Weak Galerkin Methods And Their Applications

Weak Galerkin finite element method,WGFEM for short,is a recently developed numerical method for partial differential equations.It is regarded as an extension and development of the standard finite element method.In 2011,Junping Wang and Xiu Ye firstly proposed WGFEM and applied it for a second order elliptic equation[68].Com-pared with the standard finite element method,similarly,WGFEM aims to discretize the spatial domain of the problem,and then approximate smooth function spaces of the problem with the use of piecewise defined discontinuous function spaces.However,WGFEM introduces the concepts of weak functions and weak differential operators on weak functions.The key to WGFEM is to utilize weak differential operators in place of standard differential operators(such as gradient,divergence,curl and Laplace operators,etc.)in the variational form of the problem.After WGFEM was proposed,it aroused widespread attention of international and domestic academics.A lot of work has been devoted to the development of the method during the past few years.To ensure the method is highly flexible in mesh generation and element construction,a stabilization term is introduced[48],which allows for arbi-trary piecewise polynomial shape functions in arbitrary polygonal or polyhedral elements.The weak Galerkin mixed finite element method is proposed based on the definition of discrete weak divergence operator to extend the standard mixed finite element method[69].Considering the high dimension of global stiffness matrix of weak Galerkin meth-ods,hybridized WGFEM is introduced by utilizing the degrees of freedom at element boundaries instead of the ones in elements to increase the solution speed[66].Weak Galerkin methods have also been applied to solve Helmholtz equations[21,47,51],bi-harmonic equations[46,49,65,86],Maxwell equations[50,58],Stokes equations[70,73]and Brinkman equations[45,76],etc.Additionally,weak Galerkin methods are also in-vestigated by specialists in the world from other perspectives,such as adaptive meshes based on posterior error estimates[15,87,88,89],multigrid techniques[14,61],super-convergence[4,29,75],etc.In this paper,weak Galerkin methods are applied to solve heat equations,parabolic biharmonic equations and nonlinear poroelasticity equations.According to each specific problem,we provide robust weak Galerkin schemes,establish error estimates and carry out numerical calculations.This paper is divided into five chapters:In Chapter 1,we firstly summarize the development of weak Galerkin methods,and then briefly introduce the three models in this paper and their recent research.Chapter 2 is devoted to the numerical analysis of weak Galerkin mixed finite el-ement method for solving a heat equation.After introducing the definitions of weak vector-valued function space and weak divergence operator,we provide the semi-discrete and fully-discrete numerical schemes with a stabilization term.Then we derive the s-tandard and optimal order error estimates for both continuous and discrete time weak Galerkin mixed finite element methods.Finally,two examples are supplied to illustrate our theoretical results.In Chapter 3,we shall apply WGFEM with a stabilization term for a parabolic bi-harmonic equation.After defining weak function space and weak Laplacian,we introduce the weak Galerkin finite element approximation for both continuous and discrete time.Then we develop and analyze the optimal order error estimates in L2 and an H2 type norm for both continuous and discrete time WGFEMs.Lastly,two numerical examples are provided to verify our theoretical results.In Chapter 4,weak Galerkin methods are utilized to figure out nonlinear poroe-lasticity problems.Firstly we introduce some notations,definitions and assumptions of the problem and analyze the existence and uniqueness of the weak solutions and regu-larities.Then we provide a numerical formulation with a coupling of newly developed larities.Then we provide a numerical formulation with a coupling of newly developed weak Galerkin mixed finite element method and finite element method,and study the solvability analysis of this method.After the locking-free error estimate is derived,we render two examples to demonstrate the convergence rate and the locking-free property of the coupled method.In Chapter 5,we present a summary of this paper.