Search

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 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, 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]