Discontinuous Galerkin Methods for Convection-Reaction-Diffusion systems with transmission conditions

Emmanuil Georgoulis
Department of Mathematics
University of Leicester
Leicester, England


Abstract:

We present the Discontinuous Galerkin (DG) finite element method for the solution of reaction-diffusion systems on partitioned domains. Kadem-Katchalsky-type [Biochim Biophys Acta, Volume 27, 1958] transmission conditions are imposed at the sub-domains interfaces.

The use of DG methods for time-dependent parabolic problems, initiated in the seventies by Douglas and Dupont [Lect Notes Phys, 58 (1976)] and by Wheeler [SIAM J Numer Anal Wheeler, 15 (1978)], is by now well established. Among the good properties of discontinuous methods, local conservation and the block-diagonal structure of the mass matrix makes them particularly advantageous when solving time-dependent problems. Moreover, the DG method is particularly suited to the discretization of problems with transmission conditions as these can be imposed quite simply by the addition of appropriate elemental boundary terms in the bilinear form. The use of continuous finite elements on partitioned domains was proposed by Quarteroni et al. [SIAM J Numer Anal, 39 no. 5 (2001)] for the modeling of the dynamics of solutes in the arteries.

We show that the optimal error analysis of the DG method by Suli and Lasis [SIAM J Numer Anal, 45 no. 4 (2007)] can be extended to semi-linear systems with transmission conditions under appropriate assumptions on the reaction terms growth. The problem considered is relevant to the modeling of chemical species passing through thin biological membranes. In particular, we consider applications in cellular signal transduction. A series of numerical experiments will be presented highlighting the performance of the method. This is joint work with Andrea Cangiani (Dipartimento di Matematica e Applicazioni, UniversitÓ di Milano Bicocca, ITaly) and Max Jensen (Department of Mathematical Sciences, University of Durham, UK).