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The lowest-order weak Galerkin finite element method is introduced and analysed for second order elliptic interface problems with discontinuous coefficients and solutions. Compared with the existing weak Galerkin finite element methods for interface problems, this method has a simpler formulation and superconvergence property. Hence it is suitable for more complicated interfaces and domain configurations. We derive the superconvergence error estimates in discrete H 1 and L 2 norms. Furthermore, we verify the theoretical results by numerical experiments. Optimal convergence rates are achieved for examples with low-regularity solutions. [ABSTRACT FROM AUTHOR]

We develop a temporally second-order stabilizer-free weak Galerkin (SFWG) finite element method with unequal time steps for reaction-subdiffusion equation in multiple space dimensions. In consideration of initial singularity of the solution, we prove a sharp error estimate on nonuniform time steps by two tools: discrete fractional Grönwall inequality and error convolution structure analysis. Furthermore, the fully discrete scheme achieves second-order accuracy in time by employing the graded temporal partition. Several numerical experiments are included to confirm the sharpness of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a modified weak Galerkin (MWG) finite element method with weakly imposed boundary conditions is presented for solving convection-dominated diffusion equations. The method is shown uniformly stable for all diffusion parameters. The method converges at the optimal order for large diffusion problems in the energy norm, and at half a super-convergent order for small diffusion problems. Various numerical examples are presented, showing that the method is as effective as the weak Galerkin method. [ABSTRACT FROM AUTHOR]

In this paper, a weak Galerkin (WG) finite element method for one-dimensional nonlinear convection-diffusion equation with Dirichlet boundary condition is developed. Based on a special variational form featuring two built-in parameters, the semi-discrete and fully discrete WG finite element schemes are proposed. The backward Euler method is utilized for time discretization. The WG finite element method adopts locally piecewise polynomials of degree k for the approximation of the primal variable in the interior of elements, and piecewise polynomials of degree k + 1 for the weak derivatives. Theoretically, the optimal error estimates in both discrete H 1 and standard L 2 norms are derived. Numerical experiments are performed to demonstrate the effectiveness of the WG finite element approach and validate the theoretical findings. • A special variational form possessing two built-in parameters is proposed for the one-dimensional nonlinear convection-diffusion problem. • The WG scheme, developed in this paper, is very flexible because different stencils can be obtained by tuning the built-in weighted parameters. • The optimal error estimates in both standard L 2 and discrete H 1 norms are derived for our numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we study a modified weak Galerkin finite element method (MWG-FEM) on anisotropic triangular meshes for a class of parabolic equations. Different from conventional weak Galerkin methods, the MWG-FEM replaces the boundary functions by the average of interior functions, which possesses flexibility in the approximation functions and mesh generation. Moreover, MWG-FEM is compatible with anisotropic meshes and suitable for solving problems with anisotropic property. By using the anisotropic interpolation and projection operators, the optimal order error estimate in L 2 norm is derived. Numerical experiments are performed to demonstrate the stability and efficiency of the method. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study a weak Galerkin finite element method (WG-FEM) with second-order accuracy in time for solving one type of parabolic equations. We apply a discrete weak gradient operator on discontinuous piecewise polynomials in the numerical scheme. Fully-discrete WG finite element scheme is introduced and analyzed. Optimal order error estimates are derived in the L 2 norm. Numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

• We have provided comparison of CPU time for the standard WG-FEM and the fast WG-FEM in Table 4. • All references have been updated. • We have explained the super-convergence phenomena of | | ∇ w (e h) | | in the numerical experiments. • We have re-phased the "a new application of WG-FEM" according to reviewer's sugguestion. • The definition of Pk (K) has been corrected. • All minor mistakes pointed out by the reviewers have been corrected accordingly. • A detailed response to reviewers is attached to the revised manuscript. This paper is concerned with the application of a weak Galerkin finite element method (WG-FEM) to the time-fractional diffusion equation. The WG-FEM with L1-formula and the fast evaluation scheme based on WG-FEM are designed. The optimal convergence rates for both semi-discrete and fully discrete WG-FEM schemes are obtained, and the stability analysis for the semi-discrete WG-FEM is derived. Numerical experiments are implemented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract We study weak Galerkin (WG) finite element method (FEM) for solving nonlinear convection-diffusion problems. A WG finite element scheme is presented based on a new variational form. We prove the energy conservation law and stability of the continuous time WG FEM. In particular, optimal order error estimates are established for the WG FEM approximation in both a discrete H 1-norm and L 2-norm. Numerical experiments are performed to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the multigrid algorithm for a quad-curl problem based on C 0 interior penalty method on bounded polygonal domains. It is shown that the contraction numbers of W -cycle multigrid algorithm for the C 0 interior penalty method converge uniformly with rate of m − α , where m is the number of pre-smoothing (and post-smoothing) steps, α is the index of elliptic regularity. The performance of the multigrid fast solver is illustrated by numerical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a meshless method named Moving Least Squares Particle Hydrodynamics (MLSPH) method is applied to obtain numerical solutions of the Burgers' equation with a set of initial and boundary conditions. The influence of kernel function and smoothing length on the accuracy of MLSPH method is investigated. Several various forms of one- and two- dimensional Burgers' equations are successfully calculated. It is concluded that the proposed method can obtain second-order accuracy. The way of non-uniform distribution is applied to present the desired performance near the shocks. The numerical results obtained by the proposed method are compared with the analytical solutions and the results obtained by some other methods. The satisfactorily small errors indicate the high precision of the proposed method to solve this kind of partial differential equation. [ABSTRACT FROM AUTHOR]