Robust preconditioners for $hp$-discontinuous Galerkin discretizations for elliptic problems

  • Robuste Vorkonditionierer für $hp$-Discontinuous Galerkin Methoden für elliptische Probleme

Brix, Kolja; Dahmen, Wolfgang (Thesis advisor); Canuto, Claudio (Thesis advisor)

Aachen (2018, 2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2018

Abstract

Discontinuous Galerkin (DG) methods offer a very powerful discretization tool because they not only are suited for a wide range of partial differential equations but also because they support as well local mesh refinement as varying polynomial degrees. The linear systems of equations arising from DG schemes for elliptic boundary value problems quickly become ill-conditioned such that they cannot be solved efficiently. Existing preconditioning concepts usually restrict the flexibility of DG methods by imposing strong conditions on the meshes or the distribution of the polynomial degrees such that the full capabilities of DG methods remain unused. Moreover, the condition numbers of the resulting linear systems of equations grow with the polynomial degrees employed in the discretization. This thesis aims at the construction, analysis and implementation of a preconditioner for spectral DG discretizations of elliptic boundary value problems that is fully robust in the arbitrary large and locally varying polynomial degree, i.e. under mild grading conditions the condition numbers of the preconditioned system stay uniformly bounded independent of the mesh size and the polynomial degrees. The key ingredients to achieve full robustness in the polynomial degree are Legendre-Gauss-Lobatto (LGL) grids in combination with certain equivalences that couple the norms of nodal spectral element and low order finite element functions. Since the family of LGL grids lacks some properties, e.g. they are not nested, we construct a family of nested dyadic companion grids and investigate their properties. The proposed preconditioner is composed of three stages, where in each step a different obstruction is attacked and overcome and the auxiliary space method serves as conceptual platform for the design of the preconditioners. In the first stage the spectral DG formulation is preconditioned by a corresponding conforming formulation. The second stage serves to precondition the high-order formulation on anisotropic LGL meshes to a finite-element formulation on an anisotropic dyadic companion mesh. In the third stage a multilevel preconditioner is provided that exploits the multilevel hierarchy inherent to the dyadic grids by spaces of appropriate multiwavelets. The theoretical constructions are complemented by quantitative numerical experiments, which provide some insight in the condition numbers and their dependencies on the parameters. American Mathematical Society Subject Classification (MSC2010):33C45, 34C10, 65N35, 65N55, 65N30, 65N22, 65F10, 65F08

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