Finite element analysis for a coupled bulk-surface partial differential equation

By Charles M. Elliott, Thomas R. Ranner. 2013

149 citations recorded by the source corpus.

In this paper, we define a new finite element method for numerically approximating the solution of a partial differential equation in a bulk region coupled with a surface partial differential equation posed on the boundary of the bulk domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface. Two finite element spaces are defined, one in the bulk region and one on the surface, by taking the set of all continuous functions which are also piecewise polynomial on each bulk simplex or boundary face. We study this method in the context of a model elliptic problem; in particular, we look at well-posedness of the system using a variational formulation, derive perturbation estimates arising from domain approximation and apply these to find the optimal-order error estimates. A numerical experiment is described which demonstrates the order of convergence.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/87434e44cbce8ccd205618eed6d15ea39bc501d6

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In Computational Mechanics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

CutFEM: Discretizing geometry and partial differential equations (2015, background): “Applications are found, for example, in cell membrane transport; cf. [25].”
Finite element methods for surface PDEs* (2013, methodology): “We refer to Elliott and Styles (2012) for examples suggesting the possible accuracy advantages of using the ALE version of ESFEM.”