Virtual element method for elliptic bulk-surface PDEs in three space dimensions

Abstract

In this work we present a novel bulk-surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk-surface partial differential equations (BSPDEs) in three space dimensions. The BSVEM is based on the discretisation of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. Firstly, we present a geometric error analysis of bulk-surface polyhedral meshes independent of the numerical method. Then, we show that BSVEM has optimal second-order convergence in space, provided the exact solution is $H^{2+3/4}$ in the bulk and $H^2$ on the surface, where the additional $\frac{3}{4}$ is due to the combined effect of surface curvature and polyhedral elements close to the boundary. We show that general polyhedra can be exploited to reduce the computational time of the matrix assembly. To demonstrate optimal convergence results, a numerical example is presented on the unit sphere.

arXiv comment25 pages, 4 figures, 1 table. This replacement improves figures, updates references, and avoids redundancies. arXiv admin note: substantial text overlap with arXiv:2002.11748

References in corpus (10)

• Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems
• A coupled bulk-surface model for cell polarisation
• Virtual Element Method for the Laplace-Beltrami equation on surfaces
• Pattern Formation and Oscillatory Dynamics in a Two-Dimensional Coupled Bulk-Surface Reaction-Diffusion System
• Degenerate Mobilities in Phase Field Models are Insufficient to Capture Surface Diffusion
• Bulk-surface virtual element method for systems of PDEs in two-space dimension
• High order VEM on curved domains
• A nonconforming Trefftz virtual element method for the Helmholtz problem: numerical aspects
• Lumped finite element method for reaction-diffusion systems on compact surfaces
• A hybrid finite volume -- finite element method for bulk--surface coupled problems

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• Analysis and virtual element discretisation of a Stokes/Biot--Kirchhoff bulk--surface model