Search

In this paper, an Arrow-Hurwicz iterative finite element method based on charge-conservation for solving the stationary inductionless magnetohydrodynamic equations is proposed and analyzed. The current density and electric potential are discretized by H (div , Ω) × L 2 (Ω) -conforming finite element pairs, respectively, which maintains charge-conservation. The hydrodynamic unknowns are discretized by stable velocity-pressure finite element pairs. Furthermore, we use the Arrow-Hurwicz iterative method to decouple the velocity and pressure, which leads to that there is no large-scale saddle point system that needs to be solved at each iteration step except for the determination of the initial functions. It has been proven that the proposed method converges geometrically with a contraction number independent of the mesh width. Numerical results are presented to verify the results of theoretical analysis and the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we address a two-parameter singularly perturbed convection-reaction-diffusion 2-D problem. We also consider that the convection and source terms are discontinuous in space. Due to these discontinuities and the presence of perturbation parameters, solutions to such problems show boundary and interior layers. In this study, we have carried out a numerical approach using a finite-difference technique with an appropriate layer-adapted piecewise uniform Shishkin mesh. Some examples are presented which show the best performance of the proposed method and its agreement with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use energy space regularizations in the objective functional, while the equivalent representations of the energy space norms, i.e., the H - 1 (Ω) -norm for the distributed control, the H 1 / 2 (Γ) -norm for the Dirichlet control and the H - 1 / 2 (Γ) -norm for the Neumann control, enable us to transform the optimal control problem into an elliptic variational inequality involving only the state variable. The elliptic variational inequalities are second order for the three cases, and include additional equality constraints for Dirichlet or Neumann boundary control problems. Standard C 0 finite element methods can be used to solve the resulted variational inequalities. We provide preliminary a priori error estimates for the new method for solving distributed control problems. Extensive numerical experiments are carried out to validate the accuracy of the new method. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a new semi-implicit stochastic multiscale method for the radiative heat transfer problem with additive noise fluctuation in composite materials. In the proposed method, the strong nonlinearity term induced by heat radiation is first approximated, by a semi-implicit predictor-corrected numerical scheme, for each fixed time step, resulting in a spatially random multiscale heat transfer equation. Then, the infinite-dimensional stochastic processes are modeled and truncated using a complete orthogonal system, facilitating the reduction of the model's dimensionality in the random space. The resulting low-rank random multiscale heat transfer equation is approximated and computed by using efficient spatial basis functions based multiscale method. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial multiscale properties, the high-dimensional randomness and the strong nonlinearity of the solution, so they can be overcome separately using different strategies. The convergence analysis is carried out, and the optimal rate of convergence is also obtained for the proposed semi-implicit stochastic multiscale method. Numerical experiments on several test problems for composite materials with various microstructures are also presented to gauge the efficiency and accuracy of the proposed semi-implicit stochastic multiscale method. [ABSTRACT FROM AUTHOR]

This paper proposes and investigates the two-grid stabilized lowest equal-order finite element method for the time-independent dual-permeability-Stokes model with the Beavers-Joseph-Saffman-Jones interface conditions. This method is mainly based on the idea of combining the two-grid and the two local Gauss integrals for the dual-permeability-Stokes system. In this technique, we use a difference between a consistent mass matrix and an under-integrated mass matrix for the pressure variable of the dual-permeability-Stokes model using the lowest equal-order finite element quadruples. In the two-grid scheme, the global problem is solved using the standard P 1 - P 1 - P 1 - P 1 finite element approximations only on a coarse grid with grid size H. Then, a coarse grid solution is applied on a fine grid of size h to decouple the interface terms and the mass exchange terms for solving the three independent subproblems such as the Stokes equations, microfracture equations, and the matrix equations on the fine grid. On the other hand, microfracture and matrix equations are decoupled through the mass exchange terms. The weak formulation is reported, and the optimal error estimate is derived for the two-grid schemes. Furthermore, the numerical results validate that the two-grid stabilized lowest equal-order finite element method is effective and has the same accuracy as the coupling scheme when we choose h = H 2 . [ABSTRACT FROM AUTHOR]

