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Journal of Scientific Computing

, Volume 79, Issue 1, pp 49–78 | Cite as

A High Order Mixed-FEM for Diffusion Problems on Curved Domains

  • Ricardo Oyarzúa
  • Manuel Solano
  • Paulo ZúñigaEmail author
Article
  • 260 Downloads

Abstract

We propose and analyze a high order mixed finite element method for diffusion problems with Dirichlet boundary condition on a domain \(\Omega \) with curved boundary \(\Gamma \). The method is based on approximating \(\Omega \) by a polygonal subdomain \(\mathrm {D}_{h}\), with boundary \(\Gamma _h\), where a high order conforming Galerkin method is considered to compute the solution. To approximate the Dirichlet data on the computational boundary \(\Gamma _h\), we employ a transferring technique based on integrating the extrapolated discrete gradient along segments joining \(\Gamma _h\) and \(\Gamma \). Considering general finite dimensional subspaces we prove that the resulting Galerkin scheme, which is \({\mathbf {H}}(\mathrm {div}\,; \mathrm {D}_{h})\)-conforming, is well-posed provided suitable hypotheses on the aforementioned subspaces and integration segments. A feasible choice of discrete spaces is given by Raviart–Thomas elements of order \(k\ge 0\) for the vectorial variable and discontinuous polynomials of degree k for the scalar variable, yielding optimal convergence if the distance between \(\Gamma _h\) and \(\Gamma \) is at most of the order of the meshsize h. We also approximate the solution in \(\mathrm {D}_{h}^{c}\,{:}{=}\,\Omega \backslash \overline{\mathrm {D}_{h}}\) and derive the corresponding error estimates. Numerical experiments illustrate the performance of the scheme and validate the theory.

Keywords

Curved domain High order Diffusion problem Mixed variational formulation 

Mathematics Subject Classification

65N30 65N12 65N15 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.GIMNAP-Departamento de Matemática, Facultad de CienciasUniversidad del Bío-BíoConcepciónChile
  2. 2.Centro de Investigación en Ingeniería, Matemática, CI²MAUniversidad de ConcepciónConcepciónChile
  3. 3.Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y MatemáticasUniversidad de ConcepciónConcepciónChile

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