Discrete H1 Research Articles

H(div)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(\ extrm{div})$$\\end{document}-conforming HDG methods for the stress-velocity formulation of the Stokes equations and the Navier–Stokes equations

In this paper we devise and analyze a pressure-robust and superconvergent HDG method in stress-velocity formulation for the Stokes equations and the Navier–Stokes equations with strongly symmetric stress. The stress and velocity are approximated using piecewise polynomial space of order k and H(div;Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(\ extrm{div};\\Omega )$$\\end{document}-conforming space of order k+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k+1$$\\end{document}, respectively, where k is the polynomial order. In contrast, the tangential trace of the velocity is approximated using piecewise polynomials of order k. Moreover, the characterization of the proposed schemes shows that the globally coupled unknowns are the normal trace and the tangential trace of velocity, and the piecewise constant approximation for the trace of the stress. The discrete H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}-stability is established for the discrete solution. The proposed formulation yields divergence-free velocity, but causes difficulties for the derivation of the pressure-independent error estimate given that the pressure variable is not employed explicitly in the discrete formulation. This difficulty can be overcome by observing that the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} projection to the stress space has a nice commuting property. Moreover, superconvergence for velocity in discrete H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}-norm is obtained, with regard to the degrees of freedom of the globally coupled unknowns. Then the convergence of the discrete solution to the weak solution for the Navier–Stokes equations via the compactness argument is rigorously analyzed under minimal regularity assumption. The strong convergence for velocity and stress is proved. Importantly, the strong convergence for velocity in discrete H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document}-norm is achieved. Several numerical experiments are carried out to confirm the proposed theories.

Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method’s key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete H1 and L2 norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.

The Finite Difference Method and Analysis for Simulating the Unsteady Generalized Maxwell Fluid with a Multi-Term Time Fractional Derivative

The finite difference method is used to solve a new class of unsteady generalized Maxwell fluid models with multi-term time-fractional derivatives. The fractional order range of the Maxwell model index is from 0 to 2, which is hard to approximate with general methods. In this paper, we propose a new finite difference scheme to solve such problems. Based on the discrete H1 norm, the stability and convergence of the considered discrete scheme are discussed. We also prove that the accuracy of the method proposed in this paper is O(τ+h2). Finally, some numerical examples are provided to further demonstrate the superiority of this method through comparative analysis with other algorithms.

A weak Galerkin finite element method for nonlinear convection-diffusion equation

In this paper, a weak Galerkin (WG) finite element method for one-dimensional nonlinear convection-diffusion equation with Dirichlet boundary condition is developed. Based on a special variational form featuring two built-in parameters, the semi-discrete and fully discrete WG finite element schemes are proposed. The backward Euler method is utilized for time discretization. The WG finite element method adopts locally piecewise polynomials of degree k for the approximation of the primal variable in the interior of elements, and piecewise polynomials of degree k+1 for the weak derivatives. Theoretically, the optimal error estimates in both discrete H1 and standard L2 norms are derived. Numerical experiments are performed to demonstrate the effectiveness of the WG finite element approach and validate the theoretical findings.

A weak Galerkin finite element method for 1D semiconductor device simulation models

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 H1 norm and the standard L2 norm are derived. Numerical experiments are presented to illustrate our theoretical analysis. Moreover, numerical schemes also work out for the discontinuous diffusion coefficient problems.

Weak Galerkin finite element method for linear poroelasticity problems

In this paper, we develop a weak Galerkin (WG) finite element method for a linear poroelasticity model where weak divergence and weak gradient operators defined over discontinuous functions are introduced. We establish both the continuous and discrete time WG schemes, and obtain their optimal convergence order estimates in a discrete H1 norm for the displacement and in H1 and L2 norms for the pressure. Finally, we present some numerical experiments on different kinds of meshes to illustrate the theoretical error estimates, and furthermore verify the locking-free property of our proposed method.

