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

Abstract

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 to 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 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 optimal order error estimates. A numerical experiment is described which demonstrates the order of convergence.

CutFEM: Discretizing geometry and partial differential equations

“…Here, C T is the constant of the trace inequality (12) and c G is the coercivity constant of the stability estimate (25). We conclude by choosing > 2C 2 T c 1 G , where the lower bound is independent of the mesh/boundary intersection, but not of the penalty parameter 1 in j. ; /.…”

“…Applications are found, for example, in cell membrane transport; cf. [25]. Here, we no longer have a distinct side condition and can dispense with Nitsche's method.…”

Stable finite element approximations of two-phase flow with soluble surfactant

“…See the appendix for details on how to compute radially symmetric solutions in this case. We remark that a priori error bounds for discretizations by fitted and unfitted finite element methods for related coupled linear bulk-surface elliptic partial differential equations have been obtained in [22,14,29].…”

Analysis and Simulations of Coupled Bulk-surface Reaction-Diffusion Systems on Exponentially Evolving Volumes

“…It must be observed that for the fractional-step θ method above, the computational mesh is continuously evolving according to the exponential growth and all matrices are assembled at different discretised surfaces depending on the time-level n. The number of degrees of freedom and the mesh connectivity remains constant throughout domain growth. For further details on the implementation of the evolving (bulk) surface finite element method we refer the interested reader to [2,6,7,23].…”