Multigrid on the interface for mortar mixed finite element methods for elliptic problems Original Research Article

We consider mixed finite element approximations of second order elliptic equations on domains that can be described as a union of subdomains or blocks. We assume that the subdomain grids are locally defined and need not match across the block boundaries. Specially chosen mortar finite element spaces are introduced on the interfaces for approximating the scalar variable (pressure). The mortars also serve as Lagrange multipliers for imposing flux-matching conditions. The method is implemented by reducing the algebraic system to a positive definite interface problem in the mortar spaces. This problem is then solved using a multigrid on the interface with conjugate gradient smoothing. The algorithm is very efficient in a distributed parallel computing environment as only subdomain solves are required on each conjugate gradient iteration. The standard variational assumptions for the multigrid are not satisfied, since the interface bilinear forms vary from level to level. We present theoretical results for the convergence of the V-cycle and the W-cycle. Computational results in two- and three-dimensions are given to illustrate and confirm the theory.