymptotic expansions by combining them with boundary layer solutions. Numerical simulation is performed to validate the accuracy and efficiency of the newly proposed methods, which show that the novel multiscale asymptotic expansion is more accurate than the standard one. • We introduce multiscale asymptotic expansions to solve combustion system. • Our multiscale asymptotic expansion enables derivation of more accurate solutions. • To capture the domain boundary, we propose a boundary layer problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a combined Galerkin and mixed finite element methods to analyze the fully coupled nonlinear thermo-poroelastic model problems. We design the Galerkin finite element method for the temperature, the mixed finite element method for the pressure, the Galerkin finite element method for the elastic displacement. We linearize the nonlinear convective transport term in the energy balance equation and establish the fully discrete finite element schemes. The stability and convergence of the coupled method are obtained. In particular, all previous works have required certain time step restrictions, but we unconditionally prove the optimal error estimates without certain extra restrictions on both time step and spatial meshes. Finally, some numerical examples are presented to illustrate the accuracy of the method and confirm the unconditional stability of the method. [ABSTRACT FROM AUTHOR]

Multi-physics are widely studied and used in numerical analysis and simulation. Physics-based splitting methods are introduced, in which different methods are applied to solve each individual aspect of a physical system in multiple steps. The implicit methods are typically used for temporal discretization and are unconditionally stable such that the time step size can be large during the computation. However, the implicit methods are more complicated to solve, especially for nonlinear problems. In contrast, the explicit methods are relatively easier to compute, while they may require a much smaller time step size. In this paper, we propose a partially explicit scheme with physics-based splitting. We take the convection diffusion equation as an example. In this scheme, the convection equation is solved using the exact solution, and then we solve the diffusion equation via finite element methods. Some degrees of freedom in the diffusion equation are treated implicitly while others are treated explicitly. We can change the large-scale implicit system into one smaller-scale implicit system and one smaller-scale explicit system, which provides saving in computation. We analyze the stability and the order of accuracy. We also present some numerical results to show the performance of our proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a conforming discontinuous Galerkin (CDG) finite element method for Brinkman equations. The velocity stabilizer is removed by employing the higher degree polynomials to compute the weak gradient. The theoretical analysis shows that the CDG method is actually stable and accurate for the Brinkman equations. Optimal order error estimates are established in H 1 and L 2 norm. Finally, numerical experiments verify the stability and accuracy of the CDG numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyze the modified weak Galerkin (MWG) finite element method to the linear elasticity problem in a mixed form. Comparing with weak Galerkin (WG) finite element method, the structure of modified weak Galerkin (MWG) finite element method is especial. The main idea of MWG finite element method for elasticity problem is to express to boundary stress tensor defined on the partition boundary in terms of the average of the interior stress tensor defined inside partition elements. This can substantially reduce the degrees of freedom of the linear system. The well-posedness of the solution and inf–sup condition are investigated in this paper. We obtain the optimal error estimates in both energy norm and L 2 norm. Finally, some numerical experiments are exhibited to illustrate the efficiency and accuracy of the modified weak Galerkin finite element method. • The symmetric stress tensor is constructed. • The modified weak Galerkin finite method can deal with general polygon mesh. • Comparing with weak Galerkin finite element method, the degree of freedom of the whole discrete system is reduced. • The inf–sup condition of saddle point problem is given, and the optimal convergence order of stress function and displacement function with respect to corresponding norm is obtained. [ABSTRACT FROM AUTHOR]

