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A conservative and monotone mixed-hybridized finite element approximation of transport problems in heterogeneous domains

机译：异质域输运问题的保守单调混合杂交有限元逼近

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摘要

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 result-ing 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, accu-racy and robustness of the proposed method are validated on several numerical examples motivated by applications in biology, electrophysiology and neuroelectronics.
机译：在本文中，我们讨论了在具有异质性质的物理子域之间的材料界面处发生的传输现象的数值近似。每个子域中的模型由具有扩散，对流和反应项的偏微分方程组成，每个子域之间的耦合通过Robin类型的接口传输条件实现。该问题在二维情况下的数值逼近是通过对质量通量矩阵采用数值正交的双重混合杂交有限元方法进行的。结果方法是三角形网格上的保守有限体积方案，在假设网格在域内部为Delaunay型且沿域外部边界为弱锐型的情况下，证明了离散最大原理。内部接口。通过在生物学，电生理学和神经电子学中应用的一些数值例子，验证了该方法的稳定性，准确性和鲁棒性。

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