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.

在本文中，我们考虑了二维空间中的耦合体表PDE。该模型由本体中的一个PDE组成，该PDE通过一般的非线性边界条件耦合到表面上的另一个PDE。对于这样的系统，我们提出了一种基于耦合虚拟元素方法的新颖方法（BeirãoDa Veiga等人，Math Models Methods Appl Sci 23（01）：199–214，2013年。https://doi.org/10.1051 / m2an / 2013138）到表面上的表面有限元方法（Dziuk and Elliott in Acta Numer 22：289-396，2013. https://doi.org/10.1017/s0962492913000056）。拟议的方法，我们将其称为体表面虚拟单元法作为一个特例，包括在三角形网格上的体表面有限元方法（BSFEM）（Madzvamuse和Chung in Finite Elem Anal Des 108：9–21，2016。https://doi.org/10.1016/j.finel （2015.09.002）。该方法表现出空间的二阶收敛性，条件是确切的解决方案是：本体中的\（H ^ {2 + 1/4} \）和表面上的\（H ^ 2 \），其中附加的\（\ frac {1} {4} \）仅在同时存在表面曲率和非三角形元素的情况下才需要。我们的分析中引入了两种新颖的技术：（i）保留\（L ^ 2 \）的逆跟踪算符用于边界条件的分析和（ii）Sobolev扩展作为起重操作员的替代品（Elliott和Ranner在IMA J Num Anal 33（2）：377-402，2013. https://doi.org/10.1093/imanum/drs022中获得），以提供足够平滑的精确解决方案。可以利用多边形网格的通用性来优化矩阵装配的计算时间。该方法采用优化的矩阵向量形式，也简化了已知的三角网格上BSFEM的特殊情况（Madzvamuse和Chung 2016）。三个数值示例说明了我们的发现。