Multiscale mortar expanded mixed discretization of nonlinear elliptic problems

Abstract

We consider the discretization of nonlinear second order elliptic partial differential equations by multiscale mortar expanded mixed method. This is a domain decomposition method in which the model problem is restricted to the small pieces by dividing the computational domain into the non-overlapping subdomains. An unknown (Lagrange multiplier) is introduced on the interfaces which serve as pressure Dirichlet boundary condition for local subdomain problems. A finite element space is defined on interface to approximate the pressure boundary such that the normal fluxes match weakly on interface.We demonstrated solvability of discretization, and established a priori L2-error estimates for both vector and scalar approximations. We proved the optimal order convergence rates by an appropriate choice of mortar space and polynomial degree of approximation. The uniqueness of discrete problem is shown for sufficiently small values of mesh size. An error estimate for the mortar pressure is derived via linear interface formulation with pressure dependent coefficient. We also present the analysis of the linear second order elliptic problem and prove the similar results. The computational experiments are presented to validate theory.