Error estimate of the cell-centered nonlinear positivity-preserving two-point flux approximation schemes

We present the error estimate of nonlinear positivity-preserving finite volume schemes for diffusion problems. These schemes are also named by nonlinear two-point flux approximation (NTPFA) methods. Due to their conservation and monotonicity, such schemes have attracted wide attention. Under suitable regularity assumptions on the exact solution, we prove a discrete H1 error estimate through the analysis of consistent errors. Some numerical examples are presented to demonstrate the theoretical results.

A weak Galerkin finite element method on temporal graded meshes for the multi-term time fractional diffusion equations

We consider the multi-term time fractional diffusion equation on a bounded convex domain. We analyze the L1 method on a graded mesh in time to compensate for the weak singularity of the solution and a weak Galerkin finite element method in space discretization. The stability analyses are presented for both semi-discrete and fully-discrete schemes and we prove that two schemes are unconditionally stable. Error estimates in L2-norm and a discrete H1 equivalent norm for both schemes are rigorously derived. Further we discuss the optimal spatial order error estimates in L2-norm. Finally, we give some numerical experiments to show the efficiency of the proposed method.

A weak Galerkin method for elasticity interface problems

This article introduces a weak Galerkin (WG) finite element method for linear elasticity interface problems on general polygonal/polyhedral partitions. The discrete weak divergence and gradient operators are discretized as polynomials and computed by solving inexpensive local problems on each element. The developed WG method has been proved to be stable and accurate with optimal order error estimates in the discrete H1 norm. Some numerical experiments are conducted to verify the efficiency and accuracy of the proposed WG method, in addition, and its uniform convergence independent of the jump of the interface coefficients and the incompressibility.

A stabilizer free spatial weak Galerkin finite element methods for time-dependent convection-diffusion equations

In this paper, we propose a stabilizer free spatial weak Galerkin (SFSWG) finite element method for solving time-dependent convection diffusion equations based on weak form Eq. (4). SFSWG method in spatial direction and Euler difference operator Eq. (37) in temporal direction are used. The main reason for using the SFSWG method is because of its simple formulation that makes this algorithm more efficient and its implementation easier. The optimal rates of convergence of 𝒪⁢(hk) and 𝒪⁢(hk+1) in a discrete H1 and L2 norms, respectively, are obtained under certain conditions if polynomial spaces (Pk⁢(K),Pk⁢(e),[Pj⁢(K)]2) are used in the SFSWG finite element method. Numerical experiments are performed to verify the effectiveness and accuracy of the SFSWG method.

Characteristic methods for thermal convection problems with infinite Prandtl number on nonuniform staggered grids

In this paper, two characteristic methods are introduced to solve the thermal convection problems with infinite Prandtl number on nonuniform staggered grids. We obtain spatial second order superconvergence in the discrete L2 norm for the temperature, velocity and pressure. Besides, second order superconvergence in the discrete H1 norm for both the temperature and some terms of the velocity on nonuniform grids is established. First and second order time discretizations of the thermal balance equation are provided. Numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

An analysis of weak Galerkin finite element method for a steady state Boussinesq problem

In this article, we present and analyze a weak Galerkin finite element method (WG-FEM) for the coupled Navier–Stokes/temperature (or Boussinesq) problems. In this WG-FEM, discontinuous functions are applied to approximate the velocity, temperature, and the normal derivative of temperature on the boundary while piecewise constants are used to approximate the pressure. The stability, existence and uniqueness of solution of the associated WG-FEM are proved in detail. An optimal a priori error estimate is then derived for velocity in the discrete H1 and L2 norms, pressure in the L2 norm, temperature in the discrete H1 and L2 norms, and the normal derivative of temperature in H−1/2 norm. Finally, to complete this study some numerical tests are presented which illustrate that the numerical errors are consistent with theoretical results.