In this work, we extend recent research on the fully mixed virtual element method based on Banach spaces for the stationary Boussinesq equation to suggest and analyze a new mixed-virtual element technique for the stationary generalized Boussinesq equations, where viscosity and thermal conductivity are temperature-dependent. Besides the original thermo-fluid variables, the pseudostress and vorticity tensors of the fluid and the pseudoheat vector of the thermal system are introduced as auxiliary unknowns, and the incompressibility condition is then used to remove the pressure, which is later computed using a postprocessing formula. The resulting highly coupled formulation is then written equivalently as uncoupled problems thanks to a fixed-point strategy, so that the classical Banach Theorem, combined with the corresponding Babuška–Brezzi theory, the Banach–Nečas–Babuška Theorem, smallness-of-data assumptions, and a slight higher-regularity assumption for the exact solution, are employed to establish the unique solvability of the continuous formulation. The virtual element discretization of the uncoupled problems is based on the H (div 6 / 5) - and H (div 6 / 5) -conforming virtual element techniques. The discrete analysis is conducted in a similar manner by establishing the discrete inf-sup condition and using the discrete versions of the aforementioned theorems to demonstrate the existence of the discrete solution and derive its stability estimates. In addition, a priori error estimates are established by utilizing the Céa estimate and a suitable assumption on data for all variables in their natural norms showing an optimal rate of convergence. Finally, several numerical examples are presented to illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term b ∈ [ L ∞ ⁢ (Ω) ] d and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only H 1 -regularity, which extends the result [J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 2008, 5, 2524–2550]. The theoretical results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ (h k + 1) when polynomials of degree k ≥ 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results. [ABSTRACT FROM AUTHOR]

In this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme's application is demonstrated through two test problems. [ABSTRACT FROM AUTHOR]

In this paper, we discuss singularly perturbed time-dependent convection–diffusion problems that arise in computational neuroscience. Specifically, we provide approaches for one-dimensional singularly perturbed parabolic partial differential difference equations (SPPPDDEs) with mixed shifts in the spatial variable using fitted operator spline in compression and adaptive spline. Temporal discretization is done by backward Euler's method, and spline methods with exponential fitting on uniform mesh are implemented in the spatial domain. For better approximations, the Richardson extrapolation technique is used, which is demonstrated by two numerical examples. The convergence of the proposed methods is investigated and found to be uniform with respect to the perturbation parameter. Graphical representations are provided to show how the shifts affect the proposed solution to the problem. [ABSTRACT FROM AUTHOR]

The objective of this article is to develop and analyze a parameter uniform numerical method for solving 2D singularly perturbed elliptic convection-diffusion differential-difference equations. The previous studies in this area have mainly focused on cases involving small shifts, but practical scenarios may involve shifts of arbitrary sizes, both larger and smaller. Despite this, the 2D singularly perturbed elliptic differential-difference equations involving shifts of arbitrary sizes have remained unsolved. To address this challenge, a fitted operator finite difference method is developed on a special equidistant mesh that ensures the nodal point coincides with the shifts after discretization.. The scheme is rigorously analyzed to prove parameter uniform a priori error estimate in L ∞ norm. To illustrate the theoretical findings, we provide numerical results and implement numerical experiments to investigate the effect of shifts on the solution. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the Lagrange and Crouzeix–Raviart type immersed finite element methods for elliptic interface problems with nonhomogeneous interface conditions. The solution of the method is represented as a sum of two functions: one is the so-called discontinuous bubble satisfying the interface conditions approximately, and the other is the immersed finite element solution of the elliptic interface problem with homogeneous interface conditions. The discontinuous bubble can be easily constructed, since it is a piecewise linear polynomial, supported only on the triangles intersecting with the interface, determined by the nodal values or edge averages on the triangles and the given interface conditions. We prove the optimal convergence under the piecewise H 2 regularity assumption. Several numerical experiments are provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, three-dimensional numerical algorithm is constructed to simulate the behavior of the Brinkman-Forchheimer flow and thermal fields. Numerical results of velocity, pressure and temperature are obtained by applying the efficient modified two-grid marker and cell (MAC) algorithm on staggered grids with the second-order backward difference formula (BDF2) time approximation. The modified-upwind idea is introduced to convective heat transfer equations for improving accuracy without any numerical oscillation. The second-order convergence rate can be achieved for pressure, velocity and temperature of considered three-dimensional model. Some numerical experiments are presented to illustrate the efficiency of algorithm. The numerical example with analytical solution is used to validate the effectiveness and accuracy of the algorithm by comparing with the results of traditional MAC algorithm. A time-dependent test is proposed to show a detailed sensitivity analysis to indicate the influence of parameters including the ε , Forchheimer number, Brinkman number and thermal diffusivity on the physical properties of Brinkman-Forchheimer flow and heat transfer in porous media. [ABSTRACT FROM AUTHOR]

