Discontinuous-Continuous Galerkin Methods for a Structured Model of a Biological System

Abstract

A hybrid discontinuous-continuous Galerkin finite element method is proposed to approximate the solution to a class of composite hyperbolic-parabolic PDEs which arises in the study of biological systems. We employ a discontinuous Galerkin finite element technique for the numerical approximation of the linear hyperbolic transport part of the PDE posed in an age-time domain, while we approximate the second-order elliptic operator using a standard conforming finite element method. The strong stability is established, and a priori $L^2(L^2)$-error estimates are obtained. The finite elements in the age-time domain are constructed in such a manner that the discontinuous Galerkin method is applied over a triangulation in an explicit fashion from triangle to triangle. The methods are effective in the sense that age and time steppings are easily adaptive and the computations are readily parallelizable. No restriction is imposed on the time step in connection with the mesh size in the space. The approximate solution is computed slap by slap marching in time. Some numerical examples are presented.