Two-level mortar domain decomposition preconditioners for heterogeneous elliptic problems

• We use nonoverlapping domain decomposition for mixed method approximations.
• We propose a two-level preconditioner based on the interfaces between subdomains.
• The coarse preconditioner uses the multiscale mortar domain decomposition method.
• Prolongation is defined uniquely to preserve projection onto normal velocities.
• Numerical tests of highly heterogeneous porous media show efficiency and robustness.

We consider a second order elliptic problem with a heterogeneous coefficient, which models, for example, single phase flow through a porous medium. We write this problem in mixed form and approximate it for parallel computation using the multiscale mortar domain decomposition mixed finite element method, which gives rise to a saddle point linear system. We use a relatively fine mortar space, which allows us to enforce continuity of the normal velocity flux, or nearly so in the case of nonmatching meshes. To solve the Schur complement linear system for the mortar unknowns, we propose a two-level preconditioner based on the interfaces between subdomains. The coarse preconditioner also uses the multiscale mortar domain decomposition method, but with instead a very coarse mortar space. We show that the prolongation operator of the coarse mortar to the fine is defined uniquely by the condition that the L2L2-projection of a coarse mortar agrees with its projection onto the space of normal velocity fluxes, i.e., no energy is introduced when changing mortar scales. The local smoothing preconditioner is based on block Jacobi, using blocks defined by the interfaces. We use restrictive smoothing domains that are smaller normal to the interfaces, and overlapping in the directions tangential to the interfaces. In the simplest case, the condition number of the preconditioned interface operator is bounded by a multiple of (log(1+H/h))2(log(1+H/h))2. We show several numerical examples involving strongly heterogeneous porous media to demonstrate the efficiency and robustness of the preconditioner. We see that it is often desirable, and sometimes necessary, to use a piecewise linear or higher order coarse mortar space to achieve good convergence for heterogeneous problems.