Four-Order Superconvergent Weak Galerkin Methods for the Biharmonic Equation on Triangular Meshes

A stabilizer-free weak Galerkin(SFWG)finite element method was introduced and analyzed in Ye and Zhang(SIAM J.Numer.Anal.58:2572–2588,2020)for the biharmonic equation,which has an ultra simple finite element formulation.This work is a continuation of our investigation of the SFWG method for the biharmonic equation.The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the L^(2)norm on triangular grids.This new method also keeps the formulation that is symmetric,positive definite,and stabilizer-free.Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete H^(2)norm.Superconvergence of four orders in the L^(2)norm is also derived for k≥3,where k is the degree of the approximation polynomial.The postprocessing is proved to lift a P_(k)SFWG solution to a P_(k+4)solution elementwise which converges at the optimal order.Numerical examples are tested to verify the theor ies.

A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator.Naturally,the resulting linear system is symmetric and positive definite,and thus the algorithm is easy to implement and analyze.Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh.A superconvergence result is proved when the coefficient matrix is constant or piecewise constant.Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.

A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

In this article,a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed.The idea of using the P_(k)-harmonic polynomial space instead of the full polynomial space P_(k)is to use a much smaller number of basis functions to achieve the same accuracy when k≥2.The optimal rate of convergence is derived in both H^(1)and L^(2)norms.Numerical experiments have been conducted to verify the theoretical error estimates.In addition,numerical comparisons of using the P_(2)-harmonic polynomial space and using the standard P_(2)polynomial space are presented.

Weak Galerkin Finite Element Method for the Unsteady Stokes Equation

The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.

A stabilizer-free C^(0) weak Galerkin method for the biharmonic equations

In this article, we present and analyze a stabilizer-free C^(0)weak Galerkin(SF-C^(0)WG) method for solving the biharmonic problem. The SF-C^(0)WG method is formulated in terms of cell unknowns which are C^(0)continuous piecewise polynomials of degree k + 2 with k≥0 and in terms of face unknowns which are discontinuous piecewise polynomials of degree k + 1. The formulation of this SF-C^(0)WG method is without the stabilized or penalty term and is as simple as the C1conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete H^(2)-like norm and the H^(1)norm for k≥0 are established for the corresponding WG finite element solutions. Error estimates in the L^(2)norm are also derived with an optimal order of convergence for k > 0 and sub-optimal order of convergence for k = 0. Numerical experiments are shown to confirm the theoretical results.

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space(Pk(K),P_(k−1)(∂K),[P_(k−1)(K)]^(2)).Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in L1(L2)norm.This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes.Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.

Semi-Discrete and Fully Discrete Weak Galerkin Finite Element Methods for a Quasistatic Maxwell Viscoelastic Model

This paper considers weak Galerkin finite element approximations on polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model.The spatial discretization uses piecewise polynomials of degree k(k≥1)for the stress approximation,degree k+1 for the velocity approximation,and degree k for the numerical trace of velocity on the inter-element boundaries.The temporal discretization in the fully discrete method adopts a backward Euler difference scheme.We show the existence and uniqueness of the semi-discrete and fully discrete solutions,and derive optimal a priori error estimates.Numerical examples are provided to support the theoretical analysis.

A Posteriori Error Estimate of Weak Galerkin FEM for Stokes Problem Using Auxiliary Subspace Techniques

Based on the auxiliary subspace techniques,a posteriori error estimator of nonconforming weak Galerkin finite element method(WGFEM)for Stokes problem in two and three dimensions is presented.Without saturation assumption,we prove that the WGFEM approximation error is bounded by the error estimator up to an oscillation term.The computational cost of the approximation and the error problems is considered in terms of size and sparsity of the system matrix.To reduce the computational cost of the error problem,an equivalent error problem is constructed by using diagonalization techniques,which needs to solve only two diagonal linear algebraic systems corresponding to the degree of freedom(d.o.f)to get the error estimator.Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.

Weak Galerkin Method for Second-Order Elliptic Equations with Newton Boundary Condition

The weak Galerkin(WG)method is a nonconforming numerical method for solving partial differential equations.In this paper,we introduce the WG method for elliptic equations with Newton boundary condition in bounded domains.The Newton boundary condition is a nonlinear boundary condition arising from science and engineering applications.We prove the well-posedness of the WG scheme by the monotone operator theory and the embedding inequality of weak finite element functions.The error estimates are derived.Numerical experiments are presented to verify the theoretical analysis.