This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler's scheme for temporal discretization. With the help of Ritz–Voltera and L 2 projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L 2 -norm, the Galerkin projection onto this generalized finite element space even yields ε -independent error bounds, ε being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate ε -robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers. [ABSTRACT FROM AUTHOR]

The objective of this work is to develop a conforming virtual element method for viscoelastic wave equations with variable coefficients on polygonal meshes. For problems where the coefficients are variable, the standard virtual element discrete forms do not work efficiently and require modification. For the optimal convergence estimate of the semi-discrete approximation in the L 2 norm, a special projection operator is used. In the fully discrete scheme, the implicit second-order Newmark method is employed to approximate the temporal derivatives. Numerical experiments are presented to support the theoretical results. The proposed numerical algorithm can be applied to various problems arising in the engineering and medical fields. [ABSTRACT FROM AUTHOR]

The symmetric C0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. [ABSTRACT FROM AUTHOR]

Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable and isotropic elastic solid, which is immersed in a homogeneous compressible air/fluid. By the Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the model is formulated as a boundary value problem of acoustic-elastic interaction. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated DtN boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN boundary operator, where the latter decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the first family of conforming finite element divdiv complexes on cuboid meshes is constructed. The complex exhibits exactness on a contractible domain in the sense that the kernel space of each successive discrete map is the range of the previous one. This allows for algebraic structure-preserving finite element discretization of both the biharmonic equation and the linearized Einstein–Bianchi system. The convergence of optimal order is established and validated through numerical examples. [ABSTRACT FROM AUTHOR]

We introduce an hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and hp-version a priori error analysis of the proposed method, by deriving suitable hp-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented. [ABSTRACT FROM AUTHOR]

This paper introduces an extension of the well-known Morley element for the biharmonic equation, extending its application from triangular elements to general polytopal elements using the weak Galerkin finite element methods. By leveraging the Schur complement of the weak Galerkin method, this extension not only preserves the same degrees of freedom as the Morley element on triangular elements but also expands its applicability to general polytopal elements. The numerical scheme is devised by locally constructing weak tangential derivatives and weak second-order partial derivatives. Error estimates for the numerical approximation are established in both the energy norm and the L 2 norm. A series of numerical experiments are conducted to validate the theoretical developments. [ABSTRACT FROM AUTHOR]

In this paper we extend the Banach spaces-based fully mixed approach recently developed for the coupled Stokes and Poisson–Nernst–Planck equations, to cover the coupled Navier–Stokes and Poisson–Nernst–Planck equations. In addition to the velocity and pressure of the fluid, the velocity gradient and the Bernoulli-type stress tensor are added as further unknowns. Similarly, fully mixed formulations for the Poisson and Nernst–Planck sub-problems are achieved by considering, alongside the electrostatic potential and the concentration of ionized particles, the electric current field and total ionic fluxes as new mixed variables. As a consequence, two saddle-point problems, one of them non-linear, and both involving nonlinear source terms depending on the other unknowns, along with a perturbed saddle-point problem that is in turn further perturbed by a bilinear form depending on the remaining unknowns, constitute the resulting variational formulation of the whole coupled system. Fixed-point strategies are then employed to prove, under smallness assumptions on the data, the well-posedness of the continuous and associated Galerkin schemes, the latter for arbitrary finite element subspaces under suitable stability assumptions. The main theoretical tools utilized include the Babuška–Brezzi and Banach–Nečas–Babuška theories in Banach spaces, an abstract result for perturbed saddle-point problems (also in Banach spaces), and the classical Banach and Brouwer fixed-point theorems. Strang-type lemmas are then applied to establish a priori error estimates. Next, specific finite element subspaces (defined by Raviart–Thomas elements of order k ≥ 0 and piecewise polynomials of degree ≤ k ) are shown to satisfy the required hypotheses, and this yields specific convergence rates. Finally, several numerical results are reported, confirming the theoretical findings and illustrating the good performance of the method. [ABSTRACT FROM AUTHOR]

