Bulk-surface virtual element method for systems of PDEs in two-space dimension

Abstract

In this paper we consider a coupled bulk-surface PDE in two space dimensions. The model consists of a PDE in the bulk that is coupled to another PDE on the surface through general nonlinear boundary conditions. For such a system we propose a novel method, based on coupling a virtual element method [Beirão da Veiga et al., 2013] in the bulk domain to a surface finite element method [Dziuk & Elliott, 2013] on the surface. The proposed method, which we coin the Bulk-Surface Virtual Element Method (BSVEM) includes, as a special case, the bulk-surface finite element method (BSFEM) on triangular meshes [Madzvamuse & Chung, 2016]. The method exhibits second-order convergence in space, provided the exact solution is $H^{2+1/4}$ in the bulk and $H^2$ on the surface, where the additional $\frac{1}{4}$ is required only in the simultaneous presence of surface curvature and non-triangular elements. Two novel techniques introduced in our analysis are (i) an $L^2$-preserving inverse trace operator for the analysis of boundary conditions and (ii) the Sobolev extension as a replacement of the lifting operator [Elliott & Ranner, 2013] for sufficiently smooth exact solutions. The generality of the polygonal mesh can be exploited to optimize the computational time of matrix assembly. The method takes an optimised matrix-vector form that also simplifies the known special case of BSFEM on triangular meshes [Madzvamuse & Chung, 2016]. Three numerical examples illustrate our findings.

arXiv comment39 pages, 9 figures, 5 tables

References in corpus (10)

• Evolving surface finite element method for the Cahn-Hilliard equation
• Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems
• A coupled bulk-surface model for cell polarisation
• Symmetry breaking in a bulk-surface reaction-diffusion model for signaling networks
• 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
• High order VEM on curved domains
• Lumped finite element method for reaction-diffusion systems on compact surfaces
• A hybrid finite volume -- finite element method for bulk--surface coupled problems

Cited by in corpus (5)

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• High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems
• Numerical investigations of the bulk-surface wave pinning model
• Analysis and virtual element discretisation of a Stokes/Biot--Kirchhoff bulk--surface model