Mathematical Modeling and Analysis A Mortar Mimetic Finite Difference Method on Non-Matching Grids

We consider mimetic finite difference approxi-mations to the mixed form of second order elliptic problems on non-matching quadrilateral and tri-angular multi-block grids. Mortar finite elements are employed on the non-matching interfaces to impose weak continuity of the velocity. Optimal convergence and, for certain cases, superconver-gence is established for both the pressure and the velocity. Background In this work, we consider second order linear elliptic equations that in porous medium applica-tions model single phase Darcy flow. We solve for pressure p and velocity u satisfying u =−K∇p in Ω, ∇ ·u = b in Ω, p = g on ∂Ω, where Ω ⊂ R2, is a multi-block domain, and K is a symmetric, uniformly positive definite tensor with L∞(Ω) components representing the rock permeability divided by the fluid viscosity. The Dirichlet boundary conditions are considered merely for simplicity. The mortar mixed finite element (mortar MFE) method has been studied, for example, in [1, 2]. In this method, the domain is divided into nonoverlapping subdomain blocks, and each of these subdomain blocks is discretized on a lo-cally constructed mesh. As a result, the subdo-main grids do not match at interblock boundaries. To solve this problem, Lagrange multiplier pres-sures are introduced at the interblock boundaries. This Lagrange multiplier space is called the mor-tar finite element space. It was shown in [2] that the mortar MFE method is optimally convergent, if the boundary space has one order higher ap-proximability than the normal trace of the veloc-ity space