We carry out the optimal error estimates of a scalar auxiliary variable (SAV) based the Euler semi-implicit finite element method for the Cahn–Hilliard–Navier–Stokes (CHNS) system in this paper. A SAV is introduced to reformulate the CHNS system into an equivalent system. We apply the Euler semi-implicit method and finite element method for the temporal and spatial discretizations to obtain a linear and unconditional energy stable scheme. The stability of numerical solutions under different norms and the optimal error estimates for our proposed numerical schemes are also presented. Finally, numerical examples are presented to demonstrate the accuracy of the proposed method, and the discrete energy is dissipated in numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, an energy-stable finite element method with the Crank–Nicolson type of temporal discretization scheme is developed and analyzed for a class of nonlinear fourth-order parabolic equations (including the Swift–Hohenberg (SH) equation and the extended Fisher–Kolmogorov (EFK) equation). In addition to the energy stability properties, the optimal spatial convergence properties in both L ∞ (L 2) - and L ∞ (H 1) -norm and the second-order temporal approximation rate are also obtained for numerical solutions approximating to the real solution and its Laplacian for the developed energy-stable finite element method in both semi- and fully discrete schemes. Numerical experiments are carried out to validate all attained theoretical results. • Energy-stable FEM is developed for a class of nonlinear fourth-order parabolic equations. • The model is more general than Swift–Hohenberg/extended Fisher–Kolmogorov equation. • Semi-discrete and Crank–Nicolson/fully discrete schemes are developed and analyzed. • The discrete energy is proved to be stable all the time for both schemes. • Optimal convergences in both L ∞ (H 1) - and L ∞ (L 2) -norm are obtained for both schemes. • Numerical results of 7 numerical experiments agree with all theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a SAV method for the Cahn–Hilliard type phase field model with new dynamic boundary conditions. By using the continuous finite element method in space and the backward difference method in time, the fully discrete numerical schemes preserving the energy law are constructed. Numerical examples show that the proposed scheme can simulate the phase field model well even in a relatively rough grid. [ABSTRACT FROM AUTHOR]

In this paper, we study, from both an analytical and a numerical point of view, a poro-thermoelastic problem with microtemperatures. The so-called Moore–Gibson–Thompson equation is used to model the contribution for the temperature and microtemperatures. An existence and uniqueness result is proved by using the theory of linear semigroups of contractions and, for the one-dimensional case, the exponential energy decay is found under some conditions on the constitutive coefficients. Then, a fully discrete approximation is introduced by using the finite element method and the implicit Euler scheme. We show that the discrete energy decays and we obtain some a priori error estimates from which, under some adequate additional regularity conditions on the continuous solution, we derive the linear convergence of the approximations. Finally, we perform some numerical simulations to demonstrate the accuracy of the approximations and the behavior of the discrete energy and the solution. [ABSTRACT FROM AUTHOR]

In this paper, we present a new error analysis of a class of fully discrete finite element methods for the dynamical inductionless magnetohydrodynamic (MHD) equations. The methods use the semi-implicit backward Euler scheme in time and use the standard inf–sup stable Mini/Taylor–Hood pairs to discretize the velocity and pressure, and the Raviart–Thomas for solving the current density in space. Due to the strong coupling of the system and the pollution of the lower-order Raviart–Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimates in negative norms, we establish new and optimal error estimates for all variables. In particular, we prove that the method with the lowest-order Raviart–Thomas face element and Mini element provides the optimal accuracy for the velocity in L ∞ (0 , T ; L 2 (Ω)) -norm, and the method with the lowest-order Raviart–Thomas face element and P 2 − P 1 Taylor–Hood element supplies the optimal accuracy for the velocity in L 2 (0 , T ; H 1 (Ω)) -norm and the pressure in L 1 (0 , T ; L 2 (Ω)) -norm. Moreover, we propose a simple recovery technique to obtain a new numerical current density of first-order higher accuracy in the spatial direction by re-solving a mixed Poisson equation. Numerical experiments are performed to verify the theoretical analysis. • New error analysis of a class of fully discrete finite element methods is established. • A recovery technique is proposed to obtain a higher-spatial-accuracy current density. • Numerical examples are provided to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

Generalized/extended finite element methods (GFEM/XFEM) for an interface problem are generally enriched by a distance function to the interface to handle discontinuities across the interface. Evaluations of the distance function and its gradients are needed for assembling stiffness matrices, which requires certain computational geometry algorithms. The computational complexity grows if the number of integration points is increased. Besides, the mathematical analysis on the effect of numerical integration has not been established in the field of GFEM/XFEM yet. This study designs an improved enriched function for a stable GFEM (SGFEM, a stable version of GFEM) based on the distance function, which only requires evaluating the distance function at nodes of elements that interact the interface (the gradients of distance function are not needed). The optimal convergence order (in the energy norm) of the SGFEM with such an improved enriched function is proven. In turn, we propose a perturbed variational formula, which can be integrated exactly by simple Gaussian rules designed in this paper. We prove that the SGFEM based on the improved enriched function and the perturbed variational formula (exactly integrated by the numerical integration) achieves the optimal convergence order for the interface problem. Namely, we develop a numerical integration rule for the SGFEM such that the optimal convergence order of SGFEM is maintained for the interface problem. The theoretical achievements are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