We present here a novel approach to handling curved meshes in polytopal methods within the framework of hybrid high-order methods. The hybrid high-order method is a modern numerical scheme for the approximation of elliptic PDEs. An extension to curved meshes allows for the strong enforcement of boundary conditions on curved domains and for the capture of curved geometries that appear internally in the domain e.g. discontinuities in a diffusion coefficient. The method makes use of non-polynomial functions on the curved faces and does not require any mappings between reference elements/faces. Such an approach does not require the faces to be polynomial and has a strict upper bound on the number of degrees of freedom on a curved face for a given polynomial degree. Moreover, this approach of enriching the space of unknowns on the curved faces with non-polynomial functions should extend naturally to other polytopal methods. We show the method to be stable and consistent on curved meshes and derive optimal error estimates in L 2 and energy norms. We present numerical examples of the method on a domain with curved boundary and for a diffusion problem such that the diffusion tensor is discontinuous along a curved arc. [ABSTRACT FROM AUTHOR]

We construct a nonconforming virtual element method for the approximation of singular solutions to isotropic linear elasticity problems on polygonal domains. Standard nonconforming virtual element spaces are enriched with suitable singular functions. The enrichment is based on the nonconforming structure of the discrete spaces and not on partition of unity techniques. We prove optimal convergence and assess numerically the theoretical results of the method. The proposed scheme naturally paves the way for an efficient linear elastic fracture solver. [ABSTRACT FROM AUTHOR]

In case of variable coefficients, the discrete bilinear forms in the virtual element discretization do not satisfy the consistency and stability properties. Also, for distinct damping and diffusion coefficient, it is difficult to obtain optimal convergence rate in L 2 norm. We discuss the virtual element method for the linear Sobolev equations with variable coefficients. The consistency and stability estimates for the discrete bilinear forms are derived. For optimal convergence analysis a new non-standard projection operator is introduced. We have used implicit second order Newmark scheme for the fully discrete approximation. Numerical experiments are illustrated to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Spatial network models are used as a simplified discrete representation in a wide range of applications, e.g., flow in blood vessels, elasticity of fiber based materials, and pore network models of porous materials. Nevertheless, the resulting linear systems are typically large and poorly conditioned and their numerical solution is challenging. This paper proposes a numerical homogenization technique for spatial network models which is based on the super-localized orthogonal decomposition (SLOD), recently introduced for elliptic multiscale partial differential equations. It provides accurate coarse solution spaces with approximation properties independent of the smoothness of the material data. A unique selling point of the SLOD is that it constructs an almost local basis of these coarse spaces, requiring less computations on the fine scale and achieving improved sparsity on the coarse scale compared to other state-of-the-art methods. We provide an a posteriori analysis of the proposed method and numerically confirm the method's unique localization properties. In addition, we show its applicability also for high-contrast channeled material data. [ABSTRACT FROM AUTHOR]

In this paper, the Marker and Cell scheme based on a two-grid algorithm is proposed for the two-dimensional incompressible Darcy–Brinkman–Forchheimer equations in porous media. The motivation of the two-grid Marker and Cell algorithm is figuring out a nonlinear equation on a coarse grid with mesh size H and a linear equation on a fine grid with mesh size h. A small positive parameter ε is introduced. By using it, the non-differentiable nonlinear term can be transformed into the term which is twice continuously differentiable. The error estimates of the velocity and pressure in the L 2 norms are obtained, which show O (ε + H 4 + h 2) . Second-order accuracy for some terms of velocity in the H 1 norms is also obtained. Several numerical experiments are provided to confirm the availability of this efficient second-order algorithm. Behavior of the fluid flow with different Brinkman number is considered. [ABSTRACT FROM AUTHOR]

In two-dimensional bounded Lipschitz domains, we analyze a convective Brinkman–Forchheimer problem on the weighted spaces H 0 1 (ω , Ω) × L 2 (ω , Ω) / R , where ω belongs to the Muckenhoupt class A 2 . Under a suitable smallness assumption, we prove the existence and uniqueness of a solution. We propose a finite element method and obtain a quasi-best approximation result in the energy norm à la Céa under the assumption that Ω is convex. We also develop an a posteriori error estimator and study its reliability and efficiency properties. Finally, we develop an adaptive method that yields optimal experimental convergence rates for the numerical examples we perform. [ABSTRACT FROM AUTHOR]

We introduce a numerical scheme that approximates solutions to linear PDE’s by minimizing a residual in the W - 1 , p ′ (Ω) norm with exponents p > 2 . The resulting problem is solved by regularized Kačanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents p ≫ 2 . Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion. [ABSTRACT FROM AUTHOR]

