Two‐grid weak Galerkin method for semilinear elliptic differential equations

• Published in: Mathematical methods in the applied sciences Vol. 46; no. 1; pp. 423 - 437
• Main Authors: Chen, Luoping, Wu, Fanyun, Zeng, Guoyan
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 15.01.2023
• Subjects: Elliptic differential equations; Elliptic functions; Exact solutions; Galerkin method; Mathematical analysis; semilinear elliptic differential equations; two‐grid discretization; weak Galerkin method
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.8519

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.