The motion of ions in the complex fluids widely appears in hydrodynamics, geodynamics and geophysics. Setting up suitable mathematical model, performing accurate and efficient numerical simulations are essential to understand the underlying physical principle in these phenomena. In this paper, we present a linear, second-order accurate in time, decoupled and unconditionally energy stable numerical scheme for solving an electrohydrodynamic power-law model with variable electrical conductivity. Based on a logarithm transformation for conductivity and the zero energy contribution property of the nonlinear coupling terms in the model, we derive an equivalent system by introducing a nonlocal auxiliary variable. The two-step backward differentiation scheme and the finite element method are used for the temporal and spatial discretizations, respectively. To decouple the system, the nonlocal splitting technique is employed to yield several linear subsystems, which can be solved very efficiently. Numerical simulations are carried out to demonstrate the second-order convergence in time and the unconditional energy stability. Moreover, the effects of Coulombic force and power-law fluid exponents are numerically investigated. • An electrohydrodynamic model with variable electrical conductivity is found satisfying an energy dissipation law. • The proposed scheme inherits the energy dissipation law at the fully discrete level. • The proposed scheme is a linear, second-order accurate in time and decoupled method. • The proposed scheme can be solved very efficiently with using a nonlocal splitting technique. [ABSTRACT FROM AUTHOR]

