Department of Mathematics
University of Leicester
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).