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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 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]