In this paper, we study a weak Galerkin (WG) finite element method for semiconductor device simulations. We consider the one-dimensional drift–diffusion (DD) and high-field (HF) models, which involves not only first derivative convection terms but also second derivative diffusion terms, as well as a coupled Poisson potential equation. The main difficulties in the analysis include the treatment of the nonlinearity and coupling of the models. The weak Galerkin finite element method adopts piecewise polynomials of degree k for the approximations of electron concentration and electric potential in the interior of elements, and piecewise polynomials of degree k + 1 for the discrete weak derivative space. The optimal order error estimates in a discrete H 1 norm and the standard L 2 norm are derived. Numerical experiments are presented to illustrate our theoretical analysis. Moreover, numerical schemes also work out for the discontinuous diffusion coefficient problems. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze an upwind finite volume element method on a Shishkin mesh for singularly perturbed convection–diffusion problems. We prove the stability of the method under the assumption of the convection and reaction term coefficients. The error estimate in the energy norm is presented on the Shishkin mesh, and the optimal error bound O (N − 1 (ln N) 3 / 2) is obtained. This error bound is uniformly valid with respect to the singular perturbation parameter. Numerical examples are provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop an online generalized multiscale approximation of the multipoint flux mixed finite element method (MFMFE) for Darcy flow in highly heterogeneous porous media, which can handle full tensor permeability fields and unstructured grids with local velocity elimination. Here, for illustration of the proposed method, we will mainly consider the situation that the porous medium is filled with fractures, which usually results in unstructured grids, and the extension to the situation that the permeability is represented as a full tensor is straightforward. From the fine-grid discretization by the MFMFE method, we derive the symmetric and positive definite discrete bilinear form for both the matrix and fracture pressure, and thereby obtain the corresponding discrete weak formulation that only related to pressure. In the offline stage, we compute offline basis functions which contain the important local multiscale information of each coarse-grid block to form the initial multiscale space. In the online stage, we construct and add online basis functions to enrich the multiscale space and consequently improve the accuracy of the multiscale solution. Each offline basis function and each online basis function contains multiscale information both for matrix and fracture. We give the theoretical analysis for the convergence of the online enrichment, which shows that more sufficient initial basis functions lead to faster convergence rates. A series of numerical examples that either the permeability is a full tensor or the porous medium is filled with fractures are presented to show the performance of the multiscale method. • An online generalized multiscale approximation of the MFMFE method is proposed. • The proposed method can handle full tensor permeabilities and unstructured grids. • The proposed method can simulate Darcy flow in fractured porous medium. • The discrete bilinear form for pressure is symmetric and positive definite. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly 1 for the approximate solution. The proof is based on new Hölder continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain H 1 stability, higher moment H 1 stability, L 2 stability, and higher moment L 2 stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a new type of adaptive finite element method is proposed for nonlinear eigenvalue problems in electronic structure calculations based on the multilevel correction method and adaptive multigrid method. Different from the classical adaptive finite element method for electronic structure calculations which needs to solve a nonlinear eigenvalue model directly in each adaptive finite element space, our approach only needs to solve a linear elliptic boundary value problem by adaptive multigrid method in each adaptive space, and then correct the eigenpair approximation by solving a small-scale nonlinear eigenvalue model in a low-dimensional correction space. Since solving large-scale nonlinear eigenvalue model is avoided which has exponentially increased computational time, the efficiency of electronic structure calculations can be improved evidently. In addition, the corresponding convergence analysis of the proposed adaptive multigrid algorithm are also derived theoretically and numerically. • An adaptive multigrid method is proposed for electronic structure calculations. • The new method only needs to solve some boundary value problems on adaptive meshes. • Then solve some low dimensional nonlinear eigenvalue problems in a correction space. • Only some multigrid iterations are needed for boundary value problems. • The proposed algorithm has almost the same efficiency as the boundary value problems. • Convergence analysis is derived theoretically and numerically. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a temporally second-order accurate numerical scheme for the Cahn–Hilliard-Magnetohydrodynamics system of equations. The scheme is based on a modified Crank–Nicolson-type approximation for the time discretization and a mixed finite element method for the spatial discretization. The modified Crank–Nicolson approximation enables us to carry out the mass conservation and the energy stability analysis. Error estimates are derived for the phase field in the L τ ∞ (0 , T ; H 1) norm, and for the velocity and the magnetic fields in the L τ ∞ (0 , T ; L 2) norm, respectively. Numerical examples are presented to validate the theoretical results of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we design the first residual type a posteriori error estimator for mixed interior penalty discontinuous Galerkin method for the H (curl) -elliptic problems. Then we prove that our indicator is both reliable and efficient. At last, we present some numerical experiments to validate the performance of the indicator within an adaptive mesh refinement procedure. [ABSTRACT FROM AUTHOR]

We present localized pointwise error estimates for a numerical method designed for accurate and efficient approximations for the flux variables. The method can be considered as a reduced version of the mixed finite element method. The method uses much less degrees of freedom compared to the mixed finite element methods while produces approximation solutions for the flux variable of the same accuracy. In this paper, we present optimal pointwise error estimates for the method, and they show local dependence of the error at a point and weaker dependence on global norms. [ABSTRACT FROM AUTHOR]

In this paper, we develop the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions (Dirichlet and Neumann) for the elasticity equations in high contrast media. By a special treatment of mixed boundary conditions separately, and combining the construction of the relaxed and constraint version of the CEM-GMsFEM, we discover that the method offers some advantages such as the independence of the target region's contrast from precision, while the sizes of oversampling domains have a significant impact on numerical accuracy. Moreover, to our best knowledge, this is the first proof of the convergence of the CEM-GMsFEM with mixed boundary conditions for the elasticity equations given. Some numerical experiments are provided to demonstrate the method's performance. [ABSTRACT FROM AUTHOR]

