Domain decomposition procedures combined with H~1-Galerkin mixed finite element method for parabolic equation

摘要

Non-overlapping domain decomposition procedures are considered for parabolic equation. These procedures are combined with using H~1-Galerkin mixed finite element method in the sub-domains to approximate the primary variable u and its flux σ simultaneously. Explicit calculations are built by using integral mean methods to present the inter-domain boundary conditions for the flux. Thus, the parallelism can be achieved by these procedures. Two approximation schemes are established. Time step constraints are proved necessary to preserve stability, which are less severe than that of fully explicit Galerkin finite element method. The mixed finite element spaces are allowed to be of different polynomial degrees and not subject to the LBB-consistency condition. New nonstandard elliptic projections are defined and analyzed. Optimal error estimates for the variable u in H~1-norm and its flux σ in L~2-norm and are derived for these schemes. Numerical experiments are presented to confirm the theoretical results.