论文标题
PDES系统中的散装曲面虚拟元素方法在两个空间维度
Bulk-surface virtual element method for systems of PDEs in two-space dimension
论文作者
论文摘要
在本文中，我们考虑了两个空间维度的耦合散装pde。该模型由散装中的PDE组成，该PDE通过一般的非线性边界条件耦合到表面的另一个PDE。对于这样一个系统，我们提出了一种新的方法，基于批量域中的虚拟元素方法[BeirãoDaveiga等，2013]在表面上的表面有限元方法[Dziuk＆Elliott，2013]中。提出的方法，我们将其构造的散装虚拟元素方法（BSVEM）作为一种特殊情况，包括三角形网格上的大量表面有限元法（BSFEM）[Madzvamuse＆Chung，2016]。该方法在太空中表现出二阶收敛，前提是确切的解决方案为批量的$ h^{2+1/4} $，表面上的$ h^2 $，在同时存在表面曲率和非三角元元素的情况下，仅需要额外的$ \ frac {1} {4} $。我们的分析中引入的两种新技术是（i）$ l^2 $ - 提供边界条件分析的逆痕量运算符，以及（ii）Sobolev扩展作为替代起重操作员的替代[Elliott＆Ranner，2013]，以进行足够的平滑精确解决方案。可以利用多边形网格的一般性来优化矩阵组件的计算时间。该方法采用了优化的矩阵矢量形式，该形式也简化了三角形网格中BSFEM的已知特殊情况[Madzvamuse＆Chung，2016]。三个数字示例说明了我们的发现。
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.
