Xiaoshen Wang

A modified weak Galerkin finite element method for the Stokes equations

A modified weak Galerkin (MWG) finite element method is introduced for the Stokes equations in the primary velocity-pressure formulation. This method is based on the weak Galerkin (WG) finite element method developed recently in Wang and Ye (0000). The unknowns associated with element boundaries in the WG method have been eliminated in the MWG method. As the result, the MWG method has fewer degrees of freedom. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Numerical results are presented to confirm the theory.

A modified weak Galerkin finite element method

In this paper we introduce a new discrete weak gradient operator and a new weak Galerkin (WG) finite element method for second order Poisson equations based on this new operator. This newly defined discrete weak gradient operator allows us to use a single stabilizer which is similar to the one used in the discontinuous Galerkin (DG) methods without having to worry about choosing a sufficiently large parameter. In addition, we will establish the optimal convergence rates and validate the results with numerical examples.

A weak Galerkin finite element scheme for solving the stationary Stokes equations

A weak Galerkin (WG) finite element method for solving the stationary Stokes equations in two- or three- dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed. The variational form we considered is based on two gradient operators which is different from the usual gradient-divergence operators. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Numerical results are presented to illustrate the theoretical analysis of the new WG finite element scheme for Stokes problems.

A modified weak Galerkin finite element method for a class of parabolic problems

In this paper, we consider the solution of parabolic equation using the modified weak Galerkin finite element procedure, which is named as MWG-FEM, based on the conception of the modified weak derivative over discontinuous functions with heterogeneous properties, in which the classical gradient operator is replaced by a modified weak gradient operator. Optimal order error estimates in a discrete L2 norm and H1 norm are established for the corresponding modified weak Galerkin finite element solutions. Finally, we numerically verify the convergence theory for the MWG-FEM through some examples.

Multi-level Monte Carlo weak Galerkin method with nested meshes for stochastic Brinkman problem

Abstract This paper is devoted to the numerical analysis of a multi-level Monte Carlo weak Galerkin (MLMCWG) approximation with nested meshes for solving stochastic Brinkman equations with two dimensional spatial domain. With weak gradient operator and a stabilizer at hand, the weak Galerkin (WG) technique is a high-order accurate and stable method which can easily handle deterministic partial differential equations with complex geometries, flows with jump fluid viscosity coefficients or high-contrast permeability fields given by each sample. The multi-level Monte Carlo (MLMC) technique with nested meshes balances the sampling error and the spatial approximation error, where the computational cost can be sharply reduced to log-linear complexity with respect to the degree of freedom in spatial direction. The nested meshes requirement is introduced here in order to simplify the analysis, which can be generalized to MLMC with non-nested meshes. Error estimates are derived in terms of the spatial meshsize and the number of samples. The numerical tests are provided to illustrate the behavior of the MLMCWG method and verify our theoretical results regarding optimal convergence of the approximate solutions.

Superconvergence of a modified weak Galerkin approximation for second order elliptic problems by L2 projection method

Abstract This paper derives a superconvergence result for the modified weak Galerkin (MWG) finite element method of the second order elliptic problem. The convergence rate of the MWG approximation is improved by 30% after applying a low cost L 2 projection post-processing technique. These superconvergence phenomena are proved theoretically and confirmed numerically.