In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and error estimates of the finite element solution under a resolution condition between the wave number k, the mesh size h and the polynomial degree p of the form “ k (k h) p + 1 sufficiently small” and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in h from the case p = 1 in Wu and Zou (SIAM J Numer Anal 56(3):1338–1359, 2018) can be removed for p ≥ 2 . We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data. [ABSTRACT FROM AUTHOR]

Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix–Raviart ( m = 1 ) or Morley ( m = 2 ) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated L 2 error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability. [ABSTRACT FROM AUTHOR]

We propose here a convergence analysis for virtual element discretizations of the cardiac Bidomain model, a degenerate system of parabolic reaction-diffusion equations that models the propagation of the electric signal in the cardiac tissue. The virtual element method is a recent numerical technology that generalizes finite elements by considering polytopal computational grids, thus allowing more flexibility and accuracy in approximating complex computational domains. This can be an advantage when modeling for instance damaged cardiac tissues or structural heterogeneities. A previous similar study was performed in Anaya et al. (IMA J Numer Anal 40(2):1544–1576, 2020), where the propagation was modeled by means of a scalar nonlocal FitzHugh-Nagumo reaction-diffusion model. In the present work, we extend this analysis to the full semi-discrete Bidomain system, providing extensive numerical tests that validate the theoretical result on several structured and unstructured meshes. [ABSTRACT FROM AUTHOR]

In this paper, a modified two-grid algorithm based on block-centred finite difference method is developed for the fourth-order nonlinear extended Fisher–Kolmogorov equation. To further improve the computational efficiency, an effective second-order accurate backward difference formula is considered. The modified two-grid method based on Newton iteration is constructed to linearize the nonlinear system. The method solves a miniature nonlinear system on a coarse grid accompanying a larger time step to get the numerical solution, then computes a linear system constructed by the previous result with the Taylor expansion on a fine grid accompanying a smaller time step to get the correct numerical solution. Theoretical analysis shows that the modified two-grid algorithm can achieve second-order convergence accuracy both in time and space domain. Several numerical experiments are provided to verify the theoretical result and the high efficiency of this approach. The practical problem illustrates the actual applicable value of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we locally construct a conservative flux for finite element solutions of elliptic interface problems with discontinuous coefficients. Since the Discontinuous Galerkin method has built-in conservative flux, we consider in this paper the conforming finite element method and a special type of nonconforming method with arbitrary orders. We also perform our analysis based on Nitsche’s method, which imposes the Dirichlet boundary condition weakly. The construction method is derived based on a mixed problem with one solution coinciding with the finite element solution and with the other solution being naturally used to obtain a conservative flux. We then apply the recovered flux to the a posteriori error estimation and prove the robust reliability and efficiency for conforming elements, under the assumption that the diffusion coefficient is quasi-monotone. Numerical experiments are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This article develops a conforming virtual element method for a control-constrained Dirichlet boundary optimal control problem governed by the diffusion problem. An energy-based cost functional is used to approximate the control problem which results in a smooth control in contrast to the L 2 (Γ) approach which can lead to a control with discontinuities at the corners (Gong in SIAM J Numer Anal 60:450-474, 2022). We use virtual element discretization of control, state, and adjoint variables along with a discretize-then-optimize approach to compute the optimal control is used to solve the problem. A new framework for the a priori error analysis is presented, which is optimal up to the regularity of the continuous solution. A primal-dual algorithm is used to solve the Dirichlet optimal control problem, and numerical experiments are conducted to illustrate the theoretical findings on general polygonal meshes. [ABSTRACT FROM AUTHOR]

In this article, we design and analyze a hybrid high-order (HHO) finite element approximation for a class of strongly nonlinear boundary value problems. We consider an HHO discretization for a suitable linearized problem and show its well-posedness using the G a ˚ rding type inequality. The essential ingredients for the HHO approximation involve local reconstruction and high-order stabilization. We establish the existence of a unique solution for the HHO approximation using the Brouwer fixed point theorem and contraction principle. We derive an optimal order a priori error estimate in the discrete energy norm. Numerical experiments are performed to illustrate the convergence histories. [ABSTRACT FROM AUTHOR]