A MODIFIED WEAK GALERKIN FINITE ELEMENTMETHOD FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION-REACTION PROBLEMS

In this work,a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations.The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators,respectively.We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh.The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods.Optimal order of convergences are obtained in suitable norms.We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method.Various numerical examples are presented to support the theoretical results.It is theoretically and numerically shown that the method is quite stable.

四阶线性方程极弱局部间断Galerkin法傅里叶分析

主要研究了四阶线性方程极弱局部间断Galerkin方法的傅里叶误差分析问题。首先,给出四阶线性方程的极弱局部间断Galerkin空间离散格式,并在周期边界条件及一致网格的条件下将离散格式表示为差分形式,然后,在k=2的情况下,利用傅里叶分析方法分析其稳定性及其误差估计问题,最后,利用数值实验,分别对得到的结果进行验证。

Caputo型时间分数阶变系数扩散方程的局部间断Galerkin方法

提出一种带有Caputo导数的时间分数阶变系数扩散方程的数值解法.方程的解在初始时刻附近通常具有弱正则性,采用非一致网格上的L1公式离散时间分数阶导数,并使用局部间断Galerkin(local discontinuous Galerkin,LDG)方法离散空间导数,给出方程的全离散格式.基于离散的分数阶Gronwall不等式,证明了格式的数值稳定性和收敛性,且所得结果关于α是鲁棒的,即当α→1^(-)时不会发生爆破.最后,通过数值算例验证理论分析的结果.

ROBUST GLOBALLY DIVERGENCE-FREE WEAK GALERKIN METHODS FOR STOKES EQUATIONS

This paper proposes and analyzes a class of robust globally divergence-free weak Galerkin （WG） finite element methods for Stokes equations. The new methods use the Pk/Pk-1 （k ≥ 1） discontinuous finite element combination for velocity and pressure in the interior of elements, and piecewise P1/Pk （l = k - 1, k） for the trace approximations of the ve- locity and pressure on the inter-element boundaries. Our methods not only yield globally divergence-free velocity solutions, but also have uniform error estimates with respect to the Reynolds number. Numerical experiments are provided to show the robustness of the proposed methods.

A MODIFIED WEAK GALERKIN FINITE ELEMENT METHOD FOR SOBOLEV EQUATION

For Sobolev equation, we present a new numerical scheme based on a modified weak Galerkin finite element method, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. Optimal order error estimates in discrete H^1 and L^2 norms are established for the corresponding modified weak Galerkin finite element solutions. Finally, some numerical results are given to verify theoretical results.

A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR SECOND-ORDER ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

In this paper, we present a weak Galerkin （WG） mixed finite element method for solving the second-order elliptic equations with Robin boundary conditions. Stability and a priori error estimates for the coupled method are derived. We present the optimal order error estimate for the WG-MFEM approximations in a norm that is related to the L^2 for the flux and H1 for the scalar function. Also an optimal order error estimate in L^2 is derived for the scalar approximation by using a duality argument. A series of numerical experiments is presented that verify our theoretical results.

Weak Galerkin finite element method for valuation of American options

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.

Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form

A hybridization technique is applied to the weak Galerkin finite element method （WGFEM） for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results.

The Weak Galerkin Method for Elliptic Eigenvalue Problems

This article is devoted to studying the application of the weak Galerkin(WG)finite element method to the elliptic eigenvalue problem with an emphasis on obtaining lower bounds.The WG method uses discontinuous polynomials on polygonal or polyhedral finite element partitions.The non-conforming finite element space of the WG method is the key of the lower bound property.It also makes the WG method more robust and flexible in solving eigenvalue problems.We demonstrate that the WG method can achieve arbitrary high convergence order.This is in contrast with existing nonconforming finite element methods which can provide lower bound approximations by linear finite elements.Numerical results are presented to demonstrate the efficiency and accuracy of the theoretical results.

A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation

A weak Galerkin(WG)method is introduced and numerically tested for the Helmholtz equation.This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property.At the same time,the WG finite element formulation is symmetric and parameter free.Several test scenarios are designed for a numerical investigation on the accuracy,convergence,and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains.Challenging problems with high wave numbers are also examined.Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement,and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.

A WEAK GALERKIN FINITE ELEMENT METHOD FOR THE LINEAR ELASTICITY PROBLEM IN MIXED FORM

In this paper, we use the weak Galerkin （WG） finite element method to solve the mixed form linear elasticity problem. In the mixed form, we get the discrete of proximation of the stress tensor and the displacement field. For the WG methods, we define the weak function and the weak differential operator in an optimal polynomial approximation spaces. The optimal error estimates are given and numerical results are presented to demonstrate the efficiency and the accuracy of the weak Galerkin finite element method.