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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]

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 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]

SUMMARY In modern numerical simulation of problems in energy resources and environmental science, it is important to develop efficient numerical methods for time-dependent convection-diffusion problems. On the basis of nonstandard covolume grids, we propose a new kind of high-order upwind finite volume element method for the problems. We first prove the stability and mass conservation in the discrete forms of the scheme. Optimal second-order error estimate in L2-norm in spatial step is then proved strictly. The scheme is effective for avoiding numerical diffusion and nonphysical oscillations and has second-order accuracy. Numerical experiments are given to verify the performance of the scheme. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]