A modified weak Galerkin finite element method for nonmonotone quasilinear elliptic problems

A modified weak Galerkin finite element method is studied for nonmonotone quasilinear elliptic problems. Using the contraction mapping theorem, the uniqueness of the solution to the discrete problem is proved. Moreover, optimal order a priori error estimates are established in both a discrete H1 norm and the L2 norm. Numerical experiments are conducted to confirm the theoretical results.

A study of mixed problem for second order elliptic problems using modified weak Galerkin finite element method

In this article, a modified weak Galerkin finite element method is developed for a second order elliptic problem with mixed boundary conditions. The basic idea of modified weak Galerkin method is to reduce the degrees of freedom in comparison to the standard weak Galerkin method. The optimal convergence orders in a discrete L2 norm and H1 norm are established. Numerical examples are presented to verify the theoretical estimates.

A staggered discontinuous Galerkin method for elliptic problems on rectangular grids

In this article, a staggered discontinuous Galerkin (SDG) approximation on rectangular meshes for elliptic problems in two dimensions is constructed and analyzed. The optimal convergence results with respect to discrete L2 and H1 norms are theoretically proved. Some numerical evidences to verify the optimal convergence rates are presented. Several numerical examples to the elliptic singularly perturbed problems with sharp boundary or interior layers are presented to show that the proposed SDG method is very effective, stable and accurate. Thanks to the simple structure of rectangular meshes, the discrete gradients across the boundaries of rectangular elements are easily defined, making numerical implementation much easier. The idea of using the rectangular meshes will be extended to more practical problems on a curved domain in future works.

A stabilizer-free weak Galerkin finite element method on polytopal meshes

A stabilizing/penalty term is often used in finite element methods with discontinuous approximations to enforce connection of discontinuous functions across element boundaries. Removing stabilizers from discontinuous Galerkin finite element methods will simplify formulations and reduce programming complexity significantly. The goal of this paper is to introduce a stabilizer free weak Galerkin (WG) finite element method for second order elliptic equations on polytopal meshes. This new WG method keeps a simple symmetric positive definite form and can work on polygonal/polyhedral meshes. Optimal order error estimates are established for the corresponding WG approximations in both a discrete H1 norm and the L2 norm. Numerical results are presented verifying the theorem.

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

AbstractA time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ≔ Ω × [0, T], where Ω is the interval [0, l]. Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2. In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L2(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L2(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H1(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.

Energy conservation and super convergence analysis of the EC‐S‐FDTD schemes for Maxwell equations with periodic boundaries

This paper is concerned with new energy analysis of the two dimensional Maxwell's equations and the symmetric energy‐conserved splitting finite difference time domain (EC‐S‐FDTD) method with the periodic boundary (PB) condition. New energy identities of the Maxwell's equations in terms of H1 and H2 norms are proposed and interpreted by considering the physical meanings of the H1 and H2 semi‐norms in the identities. It is found from these new identities that the first and second curls of the electromagnetic fields are conserved in terms their magnitudes. By the energy methods, the numerical energy identities of the symmetric EC‐S‐FDTD method are derived and shown to converge to the continuous energy identities of the Maxwell's equations. This proves that the symmetric EC‐S‐FDTD scheme is unconditionally stable and energy conserved in the discrete H1 and H2 norms. Also by the energy methods, it is proved that the symmetric EC‐S‐FDTD method with PB condition is of second order (super) convergence in the discrete H1 and H2 norms. Numerical experiments are carried out and confirm the analysis on energy conservation, stability and super convergence.

Stability and convergence of characteristic MAC scheme and post-processing for the Oseen equations on non-uniform grids

In this paper, we establish the LBB condition and stability for both velocity and pressure of characteristic MAC scheme for the Oseen equations on non-uniform grids. We obtain the second order convergence in discrete L2 norm for both velocity and pressure and the first order convergence in discrete H1 norm for velocity. Moreover, we construct the post-processing characteristic MAC scheme to obtain second order accuracy in discrete H1 norm for the velocity. Finally, some numerical experiments are presented to show the correctness and accuracy of the MAC scheme.