Anisotropic mesh adaptation is an efficient procedure for controlling the output error of finite element simulations, particularly when used for three-dimensional problems. In this paper, we present an enhanced computational algorithm based on an anisotropic mesh adaptation for nonlinear SP N approximations of radiative heat transfer in both two- and three-dimensional enclosures. Using an asymptotic analysis for the optical scale in the radiative transfer, the integro-differential equation is replaced by a series of partial differential equations of elliptic type. The nonlinear coupling between the heat transfer and radiation in the SP N equations does not depend on the direction ordinates. In an anisotropic participating media, internal boundary layers with different magnitudes occur for each direction and developing an efficient numerical algorithm to accurately resolve them is a challenging task. In the present study, we propose an adaptive finite element method using an efficient hierarchical error estimator. A second-order scheme is used for the time integration and a Newton-type solver is implemented for the fully coupled nonlinear system. The proposed method has the potential to use large timesteps in the simulations for radiative heat transfer in anisotropic media at high temperatures. To demonstrate the viability of the method, several examples are presented for two- and three-dimensional problems. The numerical results confirm the capability of the proposed method to efficiently solve the nonlinear SP N approximations of radiative transfer in anisotropic media. [ABSTRACT FROM AUTHOR]

This paper examines inverse Cauchy problems that are governed by a kind of elliptic partial differential equation. The inverse problems involve recovering the missing data on an inaccessible boundary from the measured data on an accessible boundary, which is severely ill-posed. By using the coupled complex boundary method (CCBM), which integrates both Dirichlet and Neumann data into a single Robin boundary condition, we reformulate the underlying problem into an operator equation. Based on this new formulation, we study the solution existence issue of the reduced problem with noisy data. A Golub–Kahan bidiagonalization (GKB) process together with Givens rotation is employed for iteratively solving the proposed operator equation. The regularizing property of the developed method, called CCBM-GKB, and its convergence rate results are proved under a posteriori stopping rule. Finally, a linear finite element method is used for the numerical realization of CCBM-GKB. Various numerical experiments demonstrate that CCBM-GKB is a kind of accelerated iterative regularization method, as it is much faster than the classic Landweber method. [ABSTRACT FROM AUTHOR]

A fitted weak Galerkin (WG) finite element scheme has been demonstrated in the present paper for solving H (div) -elliptic equation with discontinuous coefficients and interface. In both L 2 norm and H 1 norm, error estimates of optimal orders are discussed for the H (div) -elliptic interface problems. High-order convergence rates have been achieved by using suitable WG approximation spaces of higher degrees. We have performed some typical numerical tests to confirm the theoretical findings of the proposed WG algorithm. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips. [ABSTRACT FROM AUTHOR]

In this paper, we propose a mathematical model of the interaction of two languages. In our model, we consider two languages, though it can be generalized to multiple languages, which compete in a heterogeneous environment, consisting of highly varying properties related to the dynamics of the interaction. We use coupled convection–diffusion–reaction equations to describe the processes. Each equation describes the dynamics of one of the languages and contains terms related to the stand-alone dynamics and some coupling terms. The coupling terms represent the interaction between languages. We propose a numerical approach for solving the proposed model equation. In particular, we consider various inhomogeneities associated with cities and countryside, where languages are used differently (e.g., Sakha republic). These dynamics are essential for understanding the evolution of languages (one being dominant) and linguistic ecology that studies languages and their use in real/social life. Because of heterogeneities associated with geography, we use a multiscale approach. The proposed multiscale approach designs special basis functions to represent the small-scale information on larger scales. This way, we can solve the problem on a much coarser grid. Numerical results are presented that describe the dynamics and interaction of two languages. The main novelty of the paper consists of the proposed model and a multiscale algorithm. [ABSTRACT FROM AUTHOR]

In this paper we present and validate a simple adaptive surface remeshing technique to transfer history dependent variables from an old distorting mesh to a new mesh during finite element simulations of elastic surface deformation. This technique allows us to reduce the error arising from excessive mesh distortion whilst preserving information about the initial configuration of the mesh and the history dependent variables. The transfer technique presented here constructs the initial configuration of the new mesh by considering the distortion incurred by the elements of the old mesh and projecting backwards in time. Using this new initial configuration, the stress and strain over the new mesh can be easily calculated. After presenting the necessary steps to reconstruct the initial configuration, we show that this relatively simple transfer technique adds stability to finite element simulations and reduces the spatial error and the strain error across the domain. The novel transfer technique presented in this paper is easy to implement, released under an open source licence, and provides a simple strategy to add stability to simulations undergoing large deformations. • Excessive mesh distortion during finite element models corrupts simulation results. • Remeshing is essential, however it is difficult to transfer the history-dependent variables to to the new mesh. • Here we present the development of a novel history-dependent remeshing method in which we transfer the history-dependent variables between meshes through a mapping of the initial configuration to the new mesh. • Our variable transfer technique is easy to understand, and easy to implement. • This technique sucessfully avoids the error associated with excessive element distortion. [ABSTRACT FROM AUTHOR]

