A Two-Level Additive Schwarz Preconditioning Algorithm for the Weak Galerkin Method for the Second-Order Elliptic Equation

This paper proposes a two-level additive Schwarz preconditioning algorithm for the weak Galerkin approximation of the second-order elliptic equation. In the algorithm, aP1conforming finite element space is defined on the coarse mesh, and a stable intergrid transfer operator is proposed to exchange the information between the spaces on the coarse mesh and the fine mesh. With the framework of the Schwarz method, it is proved that the condition number of the preconditioned system only depends on the rate of the coarse mesh size and the overlapping size. Some numerical experiments are carried out to verify the theoretical results.