Optimal preconditioners of spectral Discontinuous Galerkin methods for elliptic boundary value problems

椭圆边值问题谱间断Galerkin方法的最优预处理器

• 批准号: 218348188
• 负责人: Professor Dr. Wolfgang Dahmen
• 金额: --
• 依托单位: Institut für Geometrie und Praktische Mathematik (IGPM)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2012
• 资助国家: 德国
• 起止时间: 2011-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/218348188?language=en
• 关键词: Optimal; preconditioners; spectral; Discontinuous; Galerkin

Fully flexible hp-Discontinuous Galerkin (DG) schemes in the sense that arbitrary local mesh refinements and arbitrary polynomial degree distributions are permitted offer a powerful discretization platform for a wide scope of partial differential equations. For elliptic boundary value problems corresponding linear systems quickly become very ill-conditioned. So far, efficient preconditioners seem to be known only under strong restrictions on meshes and degree distributions thereby severely constraining the full DG potential. This project aims at developing efficient preconditioners for the „fully flexible“ case. The auxiliary space method, in a catenated form, is used as the conceptual platform. A major new ingredient are certain hierarchies of low order finite element spaces on anisotropic dyadic meshes associated with non-nested Legendre-Gauß-Lobatto grids that offer in our opinion a chance to overcome the main obstructions to the currently used methodologies. The development of corresponding theoretical foundations is to be complemented by an appropriate software concept.Finally, the connection with recent DG hybridization techniques will be explored.

允许任意的本地网格细化和任意多项式分布的意义上，完全柔性的HP - 透视式Galerkin（DG）方案为广泛的部分微分方程提供了强大的离散平台。对于椭圆边界值问题，相应的线性系统迅速变得非常不良条件。到目前为止，有效的预处理似乎仅在对网格和程度分布的强限制下才知道，从而严重限制了全部DG潜力。该项目旨在为“完全灵活”案例开发有效的预处理。辅助空间方法以融合形式用作概念平台。一个主要的新功能是在各向异性二元网格上的低阶有限元空间的某些层次结构，与非巢穴Legendre-gauß-Lobatto网格相关，这些网格在我们看来提供了一个机会，有机会克服当前使用方法的主要对象。相应的理论基础的开发将通过适当的软件概念完成。在本文中，将探讨与最新的DG杂交技术的联系。

项目成果

New Discretization Methods for the Numerical Approximation of PDEs

偏微分方程数值逼近的新离散方法

• DOI: 10.4171/owr/2015/2
• 发表时间: 2015
• 期刊: Oberwolfach Reports
• 作者: Kolja Brix (joint with Martin Campos Pinto;Claudio Canuto;Wolfgang Dahmen)
• 通讯作者: Wolfgang Dahmen)

Nested dyadic grids associated with Legendre–Gauss–Lobatto grids

与 LegendreâGaussâLobatto 网格关联的嵌套二进网格

• DOI: 10.1007/s00211-014-0691-4
• 发表时间: 2015
• 期刊: Numerische Mathematik
• 影响因子: 2.1
• 作者: Kolja Brix;Claudio Canuto;Wolfgang Dahmen
• 通讯作者: Wolfgang Dahmen

Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods Part I: Geometrically Conforming Meshes

不连续伽辽金谱元方法的多级预处理第一部分：几何一致网格

• DOI: 10.1093/imanum/dru053
• 发表时间: 2015
• 期刊: Ima Journal of Numerical Analysis
• 影响因子: 2.1
• 作者: Kolja Brix;Martin Campos Pinto;Claudio Canuto;Wolfgang Dahmen
• 通讯作者: Wolfgang Dahmen

Refinement and Connectivity Algorithms for Adaptive Discontinuous Galerkin Methods

自适应间断伽辽金法的细化和连通算法

• DOI: 10.1137/090767418
• 发表时间: 2011
• 期刊: SIAM J. Sci. Comput.
• 作者: Kolja Brix;Ralf Massjung;Alexander Voss
• 通讯作者: Alexander Voss