BPS preconditioners for a weak Galerkin finite element method for 2D diffusion problems with strongly discontinuous coefficients

We analyze two BPS preconditioners for a weak Galerkin (WG) finite element method for 2D diffusion equations with strongly discontinuous coefficients. The first preconditioner uses nonconforming linear elements on the coarse mesh, and the corresponding condition number is bounded from above by C(1+ln(H∕h))3 with C independent of the coefficients; the second one uses H1-conforming linear elements on the coarse mesh, and the corresponding condition number is bounded from above by C(1+ln(H∕h))2 with C depending on the coefficients. In addition, we construct and analyze a preconditioner for the sub-problems encountered in the procedure of applying the two preconditioners. The condition number of the preconditioned system is bounded from above by C(1+ln(H∕h))2 with C independent of the coefficients. Finally we carry out some numerical experiments to verify our theoretical results.