 
Summary: MULTISCALE MODEL. SIMUL. c 2007 Society for Industrial and Applied Mathematics
Vol. 6, No. 1, pp. 319346
A MULTISCALE MORTAR MIXED FINITE ELEMENT METHOD
TODD ARBOGAST, GERGINA PENCHEVA, MARY F. WHEELER§, AND
IVAN YOTOV
Abstract. We develop multiscale mortar mixed finite element discretizations for second order
elliptic equations. The continuity of flux is imposed via a mortar finite element space on a coarse
grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid
scale. The polynomial degree of the mortar and subdomain approximation spaces may differ; in fact,
the mortar space achieves approximation comparable to the fine scale on its coarse grid by using
higher order polynomials. Our formulation is related to, but more flexible than, existing multiscale
finite element and variational multiscale methods. We derive a priori error estimates and show, with
appropriate choice of the mortar space, optimal order convergence and some superconvergence on
the fine scale for both the solution and its flux. We also derive efficient and reliable a posteriori
error estimators, which are used in an adaptive mesh refinement algorithm to obtain appropriate
subdomain and mortar grids. Numerical experiments are presented in confirmation of the theory.
Key words. multiscale, mixed finite element, mortar finite element, error estimates, a posteriori,
superconvergence, multiblock, nonmatching grids
AMS subject classifications. 65N06, 65N12, 65N15, 65N22, 65N30
DOI. 10.1137/060662587
