Multilevel preconditioning of discontinuous Galerkin spectral element methods. Part I: geometrically conforming meshes

Abstract

This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of the condition numbers that is logarithmic in the polynomial degrees when all degrees are equal and quadratic otherwise, our main objective is to realize full robustness with respect to arbitrarily large locally varying polynomial degrees degrees, i.e., under mild grading constraints condition numbers stay uniformly bounded with respect to the mesh size and variable degrees. The conceptual foundation of the envisaged preconditioners is the auxiliary space method. The main conceptual ingredients that will be shown in this framework to yield "optimal" preconditioners in the above sense are Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic nested dyadic grids as well as specially adapted wavelet preconditioners for the resulting low order auxiliary problems. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations. One of the components can, for instance, be conveniently combined with domain decomposition, at the expense though of a logarithmic growth of condition numbers. Our analysis is complemented by quantitative experimental studies of the main components.

A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems

“…A study of a BDDC scheme in the case of hp-spectral DG methods is addressed in [16], where the DG framework is reduced to the conforming one via the ASM. The ASM framework is employed also in [11], where the high order conforming space is employed as auxiliary subspace, and a uniform multilevel preconditioner is designed for hp-DG spectral element methods in the case of locally varying polynomial degree. To the best of our knowledge, this preconditioner is the only uniform preconditioner designed for high order DG discretizations.…”

Robust Preconditioners for DG-Discretizations with Arbitrary Polynomial Degrees

“…The resulting matrix B is diagonal so that the application of C B := B −1 requires only O(N) operations. It is shown in [3] that all ASM conditions are satisfied for this choice of b(·, ·). Numerical experiments show that the parameters β 1 and ρ 1 can by and large be optimized independently of the polynomial degrees.…”

“…How to ensure nestedness while keeping the grid size under control is shown in [3]. The key quality of the associated dyadic grids G D,p is that mutual low order piecewise multi-linear interpolation between the low order finite element spaces on G p (R), G D,p (R) is uniformly H 1 -stable, see [3] for the proofs. Denoting by V h,D,p (R) the space of piecewise multi-linear conforming finite elements on G D,p (R), we now take V :…”

“…The operatorQ is defined analogously with the roles of dyadic and LGL-grids exchanged, see [3]. To finally address (3), for the structure of the form b(·, ·) from the first stage the direct estimates in the ASM conditions are no longer valid.…”

“…We are content here for most part of the paper with brief pointers to the detailed analysis in [3], [4] and [2] to an extent needed for the present discussion.…”