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. in Math Models Methods Appl Sci 23(01):199–214, 2013. https://doi.org/10.1051/m2an/2013138) in the bulk domain to a surface finite element method (Dziuk and Elliott in Acta Numer 22:289–396, 2013. https://doi.org/10.1017/s0962492913000056) on the surface. The proposed method, which we coin the bulk-surface virtual element method includes, as a special case, the bulk-surface finite element method (BSFEM) on triangular meshes (Madzvamuse and Chung in Finite Elem Anal Des 108:9–21, 2016. https://doi.org/10.1016/j.finel.2015.09.002). 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 and Ranner in IMA J Num Anal 33(2):377–402, 2013. https://doi.org/10.1093/imanum/drs022) 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 and Chung 2016). Three numerical examples illustrate our findings.
Highlights
An interesting class of PDE problems that is recently drawing attention in the literature is that of coupled bulk-surface partial differential equations (BSPDEs)
Given a number d ∈ N of space dimensions, a BSPDE is a system of m ∈ N PDEs posed in the bulk Ω ⊂ Rd coupled with n ∈ N PDEs posed on the surface Γ := ∂Ω through either linear or non-linear coupling, see for instance [37]
The purpose of the present paper is to introduce a bulk-surface virtual element method (BSVEM) for the spatial discretisation of a coupled system of BSPDEs in two space dimensions
Summary
An interesting class of PDE problems that is recently drawing attention in the literature is that of coupled bulk-surface partial differential equations (BSPDEs). The Sobolev extension of a function has the property of preserving its W m,p class, while its lift does not because it is not C k for any positive integer k This property, which is crucial in our analysis, is potentially beneficial to the error analysis of bulk-only or BSVEM approximation of more general PDEs, where the boundary curvature was not accounted for, see for instance [5,8,31,39,45,46]. 3, we introduce polygonal BS meshes, analyse geometric error, define suitable function spaces, analyse their approximation properties and present the spatial discretisation of the considered BSPDE problems. We shall assume that the weak parabolic problem (5) has a unique and sufficiently regular solution
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