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Domain decomposition methods for nonlinear elliptic and parabolic equations

In English

Författare

Summary, in English

Nonoverlapping domain decomposition methods have been utilized for a long time to solve linear and nonlinear elliptic problems, and more recently, parabolic problems. Despite this, there is no convergence theory for nonlinear elliptic and parabolic equations on general Lipschitz domains in Rd, d ≥ 2. We therefore develop a Steklov–Poincaré theory for nonlinear elliptic and parabolic problems and study the properties of the Steklov–Poincaré operators. In the elliptic case, we show that the two standard methods, the Dirichlet–Neumann and Robin–Robin methods converge. We demonstrate with numerical results that the Neumann-Neumann method does not converge in some cases and instead develop two modified Neumann–Neumann methods and prove their convergence. In the parabolic case, we show convergence of the Robin–Robin method and introduce two modified Dirichlet–Neumann methods, for which we show convergence. We also discuss how the results can be applied to an equation after it has been discretized by a finite element method and give numerical results for each method.

Publiceringsår

2025-01-30

Språk

Engelska

Dokumenttyp

Doktorsavhandling

Förlag

Centre for Mathematical Sciences, Lund University

Ämne

  • Computational Mathematics

Nyckelord

  • Nonoverlapping domain decomposition
  • Steklov-Poincaré operator
  • convergence
  • Nonlinear elliptic equation
  • Nonlinear parabolic equation

Aktiv

Published

Projekt

  • Next generation numerical partitioning schemes for time dependent PDEs
  • Moving domain decomposition methods for parabolic PDEs

Forskningsgrupp

  • Applied Mathematics

ISBN/ISSN/Övrigt

  • ISBN: 978-91-8104-348-8
  • ISBN: 978-91-8104-347-1

Försvarsdatum

14 mars 2025

Försvarstid

13:15

Försvarsplats

Lecture Hall MH:Riesz, Centre of Mathematical Sciences, Sölvegatan 18 A, Faculty of Engineering LTH, Lund University, Lund.

Opponent

  • Axel Målqvist (Prof.)