Discontinuous Galerkin Finite Element Convergence for Incompressible Miscible Displacement Problems of Low Regularity

Sören Bartels , Max Jensen , Rüdiger Müller

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2, 20

MSC 2000

65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

76S05 Flows in porous media; filtration; seepage

Abstract
In this article we analyse the numerical approximation of incompressible miscible displacement problems with a combined mixed finite element and discontinuous Galerkin method under minimal regularity assumptions. The main result is that sequences of discrete solutions weakly accumulate at weak solutions of the continuous problem. In order to deal with the non-conformity of the method and to avoid overpenalisation of jumps across interelement boundaries, the careful construction of a reflexive subspace of the space of bounded variation, which compactly embeds into $L^2(\Omega)$, and of a lifting operator, which is compatible with the nonlinear diffusion coefficient, are required. An equivalent skew-symmetric formulation of the convection and reaction terms of the nonlinear partial differential equation allows to avoid flux limitation and nonetheless leads to an unconditionally stable and convergent numerical method. Numerical experiments underline the robustness of the proposed algorithm.