Research On Weak Galerkin Finite Element Methods For Several Kinds Of Partial Differential Equations

The weak Galerkin finite element method is an efficient numerical method developed recently for solving partial differential equations.The main idea of the method is to replace the usual differential operator by the weak differential operator,and apply the weak differential operator to the usual variational form to numerically solve the partial differential equation.The approximation function of the weak Galerkin finite element method is piecewise discontinuous polynomial.The connection of approximation functions between the elements is given by a specific polynomial on the boundary of the elements.Since the weak Galerkin finite element method was proposed by Junping Wang and Xiu Ye[1]in 2011,it has been successfully applied to solve various partial differential equations.In this paper,we mainly present some new weak Galerkin finite element methods for several kinds of partial differential equations.In Chapter 3,by introducing a skew symmetric matrix,a new weak Galerkin finite element scheme for the reaction-convection-diffusion equation is established.The linear system from the scheme is positive definite and one might easily get the well-postedness of the system.In this scheme,WG elements are designed to have the form of?Pk?T?,Pk₁?e??.That is,we choose the polynomials of degree k>1 on each element and polynomials of degree k-1 on the edge/face of each element.As a result,fewer degrees of freedom are generated in the new WG finite element scheme.Through error analysis,the error estimates are given under discrete H1 norm and L2 norm,and corresponding numerical examples are also s1own in this chapter.The theoretical analysis and numerical experiments show that the new WG scheme reaches the optimal order in the sense of the above norms.That is,for elements of degree k,the convergence rate is O?hk?under discrete H1 norm,and the convergence rate is O?hk+1?under L2 norm.In Chapter 4,inspired by DG finite element methods,a new MWG finite element method for reaction-diffusion equations is proposed by introducing the average:{·} and the jump:[[·]]on the element boundary.On the one hand,this method produces fewer degrees of freedom than that of the traditional WG method.On the other hand,compared with some existing MWG methods,the finite element space and the test function space are the same piecewise discontinuous polynomial spaces.That is,the two spaces are give by Vh={v:v|T ?Pk?T?,T ?Th,k?1}.The advantage is that the error eh=uh-Q0U belongs to the test function space Vh,thus facilitating the error analysis of the MWG scheme.The theoretical analysis and numerical experiments show that the new MWG scheme reaches the optimal order in the sense of discrete H1 norm and L2 norm.That is,for elements of degree k,the convergence rate is O?hk?under discrete H1 norm,and the convergence rate is O?hk+1?under L2 norm.In Chapter 5,by using the MWG method proposed in Chapter 4,a MWG finite element scheme for the reaction-convection-diffusion equation is construct-ed.In this scheme,by introducing a skew symmetric matrix,the convection term is handled and it guarantees the positive definite of the scheme.Through error analysis,the error estimates are given under discrete H1 norm and L2 norm,and corresponding numerical examples are also shown in this chapter.The theoretical analysis and numerical experiments show that the MWG scheme reaches the op-timal order in the sense of the above norms.That is,for elements of degree k,the convergence rate is O?hk?under discrete H1 norm,and the convergence rate is O?hk+1?under L2 norm.In Chapter 6,by means of the average:{·} on the boundary of the element,we construct a new form of the weak function,give a new definition of the weak Laplace,and propose a MWG finite element scheme for the Biharmonic equation.In this method,the finite element space and the test function space are the same piecewise discontinuous polynomial spaces.That is,the two spaces are given by Sh={v:v|T E Pk?T?,T?Th,k?2}.Through error analysis,the error estimates are given in the sense of discrete H2 lorm and L2 norm.ThG theoretical analysis shows that in the sense of discrete H2 norm,the MWG scheme reaches the optimal order,that is,the convergence rate is O?hk?for elements of degree k;In the sense of L2 norm,the MWG scheme reaches the sub-optimal order for the MWG element of degree 2?that is,the convergence rate is O?h2??,and reaches the optimal order for the MWG element of degree k?k?3??that is,the convergence rate is O?hk+1??.