Search

We present some numerical schemes based on the weak Galerkin finite element method for one class of Sobolev equations, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. The proposed schemes will be proved to have good numerical stability and high order accuracy when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme. Finally, some numerical results are given to verify our analysis for the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we study a stabilizer-free weak Galerkin (SFWG) finite element method with second-order accuracy in time for solving time-fractional diffusion equation. We apply the SFWG finite element method to discretize the Laplace operator and the L 2 - 1 σ formula for the Caputo fractional derivative. Optimal convergence orders of semi-discrete scheme in L 2 norm and H 1 norm and fully discrete L 2 - 1 σ -SFWG scheme in L 2 norm are obtained. Numerical experiments are performed to verify 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]

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 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]

We continue our effort in Li et al. (2024) to develop a stabilizer-free weak Galerkin finite element method (SFWG-FEM) for the two-dimensional unsteady-state drift-diffusion (DD) model. The transient state equation includes both the first derivative convection and the second derivative diffusion terms, and it couples with a Poisson potential equation. The piecewise P k (k ≥ 1) polynomials elements in the interior and on the boundaries are utilized for both electron concentration and electric potential functions. The vector function of [ P j (K) ] 2 (j ≥ j 0) for the discrete weak gradient space is used on each element K. The main difficulty is the treatment of the non-linearity and coupling of the system. Based on an introduced Ritz projection and two L 2 projection operators, optimal error estimates of O (Δ t + h k + 1) in the L 2 norm and O (Δ t + h k) in the energy-like norm are derived. Numerical experiments are presented to illustrate the theoretical analysis. • Optimal estimate of stabilizer-free WG method for drift-diffusion model is derived. • The method is flexible while reducing complexity in analyzing and programming. • A Ritz projection that can be applied to estimate other problems is introduced. [ABSTRACT FROM AUTHOR]