Domain decomposition procedures combined with H1H1-Galerkin mixed finite element method for parabolic equation

Non-overlapping domain decomposition procedures are considered for parabolic equation. These procedures are combined with using H1H1-Galerkin mixed finite element method in the sub-domains to approximate the primary variable uu and its flux σσ 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 uu in H1H1-norm and its flux σσ in L2L2-norm and are derived for these schemes. Numerical experiments are presented to confirm the theoretical results.