Multiscale mortar mixed domain decomposition approximations of nonlinear parabolic equations

In this paper, nonlinear parabolic partial differential equations are considered to approximate by multiscale mortar mixed method. The key idea of the multiscale mortar mixed approach is to decompose the domain into the smaller subregions separated by the interfaces with the Dirichlet pressure boundary condition. Each subdomain is partitioned independently on the fine scale and the standard mixed methods are used to solve each local problem. Each interface is partitioned on coarse scale and a finite element space is defined to enforce the weak continuity of flux across the mortar interface. We consider both the continuous time and discrete time settings, and derive optimal error estimates for both scalar and flux unknowns. An error estimate for the mortar pressure is also presented. Several numerical results are presented to justify the theoretical convergence estimates.

Automated summary

This paper considers the approximation of nonlinear parabolic partial differential equations using a multiscale mortar mixed method, which decomposes the domain into subregions separated by interfaces with the Dirichlet pressure boundary condition. The method solves local problems on a fine scale and enforces weak continuity of flux across mortar interfaces on a coarse scale. Optimal error estimates for scalar and flux unknowns, as well as mortar pressure, are derived and supported with numerical results.