Two-grid weak Galerkin method for semilinear elliptic differential equations

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$$ H $$, 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$$ h $$), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u$$ u $$ has sufficient regularity and h=H2$$ h={H}<^>2 $$, the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

Automated summary

This paper investigates a two-grid weak Galerkin method for solving semilinear elliptic differential equations. The method involves solving the equation on a coarse mesh and then linearizing it on a fine mesh. Theoretical analysis shows that the method achieves the same convergence accuracy as the weak Galerkin method when certain conditions are met.