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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]

We study a fully‐discrete stabilizer‐free weak Galerkin (SFWG) finite element method for the initial‐boundary value problem of variable‐order Caputo time‐fractional diffusion equation. Optimal convergence order in L2$$ {L}_2 $$ norm for L1$$ L1 $$ fully‐discrete SFWG scheme is obtained. Furthermore, we develop a fast evaluation algorithm to improve the computational efficiency. Numerical experiments are performed to verify theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we consider a weak Galerkin finite element method and a nonuniform two‐step backward differentiation formula scheme for solving the Allen–Cahn equation. A discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme. It is well‐known that the Allen–Cahn equations have a nonlinear stability property, that is, the free‐energy functional decreases with respect to time. A corresponding energy stability analysis of the discretized system is developed. Optimal order of L2‐norm error estimates are derived. Numerical experiments are preformed to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a robust weak Galerkin discretization for the incompressible miscible displacement problem in porous media, which is a nonlinear time-dependent system. The scheme we constructed is positive definite and flexible by arbitrarily shaped polygonal mesh without imposing extra conditions on the stabilization. The other feature of our method is that there is no limitation on the ratio of time-step and spatial mesh size. Error estimates are derived for concentration and pressure in the L 2 norm and energy norm, respectively. Extensive numerical experiments, including realistic test cases, have been conducted to verify our theoretical results and the efficiency of the 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]

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 work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm. [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]

A new conforming discontinuous Galerkin method, which is based on weak Galerkin finite element method, is introduced for solving second order elliptic interface problems with discontinuous coefficient. The numerical method studied in this paper has no stabilizer and fewer unknowns compared with the known weak Galerkin algorithms. The error estimates in H 1 and L 2 norms are established, which are the optimal order convergence. Numerical experiments demonstrate the performance of the method, confirm the theoretical results of accuracy. [ABSTRACT FROM AUTHOR]

Abstract: 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 norm and 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. [Copyright &y& Elsevier]

• 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]

In this article, we consider a weak Galerkin finite element method (WG-FEM) for solving one type of viscoelastic wave equation. A discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme. We introduce both the semi- and fully-discrete WG-FEMs and establish the corresponding stability estimates. Optimal order error estimate in H 1 -norm is derived. Numerical experiments are preformed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a SAV method for the Cahn–Hilliard type phase field model with new dynamic boundary conditions. By using the continuous finite element method in space and the backward difference method in time, the fully discrete numerical schemes preserving the energy law are constructed. Numerical examples show that the proposed scheme can simulate the phase field model well even in a relatively rough grid. [ABSTRACT FROM AUTHOR]

In this paper, we study a weak Galerkin (WG) finite element method for semiconductor device simulations. We consider the one-dimensional drift–diffusion (DD) and high-field (HF) models, which involves not only first derivative convection terms but also second derivative diffusion terms, as well as a coupled Poisson potential equation. The main difficulties in the analysis include the treatment of the nonlinearity and coupling of the models. The weak Galerkin finite element method adopts piecewise polynomials of degree k for the approximations of electron concentration and electric potential in the interior of elements, and piecewise polynomials of degree k + 1 for the discrete weak derivative space. The optimal order error estimates in a discrete H 1 norm and the standard L 2 norm are derived. Numerical experiments are presented to illustrate our theoretical analysis. Moreover, numerical schemes also work out for the discontinuous diffusion coefficient problems. [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 provide an expanded mixed finite element method for a class of two-dimensional Sobolev equations. The optimal error estimates for a semi-discrete scheme and a fully discrete scheme are obtained. Also numerical examples are stated to verify our theoretical results. [ABSTRACT FROM AUTHOR]