Doctoral thesis
Open access

Schwarz Methods, Schur Complements, Preconditioning and Numerical Linear Algebra

ContributorsOutrata, Michalorcid
Number of pages235
Imprimatur date2022-12-13
Defense date2022-12-13

In the first part we look at (optimized) Schwarz methods, mainly interested in the interface conditions (ICs) we impose to speed up the convergence rate. We show that low-rank formats are not well-suited for the classical analysis of the rate of convergence and proceed with numerical study of data-sparse formats (mainly low-rank and HODLR) for our test problem. We continue by looking at particular absorbing boundary condition (ABC; often used as a type of ICs) and we show its interesting and important approximation properties with respect to the optimal IC. Consequently we propose new, better ABCs and study these numerically. Finally, we turn our attention to a recently introduced family of preconditioners for stage equations of (fully) implicit Runge-Kutta methods and we give a full analysis of their performance in their simplest instance. Using the insight obtained, we propose a new type of preconditioners and numerically test and compare these.

  • Schwarz methods
  • Schur complement
  • Dirichlet-to-Neumann map
  • Laplacian
  • Optimized
  • Data-sparsity
  • Absorbing boundary conditions
  • Continued fractions
  • Implicit Runge-Kutta methods
  • Stage equations
  • Preconditioning
Citation (ISO format)
OUTRATA, Michal. Schwarz Methods, Schur Complements, Preconditioning and Numerical Linear Algebra. 2022. doi: 10.13097/archive-ouverte/unige:166614
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