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.
翻译:在这项工作中,我们为三个空间维度的椭圆散表面部分差异方程式(BSPDEs)的数字近似值提出了一个新的散状表面虚拟要素方法(BSVEM)。BSVEM的基础是将散列域分解成多面的多元元素。散列的多面近似可诱发表面的多边近似值。首先,我们提出一个独立于数字方法的散列表面多面间环形的几何错误分析。然后,我们表明,BSVEM在空间有最佳的二阶趋同,只要准确的解决方案是散数$H2+3/4美元和表面$H2美元,因为额外的美元是靠近边界的表面曲度和多面元素的综合效应。我们表明,一般的多面体可以用来减少矩阵组装的计算时间。为了证明最佳趋同结果,一个数字例子出现在单元域上。