A stabilizer free weak Galerkin (WG) finite element method on polytopal mesh has been introduced in Part I of this paper (Ye and Zhang (2020)). Removing stabilizers from discontinuous finite element methods simplifies formulations and reduces programming complexity. The purpose of this paper is to introduce a new WG method without stabilizers on polytopal mesh that has convergence rates one order higher than optimal convergence rates. This method is the first WG method that achieves superconvergence on polytopal mesh. Numerical examples in 2D and 3D are presented verifying the theorem. [ABSTRACT FROM AUTHOR]

In recent papers (see e.g. Ruas (2020a) and Ruas (2020b)) a nonparametric technique of the Petrov–Galerkin type was analyzed, whose aim is the accuracy enhancement of higher order finite element methods to solve boundary value problems with Dirichlet conditions, posed in smooth curved domains. In contrast to parametric elements, it employs straight-edged triangular or tetrahedral meshes fitting the domain. In order to attain best-possible orders greater than one, piecewise polynomial trial-functions are employed, which interpolate the Dirichlet conditions at points of the true boundary. The test-functions in turn are defined upon the standard degrees of freedom associated with the underlying method for polytopic domains. As a consequence, when the problem at hand is self-adjoint a non symmetric linear system has to be solved. This paper is primarily aimed at showing that in this case, an efficient symmetrization of the solution procedure can be achieved by means of a fast converging iterative method. In order to illustrate the great generality of our nonparametric approach, experimentation is presented with a finite element method having degrees of freedom other than nodal values. More specifically we consider a nonconforming quadratic element in the solution of the three-dimensional Poisson equation. The performance evaluation however is conducted as well for two versions of the classical conforming quadratic method, namely, the nonparametric Petrov–Galerkin formulation considered in Ruas (2020b) and the standard isoparametric one. The study of this symmetrization is completed by an optimal error estimation in the broken H 1 -norm for the nonparametric version of the nonconforming method, which had not been addressed in previous work. [ABSTRACT FROM AUTHOR]

Mass-produced electrical machines are subjected to manufacturing uncertainties in terms of geometry. A robust design is inevitable to ensure the consistent performance of the electric motor. Some parts of the rotor geometry are often simplified, like the flux barrier at the end of the magnets. This paper presents a design optimisation regarding the torque ripple and the average torque. The aim is to assess the effects of the flux barrier on the main properties of a permanent magnet synchronous motor. Also, robust design analysis is presented on the flux barrier. The computational burden of the robust design analysis is immense, even if uniform uncertainties are assumed. In this case, different Design of Experiment (DoE) methods reduce the number of simulations. The efficiency of the DoE methods is compared in terms of simulation number and extreme value approximation. We found that the Central Composite method is the most accurate, while the Plackett–Burman is the most efficient in this particular case. [ABSTRACT FROM AUTHOR]

This paper is interested in the numerical approximation of a two-dimensional fluid–structure interaction problem. A special attention is paid to the numerical realization of aerodynamic force (AF) calculations. Three different AF computations within finite element method framework are described and compared. First, the extrapolation of fluid stress tensor components from the interior of the fluid domain is the most straightforward method; however, this approach is of lower theoretical accuracy due to the explicit computation of velocity gradient. Second, the local reconstruction method applied on the velocity gradient is based idea of the least square fit of the gradients evaluated at finite element centroids from local patch. Third, the weak reformulation technique using the weak formulation of the fluid flow problem leads to the expression of AF in an integral form. As it avoids direct velocity gradient computation it is of higher accuracy compared to previous methods. The experimental convergence of all mentioned methods for the drag and lift coefficients is tested on the benchmark of a static cylinder in a cross-flow and further extended to the case with prescribed motion of the cylinder. Further, two FSI simulations compare the transferred energy from the fluid to the structure for the case of structure vibrations submerged in the viscous fluid and also for the flow-induced structural vibrations. The critical inlet velocity of flutter instability is determined and it is shown that for a given numerical setting the choice of AF computation can lead to the different FSI behavior. [ABSTRACT FROM AUTHOR]

