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.

完全灵活的hp-间断Galerkin（DG）格式允许任意局部网格细化和任意多项式次数分布，为广泛的偏微分方程提供了一个强大的离散平台。对于椭圆边值问题，相应的线性方程组很快就会变得非常病态。到目前为止，有效的预条件似乎是已知的网格和度分布，从而严重限制了充分DG潜力的强限制。该项目旨在为“完全灵活”的情况开发高效的预处理器。辅助空间方法，在一个连锁的形式，被用来作为概念平台。一个主要的新成分是某些层次的低阶有限元空间的各向异性并矢网格与非嵌套Legendre-Gaussian-Lobatto网格，在我们看来，提供了一个机会，以克服目前使用的方法的主要障碍。相应的理论基础的发展是由一个适当的软件概念的补充。最后，与最近的DG杂交技术的连接将进行探讨。

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