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

允许任意局部网格细化和任意多项式次分布的全柔性hp-不连续伽辽金格式为广泛范围的偏微分方程提供了一个强大的离散化平台。对于椭圆型边值问题，相应的线性系统很快变得非常病态。到目前为止，有效的预调节器似乎只在网格和度分布的严格限制下才知道，从而严重限制了DG的全部潜力。该项目旨在为“完全灵活”的情况开发高效的预调节器。辅助的空间方法，以链式的形式，被用作概念平台。一个主要的新成分是与非嵌套的legende - gau - lobatto网格相关的各向异性二元网格上的低阶有限元空间的某些层次结构，我们认为这为克服当前使用方法的主要障碍提供了机会。相应的理论基础的发展需要适当的软件概念的补充。最后，将探讨与最近的DG杂交技术的联系。

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.