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Foundations of Computational Mathematics

, Volume 17, Issue 3, pp 627–674 | Cite as

Analysis of the Diffuse Domain Method for Second Order Elliptic Boundary Value Problems

  • Martin Burger
  • Ole Løseth Elvetun
  • Matthias SchlottbomEmail author
Article

Abstract

The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper, we study the diffuse domain method for approximating second order elliptic boundary value problems posed on bounded domains and show convergence and rates of the approximations generated by the diffuse domain method to the solution of the original second order problem when complemented by Robin, Dirichlet or Neumann conditions. The main idea of the diffuse domain method is to relax these boundary conditions by introducing a family of phase-field functions such that the variational integrals of the original problem are replaced by a weighted average of integrals of perturbed domains. From a functional analytic point of view, the phase-field functions naturally lead to weighted Sobolev spaces for which we present trace and embedding results as well as various types of Poincaré inequalities with constants independent of the domain perturbations. Our convergence analysis is carried out in such spaces as well, but allows to draw conclusions also about unweighted norms applied to restrictions on the original domain. Our convergence results are supported by numerical examples.

Keywords

Diffuse domain method Weighted Sobolev spaces Domain perturbations Elliptic boundary value problems 

Mathematics Subject Classification

35J20 35J70 46E35 65N85 

Notes

Acknowledgments

MB and MS acknowledge support by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse. MB acknowledges support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Münster, Germany. OLE acknowledges support by DAAD for his 1 year research stay at WWU Münster.

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Copyright information

© SFoCM 2015

Authors and Affiliations

  • Martin Burger
    • 1
    • 2
  • Ole Løseth Elvetun
    • 3
  • Matthias Schlottbom
    • 1
    Email author
  1. 1.Institute for Computational and Applied MathematicsUniversity of MünsterMünsterGermany
  2. 2.Cells in Motion Cluster of ExcellenceUniversity of MünsterMünsterGermany
  3. 3.Department of Mathematical Sciences and TechnologyNorwegian University of Life SciencesÅsNorway

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