Block Preconditioning Methods for Asymptotic Preserving Scheme Arising in Anisotropic Elliptic Problems

Efficient and robust iterative solvers for strongly anisotropic elliptic equations are very challenging. Indeed, the discretization of this class of problems gives rise to a linear system with a condition number increasing with anisotropic strength. This weakness is addressed clearly by adopting the asymptotic-preserving (AP) discretizations. In this paper a block preconditioning method is introduced to solve the linear algebraic systems of a class of micro–macro asymptotic-preserving (MMAP) scheme. The MMAP method was developed by Degond et al. in 2012 where its corresponding discrete matrix has a \(2\times 2\) $2\times 2$ block structure. Motivated by approximate Schur complements, a series of block preconditioners are constructed. We first analyze a natural approximate Schur complement that is the coefficient matrix of the original Non-AP discretization. However it tends to be singular for very small anisotropic parameters. We then improve it by using more suitable approximation for boundary rows of the exact Schur complement. With these block preconditioners, a preconditioned GMRES iterative method is developed to solve the discrete equations. Several numerical tests show that block preconditioning methods can be a practically useful strategy with respect to grid refinement and anisotropic strengths.