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

• Applicant: Professor Dr. Wolfgang Dahmen
• Subject Area: Mathematics
• Term: from 2012 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 218348188

• Final Report Year: 2016

Final Report Abstract

The central objective of this project has been to develop and analyze “optimal” preconditioners for “fully flexible” DG discretizations of second order elliptic boundary value problems. By “fully flexible” we mean that arbitrary local mes refinements and variable degrees are permitted under mild grading conditions. “Optimal” then means that the condition numbers remain uniformly bounded. In particular, spectral methods are covered as a special case. Employing a cascaded Auxiliary Space Method as a central strategy, this goal has been achieved for geometrically conforming meshes and arbitrary varying polynomial degrees. The core obstructions that had so far prevented the availability of optimal preconditioners ultimately stem from the non-nestedness of LGL-grids and the anisotropy of corresponding low order finite element auxiliary spaces. Corresponding conceptional remedies have been developed in terms of associated dyadic grid hierarchies and certain anisotropic multi-waveletpreconditoners. The theoretical investigations have been accompanied and complemented by a systematic software development which, on the one hand, provided numerical realizations of the methods and, on the other hand, aimed to gaining deeper insight by monitoring and quantifying the performance of the various algorithmic constituents. An extension to covering as well local refinements with hanging nodes is currently in progress and the main theoretical ingredients are already available.

Publications

• Refinement and Connectivity Algorithms for Adaptive Discontinuous Galerkin Methods, SIAM Journal on Scientific Computing 33 (1), 2011, pp. 66-101
  Kolja Brix, Ralf Massjung, and Alexander Voss
  (See online at https://doi.org/10.1137/090767418)
• Robust Preconditioners for DG-Discretizations with Arbitrary Polynomial Degrees, in: J. Erhel et al. (eds.), Domain Decomposition Methods in Science and Engineering XXI (Proceedings of 21st International Conference on Domain Decomposition Methods, INRIA Rennes-Bretagne-Atlantique, Rennes, France, June 25-29, 2012), Lecture Notes in Computational Science and Engineering, Vol. 98, Springer, Heidelberg, pp. 537-545, 2014
  Kolja Brix, Claudio Canuto, and Wolfgang Dahmen
  (See online at https://doi.org/10.1007/978-3-319-05789-7-51)
• Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, in: Oberwolfach Reports, Report 2/2015, Vol. 12 (1), 2015, pp. 102-105
  Kolja Brix (joint with Martin Campos Pinto, Claudio Canuto, and Wolfgang Dahmen)
  (See online at https://doi.org/10.4171/OWR/2015/2)
• Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes. IMA Journal of Numerical Analysis, 35 (4), 2015, pp. 1487-1532
  Kolja Brix, Martin Campos Pinto, Claudio Canuto, and Wolfgang Dahmen
  (See online at https://doi.org/10.1093/imanum/dru053)
• Nested dyadic grids associated with Legendre–Gauss–Lobatto grids, Numerische Mathematik 131 (2), 2015, pp. 205-239
  Kolja Brix, Claudio Canuto, and Wolfgang Dahmen
  (See online at https://doi.org/10.1007/s00211-014-0691-4)