A Conservative and Monotone Mixed-Hybridized Finite Element Approximation
of Transport Problems in Heterogeneous Domains
By: Marco Brera, Joseph W. Jerome, Yoichiro Mori, and Riccardo Sacco
In this article, we discuss the numerical approximation of transport
phenomena occurring at material interfaces between physical subdomains with
heterogenous properties. The model in each subdomain consists of a partial
differential equation with diffusive, convective
and reactive terms, the coupling between each subdomain being realized
through an interface transmission condition of Robin type. The numerical
approximation of the problem in the two–dimensional case is carried out
through a dual mixed–hybridized finite element method with numerical
quadrature of the mass flux matrix. The resulting method is
a conservative finite volume scheme over triangular grids, for which a
discrete maximum principle is proved under the assumption that the mesh is
of Delaunay type in the interior of the domain and of weakly acute type
along the domain external boundary and internal interface. The stability,
accuracy and robustness of the proposed method are validated on
several numerical examples motivated by applications in Biology,
Electrophysiology and Neuroelectronics.
This article has appeared: CMAME 199 (2010), 2709--2720:
DOI:10.1016/j.cma.2010.05.016