The Application Of The Weak Galerkin Finite Element Method For The Second-order Elliptic And Brinkman Problems

The weak Galerkin(WG)finite element method is a newly proposed numerical scheme for solving partial differential equations.The key of this method is that it uses the discrete weak gradient operators or the discrete weak divergence operators to replace the traditional differential operators.In addition,the stabilizer can be applied to ensure the weak contimuous of the numerical solution.The weak Galerkin finite element method is based on discontinuous polynomials,the well-posedness of the corresponding discrete system is independent of any parameters.The numerical solutions maintain mass conser?vation in the system.Furthermore.the weak Galerkin finite element method is a stable and consistent method for an)y polygon or polyhedron with regular shape.In this paper f:irstly.we use weak Galerkin finite element met.hod(without,stabilizer)for solving mixed second order elliptic equations.The numerical schemes and error analysis of this method are concise.The optimal order convergence is given for the vector function q in H1 norm and for the scalar function u in H1 norm and L2 norm·respectively.Secondly.the weak Galerkin finite element method with stabilizer is applied to solve the time-dependent Brinkman equations.The corresponding error equations and error estimates are estab-lished in both semi-discrete scheme and full-discrete scheme.respectiveh'.We obtain the optimal convergence rates in H1 norm and L2 norm for velocity function u and H1 nor-m for pressure function p?respectively.Unfortunately.the weak Galerkin finite element method has a lager discrete system compared to the traditional finite element method.It motivates the modified weak Galerkin(MWG)finite element method is proposed.The idea of this method is that it replace the boundary function by the average of interior functions.Therefore,the degrees of freedom of the modified weak Galerkin finite element method are far less than the weak Galerkin finite element method.Finally,the modified weak Galerkin finite element method is introduced to solve the Brinkman equations.The optimal order convergence is estimated in H1 norm and L2 norm for velocity function u and H1 norm for pressure function p.The numerical experiment is presented to checkout the numerical scheme is stable,consistent and convergent.