With the aim to continue developing a hybridizable discontinuous Galerkin (HDG) method for problems arisen from photovoltaic cells modeling, in this manuscript we consider the time harmonic Maxwell’s equations in an inhomogeneous bounded bi-periodic domain with quasi-periodic conditions on part of the boundary. We propose an HDG scheme where quasi-periodic boundary conditions are imposed on the numerical trace space. Under regularity assumptions and a proper choice of the stabilization parameter, we prove that the approximations of the electric and magnetic fields converge, in the L 2 -norm, to the exact solution with order h k + 1 and h k + 1 / 2 , resp., where h is the meshsize and k the polynomial degree of the discrete spaces. Although, numerical evidence suggests optimal order of convergence for both variables. An a posteriori error estimator for an energy norm is also proposed. We show that it is reliable and locally efficient under certain conditions. Numerical examples are provided to illustrate the performance of the quasi-periodic HDG method and the adaptive scheme based on the proposed error indicator. [ABSTRACT FROM AUTHOR]

The Miura ori is a very classical origami pattern used in numerous applications in engineering. A study of the shapes that surfaces using this pattern can assume is still lacking. A constrained nonlinear partial differential equation (PDE) that models the possible shapes that a periodic Miura tessellation can take in the homogenization limit has been established recently and solved only in specific cases. In this paper, the existence and uniqueness of a solution to the unconstrained PDE is proved for general Dirichlet boundary conditions. Then an H 2 -conforming discretization is introduced to approximate the solution of the PDE coupled to a Newton method to solve the associated discrete problem. A convergence proof for the method is given as well as a convergence rate. Finally, numerical experiments show the robustness of the method and that nontrivial shapes can be achieved using periodic Miura tessellations. [ABSTRACT FROM AUTHOR]

This article focuses on the investigation of two-dimensional elliptic singularly perturbed problems that incorporate positive and negative shifts, the solution of this class of problems may demonstrate regular/parabolic/degenerate or interior boundary layers. The goal of this article is to establish the development of numerical techniques for two-dimensional elliptic singularly perturbed problems with positive and negative shifts having regular boundary layers. The three numerical schemes are proposed to estimate the solution of this class of problems based on the fitted operator and fitted mesh finite-difference methods. The fitted operator finite difference method is analysed for convergence. The effect of shift terms on the solution behaviour is demonstrated through numerical experiments. The paper concludes by providing several numerical results that demonstrate the performance of proposed numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper, two new effective operator splitting methods (SS-MD and SM-MD) for the Maxwell's equations for dispersive media in two dimensions transverse electric polarization (the 2D Maxwell–Debye TE model) are presented and analyzed. The splitting schemes consist of two sub-stages in each time step, each of which requires solving a number of 1D discrete sub-problems. The Crank–Nicolson approach is used to solve each sub-problem's time discretization. Both splitting methods satisfy the energy decay and are unconditionally stable. The convergence result of the SS-MD scheme is shown to be of first order in time and of second order in space based on the energy technique, whereas the SM-MD scheme is of second order in both time and space. We also analyze numerical dispersion analysis to obtain two identities of the discrete numerical dispersion relations of both splitting schemes. Examples and numerical experiments are provided to demonstrate and support our theoretical results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze a virtual element method for the two dimensional elasticity spectral problem, where the polygonal meshes allow for the presence of small edges. Under this approach and with the aid of the theory of compact operators, we prove convergence for the proposed VEM and error estimates. We report a series of numerical tests in order to assess the performance of the method where we analyze the effects of the Poisson ratio on the computation of the order of convergence, together with the effects of the stabilization term on the arising of spurious eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approach for the time-discretization of the incompressible stochastic Stokes equations with multiplicative noise. Our new strategy is based on the classical 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 both velocity and pressure approximations. The proof is based on a new Hölder continuity estimate of the velocity solution. While the errors of the velocity approximation are estimated in the standard L 2 - and H 1 -norms, the pressure errors are carefully analyzed in a special norm because of the low regularity of the pressure solution. In addition, a new interpretation of the pressure solution, which is very useful in computation, is also introduced. Numerical experiments are also provided to validate the error estimates and their sharpness. [ABSTRACT FROM AUTHOR]

