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Parallel Galerkin domain decomposition procedures for parabolic equation on general domain are given. These procedures use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the subdomains and across interboundaries. The explicit nature of the flux prediction induces a time-step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. factor. Experimental results are presented to confirm the theoretical results.

MSC: 65N30 65N12Keywords: Incompressible miscible displacement Characteristics-mixed finite element method Mixed finite element method Post-processing step a b s t r a c t An approximation scheme is defined for incompressible miscible displacement in porous media. This scheme is constructed by using two methods. Standard mixed finite element is used for the Darcy velocity equation. A characteristics-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest-order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L 2 -norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. This scheme conserves mass globally; in fact, on the discrete level, fluid is transported along the approximate characteristics. Numerical experiments are presented finally to validate the theoretical analysis.

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A Discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi-discrete and a family of fully-discrete time approximate schemes are formulated. These schemes are symmetric. Hp-version error estimates are analyzed for these schemes. For the semi-discrete time scheme a priori L ∞ (H 1 ) error estimate is derived and similarly, l ∞ (H 1 ) and l 2 (H 1 ) for the fully-discrete time schemes. These results indicate that spatial rates in H 1 and time truncation errors in L 2 are optimal.