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

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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.

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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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Authors and Affiliations

  1. Institute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, 48149, Münster, Germany

    Martin Burger & Matthias Schlottbom

  2. Cells in Motion Cluster of Excellence, University of Münster, Münster, Germany

    Martin Burger

  3. Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Ås, Norway

    Ole Løseth Elvetun

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  1. Martin Burger
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  2. Ole Løseth Elvetun
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  3. Matthias Schlottbom
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Correspondence to Matthias Schlottbom.

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Communicated by Douglas N. Arnold.

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Burger, M., Elvetun, O.L. & Schlottbom, M. Analysis of the Diffuse Domain Method for Second Order Elliptic Boundary Value Problems. Found Comput Math 17, 627–674 (2017). https://doi.org/10.1007/s10208-015-9292-6

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