In this paper, we investigate a two-grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semilinear elliptic equation on the coarse mesh with mesh size H, then, we use the coarse mesh solution as an initial guess to linearize the semilinear equation on the fine mesh, that is, on the fine mesh (with mesh size h), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u has sufficient regularity and h = H 2 , the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

In this paper, we study the rectified linear unit (ReLU) artificial neural network (ANN) for grid adaptation in finite element method, which is used for solving differential equations (DEs) with initial/boundary condition. Compared with the classical adaptive finite element method (AFEM), ReLU ANN based on finite element method can keep the number of grid-points constant but change their relative location. Our numerical experiments show that approximate solutions obtained from the classical finite element method by ReLU ANN are more accurate than those obtained by AFEM.

In this paper, we investigate a two-grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semi-linear elliptic equation on the coarse mesh with mesh size H, then, we use the coarse mesh solution as a initial guess to linearize the semilinear equation on the fine mesh, i.e., on the fine mesh (with mesh size $h$), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u has sufficient regularity and $h=H^2$, the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.