Virtual element method for elliptic bulk-surface PDEs in three space dimensions
Virtual element method for elliptic bulk-surface PDEs in three space dimensions
复制标题
三维空间椭圆体表面偏微分方程的虚元法
DOI：
10.1002/num.23040
复制
发表时间：
2023
影响因子：
3.9
通讯作者：
Frittelli M
中科院分区：
文献类型：
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作者：
Frittelli M
在这项工作中，我们提出了一种新的体-面虚元方法（BSVEM）的数值逼近椭圆体-面偏微分方程在三维空间。BSVEM是基于离散的散装域到多面体元素与任意多个面。块体的多面体近似导致表面的多边形近似。我们提出了一种独立于数值方法的体表面多面体网格的几何误差分析。然后，我们证明了BSVEM在空间上具有最佳二阶收敛性，前提是精确解在体上为H2+ 34 $$ {H}^{2+ 34} $$，在表面上为H2$$ {H}^2 $$，其中额外的34$$ \frac{3}{4} $$是由于表面曲率和靠近边界的多面体元素的综合影响。我们表明，一般的多面体可以利用，以减少矩阵组装的计算时间。单位球面上和Dupin环圈线上的两个数值例子证实了收敛结果。
In this work we present a novel bulk‐surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk‐surface partial differential equations in three space dimensions. The BSVEM is based on the discretization of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. 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 H2+3/4$$ {H}^{2+3/4} $$ in the bulk and H2$$ {H}^2 $$ on the surface, where the additional 34$$ \frac{3}{4} $$ 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. Two numerical examples on the unit sphere and on the Dupin ring cyclide confirm the convergence result.