On integral equation methods for the first Dirichlet problem of the biharmonic and modified biharmonic equations in nonsmooth domains

Johan Helsing, Shidong Jiang

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

Despite important applications in unsteady Stokes flow, a Fredholm second kind integral equation formulation modeling the first Dirichlet problem of the modified biharmonic equation in the plane has been derived only recently. Furthermore, this formulation becomes very ill-conditioned when the boundary is not smooth, say, having corners. The present work demonstrates numerically that a method called recursively compressed inverse preconditioning (RCIP) can be effective when dealing with this geometrically induced ill-conditioning in the context of Nystr\"om discretization. The RCIP method not only reduces the number of iterations needed in iterative solvers but also improves the achievable accuracy in the solution. Adaptive mesh refinement is only used in the construction of a compressed inverse preconditioner, leading to an optimal number of unknowns in the linear system in the solve phase.

Original languageEnglish (US)
Pages (from-to)A2609-A2630
JournalSIAM Journal on Scientific Computing
Volume40
Issue number4
DOIs
StatePublished - 2018

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Biharmonic equation
  • Modified biharmonic equation
  • RCIP method
  • Second kind integral equation

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