In this paper, we present an adaptive spectral projection based finite element method to numerically approximate the solution of the wave equation with memory. The adaptivity is not restricted to the mesh (h p -adaptivity), but it is also applied to the size of the computed spectrum (k -adaptivity). The meshes are refined using a residual based error estimator, while the size of the computed spectrum is adapted using the L 2 norm of the error of the projected data. We show that the approach can be very efficient and accurate. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the notion of an R ν -generalized solution of the Oseen problem in rotation form in two-dimensional polygonal domain with a reentrant corner at the boundary. Its existence and uniqueness in the nonsymmetric variational formulation of the problem is established. An estimate related to the conservation of the energy balance of the approximation velocity field for the Navier–Stokes problem in the rotation form is obtained. A weighted finite element method is constructed. A series of numerical experiments of test examples based on the proposed method is carried out. A comparative analysis of the approach with the classical FEM is performed. The results of numerical experiments showed a significant advantage of the proposed approach. • Existence and uniqueness of an R ν -generalized solution of Oseen problem is proved. • Crank–Nicolson scheme for solving Navier–Stokes problem is constructed. • Estimate related an energy balance of an approximation velocity field is obtained. • Weighted finite element method is constructed. • Series of numerical experiments of test examples is carried out. [ABSTRACT FROM AUTHOR]

In this paper, we study, analyze and numerically validate a conservative relaxation-type linear finite element method (FEM) for the nonlinear Schrödinger equation. The method avoids solving the complex nonlinear system and preserves the discrete mass and energy. A key to our analysis is that the errors are split into the temporal error and the spatial error by introducing the corresponding time-discrete system. Therefore, we derive the optimal L 2 and semi- H 1 error estimates without any coupling condition between time step τ and space size h. Some numerical experiments not only demonstrate the method's optimal convergence rates but also confirm the conservation laws during long time simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study a partial integral differential equation with p -Laplacian using a mixed finite element method. Two stable and convergent fixed point schemes are proposed to solve the nonlinear algebraic system. Using the implementation of the method in Matlab environment, we numerically analyse the convergence with an example. Some other examples, which illustrate several asymptotic behaviours and some localization effects of the solutions, are presented. [ABSTRACT FROM AUTHOR]

This paper proposes a C 0 (non-Lagrange) primal finite element approximation of the linear elliptic equations in non-divergence form with oblique boundary conditions in planar, curved domains. As an extension of [Calcolo, 58 (2022), No. 9], the Miranda–Talenti estimate for oblique boundary conditions at a discrete level is established by enhancing the regularity on the vertices. Consequently, the coercivity constant for the proposed scheme is exactly the same as that from PDE theory. The quasi-optimal order error estimates are established by carefully studying the approximation property of the finite element spaces. Numerical experiments are provided to verify the convergence theory and to demonstrate the accuracy and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper focus on the numerical approximation of a simplified two dimensional aeroelastic problem, where the flexibly supported airfoil with a control section is considered. In this case the coupling of the control section with other mode can cause a decrease of the flutter velocity. The problem is mathematically described using the fully coupled formulation of the incompressible viscous fluid flow over a flexibly structure, whose motion is modeled by system of ordinary differential equations. The incompressible flow is described by the system of Navier–Stokes equations written in the Arbitrary Lagrangian Eulerian form in order to treat the motion of the computational domain. The numerical approximation is treated using the variational multiscale approach using the algebraic subgrid scale model. The construction of the ALE mapping is based on a robust mesh deformation method. The numerical results are presented. • Variational multiscale method is used for fluid–structure interaction problems. • Evaluation of aerodynamical forces realized with the variational multiscale approach. • Robust mesh deformation algorithm used with the non-linear stiffness. • Numerical results are shown confirming the good performance of the method. [ABSTRACT FROM AUTHOR]