Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains
Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains
复制标题
复杂域偏微分方程间断伽辽金有限元方法综述
DOI：
--
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
P. Houston
中科院分区：
文献类型：
--
作者：
P. Antonietti;A. Cangiani;Joe Collis;Zhaonan Dong;E. Georgoulis;Stefano Giani;P. Houston
复杂几何体上的偏微分方程（PDE）的数值逼近是一个具有挑战性的计算问题，其中包括大量的小几何特征或微结构。事实上，使用标准网格生成器，例如采用单纯形或张量积元素，自然会导致非常精细的有限元网格，因此数值近似基本PDE问题所需的计算工作量可能非常昂贵。作为一种替代方法，在这篇文章中，我们提出了一个审查的复合/凝聚不连续伽辽金有限元方法（DGFEM），采用一般多面体元素。在这里，元素通常被构造为标准元素形状的并集;以这种方式，底层复合有限元空间的最小维度与几何特征的数量无关。特别是，我们提供了一个概述的HP版本的逆估计和近似结果一般多面体元素，这是尖锐的元素刻面退化。在此基础上，推导了hp-DGFEM对二阶椭圆和一阶双曲偏微分方程的逼近误差的先验界。最后，我们提出了数值实验，突出的实际应用DGFEM网格组成的一般多面体元素。
The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements.