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Convergence of weak galerkin Finite Element method for Second Order Linear Wave Equation in Heterogeneous Media

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Numerical Mathematics(Theory,methods and Applications) 2023年第2期16卷 323-347页

作者： Bhupen Deka Papri Roy Naresh Kumar Raman Kumar Department of Mathematics Indian Institute of Technology GuwahatiGuwahati 781039India

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.

关键词： Wave equation heterogeneous medium finite element method weak galerkin method semidiscrete and fully discrete schemes optimal error estimates

A weak galerkin Mixed Finite Element method for Second Order Elliptic Equations on 2D Curved Domains

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Communications in Computational Physics 2022年第9期32卷 1094-1128页

作者： Yi Liu Wenbin Chen Yanqiu Wang School of Mathematical Sciences Nanjing Normal UniversityNanjing 210023China School of Mathematical Sciences Fudan UniversityShanghai 200433China Shanghai Key Laboratory for Contemporary Applied Mathematics Fudan UniversityShanghai 200433China Jiangsu Key Laboratory for NSLSCS Nanjing Normal UniversityNanjing 210023China

This article concerns the weak galerkin mixed finite element method(WGMFEM)for second order elliptic equations on 2D domains with curved boundary.The Neumann boundary condition is considered since it becomes the essential boundary condition in this case.It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretizationwith polynomial order a>1.The purpose of this paper is two-fold.First,we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains,which exhibits an O(h^(1/2))convergence for all a≥1.It is a little surprising to see that even the lowest-order WG-MFEM(a=1)experiences a loss of accuracy.This is different from known results for the finite element method(FEM)or the mixed FEM,and appears to be a combined effect of the WG-MFEM design and the fact that the outward normal vector on the polygonal approximation domain is different from the one on the curved domain.Second,we propose a remedy to bring the approximation rate back to optimal by employing two techniques.One is a specially designed boundary correction technique.The other is to take full advantage of the nice feature that weak galerkin discretization can be defined on polygonal meshes,which allows the curved boundary to be better approximated by multiple short edges without increasing the total number of mesh elements.Rigorous analysis shows that a combination of the above two techniques renders optimal convergence for all a.Numerical results further confirm this conclusion.

关键词： weak galerkin method polygonal mesh curved domain mixed formulation

A C^0-weak galerkin FINITE ELEMENT method FOR THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS IN STREAM-FUNCTION FORMULATION

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Journal of Computational Mathematics 2020年第2期38卷 310-336页

作者： Baiju Zhang Yan Yang Minfu Feng School of Mathematics Sichuan UniversityChengdu 610064China School of Sciences Southwest Petroleum UniversityChengdu 610500China

We propose and analyze a C^0-weak galerkin(WG)finite element method for the numerical solution of the Navier-Stokes equations governing 2D stationary incompressible flows.Using a st ream-function formulation,the system of Navier-Stokes equations is reduced to a single fourth-order nonlinear partial differential equation and the incompressibility constraint is automatically satisfied.The proposed method uses continuous piecewisepolynomial approximations of degree k+2 for the stream-function 0 and discontinuous piecewise-polynomial approximations of degreek+1 for the trace of V-0 on the interelement boundaries.The existence of a discrete solution is proved by means of a topological degree argument,while the uniqueness is obtained under a data smallness condition.An optimal error estimate is obtained in L^2-norm,H^1-norm and brokenH^2-norm.Numerical tests are presented to demonstrate the theoretical results.

关键词： weak galerkin method Navier-S to kes equations St ream-function formulation.

Discrete Maximum Principle for the weak galerkin method for Anisotropic Diffusion Problems

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Communications in Computational Physics 2015年第6期18卷 65-90页

作者： Weizhang Huang Yanqiu Wang Department of Mathematics The University of KansasLawrenceKS 66045U.S.A Department of Mathematics Oklahoma State UniversityStillwaterOK 74078U.S.A

A weak galerkin discretization of the boundary value problem of a general anisotropic diffusion problem is studied for preservation of the maximum principle.It is shown that the direct application of the M-matrix theory to the stiffness matrix of the weak galerkin discretization leads to a strong mesh condition requiring all of the mesh dihedral angles to be strictly acute(a constant-order away from 90 degrees).To avoid this difficulty,a reduced system is considered and shown to satisfy the discrete maximum principle under weaker mesh conditions.The discrete maximum principle is then established for the full weak galerkin approximation using the relations between the degrees of freedom located on elements and edges.Sufficient mesh conditions for both piecewise constant and general anisotropic diffusion matrices are obtained.These conditions provide a guideline for practical mesh generation for preservation of the maximum principle.Numerical examples are presented.

关键词： Discrete maximum principle weak galerkin method anisotropic diffusion

Constructing Order Two Superconvergent WG Finite Elements on Rectangular Meshes

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Numerical Mathematics(Theory,methods and Applications) 2023年第1期16卷 230-241页

作者： Xiu Ye Shangyou Zhang Department of Mathematics University of Arkansas at Little RockLittle RockAR 72204USA Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

In this paper,we introduce a stabilizer free weak galerkin(SFWG)finite element method for second order elliptic problems on rectangular meshes.With a special weak Gradient space,an order two superconvergence for the SFWG finite element solution is obtained,in both L 2 and H1 norms.A local post-process lifts such a Pk weak galerkin solution to an optimal order Pk+2 solution.The numerical results confirm the theory.

关键词： Finite element weak galerkin method stabilizer free rectangular mesh

Semi-Discrete and Fully Discrete weak galerkin Finite Element methods for a Quasistatic Maxwell Viscoelastic Model

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Numerical Mathematics(Theory,methods and Applications) 2023年第1期16卷 79-110页

作者： Jihong Xiao Zimo Zhu Xiaoping Xie School of Mathematics Sichuan UniversityChengdu 610064China Mathematics Department of Jinjiang College Sichuan UniversityPengshan 620860China

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.

关键词： Quasistatic Maxwell viscoelastic model weak galerkin method semi-discrete scheme fully discrete scheme error estimate