In this paper we introduce and analyze new Banach spaces-based mixed finite element methods for the stationary nonlinear problem arising from the coupling of the convective Brinkman-Forchheimer equations with a double diffusion phenomenon. Besides the velocity and pressure variables, the symmetric stress and the skew-symmetric vorticity tensors are introduced as auxiliary unknowns of the fluid. Thus, the incompressibility condition allows to eliminate the pressure, which, along with the velocity gradient and the shear stress, can be computed afterwards via postprocessing formulae depending on the velocity and the aforementioned new tensors. Regarding the diffusive part of the coupled model, and additionally to the temperature and concentration of the solute, their gradients and pseudoheat/pseudodiffusion vectors are incorporated as further unknowns as well. The resulting mixed variational formulation, settled within a Banach spaces framework, consists of a nonlinear perturbation of, in turn, a nonlinearly perturbed saddle-point scheme, coupled with a usual saddle-point system. A fixed-point strategy, combined with classical and recent solvability results for suitable linearizations of the decoupled problems, including in particular, the Banach-Nečas-Babuška theorem and the Babuška-Brezzi theory, are employed to prove, jointly with the Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. Both PEERS and AFW elements of order ℓ ⩾ 0 for the fluid variables, and piecewise polynomials of degree ⩽ ℓ together with Raviart-Thomas elements of order ℓ for the unknowns of the diffusion equations, constitute feasible choices for the Galerkin scheme. In turn, optimal a priori error estimates, including those for the postprocessed unknowns, are derived, and corresponding rates of convergence are established. Finally, several numerical experiments confirming the latter and illustrating the good performance of the proposed methods, are reported. [ABSTRACT FROM AUTHOR]

We propose and analyze a new mixed finite element method for the nonlinear problem given by the stationary convective Brinkman–Forchheimer equations. In addition to the original fluid variables, the pseudostress is introduced as an auxiliary unknown, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the aforementioned tensor and the velocity. As a consequence, we obtain a mixed variational formulation consisting of a nonlinear perturbation of, in turn, a perturbed saddle point problem in a Banach spaces framework. In this way, and differently from the techniques previously developed for this model, no augmentation procedure needs to be incorporated into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that recently established solvability results for perturbed saddle-point problems in Banach spaces, along with the well-known Banach–Nečas–Babuška and Banach theorems, are applied to prove the well-posedness of the continuous and discrete systems. The finite element discretization involves Raviart–Thomas elements of order k ≥ 0 for the pseudostress tensor and discontinuous piecewise polynomial elements of degree ≤ k for the velocity. Stability, convergence, and optimal a priori error estimates for the associated Galerkin scheme are obtained. Numerical examples confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method. In particular, the case of flow through a 2D porous media with fracture networks is considered. [ABSTRACT FROM AUTHOR]

An a priori analysis for generalized local projection stabilized finite element approximations for the solution of an advection–reaction equation is presented in this article. The stability and a priori error estimates are derived for the conforming, and nonconforming (Crouzeix–Raviart) approximations in the local projection streamline derivative norm. Finally, the validation of the proposed stabilization scheme and verification of the derived estimates are presented with appropriate numerical experiments. [ABSTRACT FROM AUTHOR]

We present a local projection stabilization for the virtual element discretization of the nonlinear convection–diffusion–reaction equation. We consider a streamline-derivative-based term for stabilization in the discrete formulation that preserves the stability property. We prove the optimal convergence estimates in the H 1 - and L 2 -norms. Numerical experiments consisting of different types of interior and boundary layers are conducted. The proposed numerical method captures the solution very well away from the layers. The numerical results show the proposed method's better performance in comparison with the gradient-based local projection stabilization method. [ABSTRACT FROM AUTHOR]

We present an efficient a posteriori error control strategy for an energy-based concurrent multiscale method with sharp interface in molecular mechanics, namely the geometric reconstruction based atomistic/continuum coupling method (GRAC). For the problem of the adaptive simulation of a range of crystal defects with practical importance, we develop an adaptive algorithm for the GRAC method based on a theory based modification of the residual based a posteriori error estimator and a specially designed mesh structure. In particular, by exploiting the decay of the displacement of the crystal caused by the defects, we are able to prove that the numerically expensive modeling residual estimator, which often exists in concurrent multiscale schemes, can be appropriately approximated by the inexpensive coarsening residual estimator in certain region which results in a significantly improvement of the efficiency of the adaptivity. The numerical experiments show that our adaptive strategy based on the modified error estimator and special mesh structure with properly chosen parameters leads to the same optimal convergence rate of the error but reduces the computational cost measured in CPU times by one order with respect to the number of degrees of freedom compared with that based on the original residual based error estimator. [ABSTRACT FROM AUTHOR]