Error Analysis of an Unfitted HDG Method for a Class of Non-linear Elliptic Problems

Nestor Sánchez, Tonatiuh Sánchez-Vizuet, Manuel Solano

Research output: Contribution to journalArticlepeer-review

Abstract

We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter.

Original languageEnglish (US)
Article number92
JournalJournal of Scientific Computing
Volume90
Issue number3
DOIs
StatePublished - Mar 2022

Keywords

  • Curved boundary
  • Hybridizable discontinuous Galerkin
  • Non-linear boundary value problems
  • Transfer paths
  • Unfitted mesh

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

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