Robust approaches to handling complex geometries with Galerkin difference methods
Robust approaches to handling complex geometries with Galerkin difference methods
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使用伽辽金差分法处理复杂几何形状的稳健方法
DOI：
10.1016/j.jcp.2019.04.031
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
J. Banks
中科院分区：
文献类型：
--
作者：
J. Kozdon;L. Wilcox;T. Hagstrom;J. Banks
伽辽金差分（GD）基是一组连续的、分段的多项式，使用类似有限差分的自由度网格来定义。利用四边形元素上的张量积构造，将一维GD基函数自然地扩展到多维。GD基可以用来定义偏微分方程不连续Galerkin有限元离散化的解空间。在这项工作中，我们提出了两种处理复杂几何形状的方法：(1)使用不一致的曲线GD元素和(2)耦合仿射GD元素与曲线简单元素。在这两种情况下，（半离散）不连续伽辽金方法即使在发生变分犯罪时也是能量稳定的。此外，对于这两种元素类型都使用了权重调整质量矩阵，这确保了只有参考质量矩阵必须被反转。我们还给出了对曲线型非一致性GD单元的度量项处理的充分条件，以保证该格式既保持常数又保持保守性。数值实验验证了该耦合方案的稳定性和准确性。
The Galerkin difference (GD) basis is a set of continuous, piecewise polynomials defined using a finite-difference-like grid of degrees of freedom. The one dimensional GD basis functions are naturally extended to multiple dimensions using a tensor product construction on quadrilateral elements. The GD basis can be used to define the solution space for a discontinuous Galerkin finite element discretization of partial differential equations. In this work we propose two approaches to handling complex geometries within this setting: (1) using nonconforming, curvilinear GD elements and (2) coupling affine GD elements with curvilinear simplicial elements. In both cases the (semidiscrete) discontinuous Galerkin method is provably energy stable even when variational crimes are committed. Additionally, for both element types a weight-adjusted mass matrix is used, which ensures that only the reference mass matrix must be inverted. We also present sufficient conditions on the treatment of metric terms for the curvilinear, nonconforming GD elements to ensure that the scheme is both constant preserving and conservative. Numerical experiments confirm the stability results and demonstrate the accuracy of the coupled schemes.
影响因子：
2.5
作者：
Lucas Friedrich;D. C. D. R. Fernández-D.-C.-D.-R.-Fernández-145704654;A. R. Winters;G. Gassner;D. Zingg;Jason E. Hicken
通讯作者：
Lucas Friedrich;D. C. D. R. Fernández-D.-C.-D.-R.-Fernández-145704654;A. R. Winters;G. Gassner;D. Zingg;Jason E. Hicken