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

Burger M, Elvetun OL, Schlottbom M (2015)

Publication Language: English

Publication Type: Journal article

Publication year: 2015

Journal

Foundations of Computational Mathematics Springer Verlag (Germany)

Publisher: Springer New York LLC

Book Volume: null

Issue: null

DOI: 10.1007/s10208-015-9292-6

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.

Authors with CRIS profile

Martin Burger

Involved external institutions

Westfälische Wilhelms-Universität (WWU) Münster DE Germany (DE) Norwegian University of Life Sciences (NMBU) NO Norway (NO)

How to cite

APA:

Burger, M., Elvetun, O.L., & Schlottbom, M. (2015). Analysis of the Diffuse Domain Method for Second Order Elliptic Boundary Value Problems. Foundations of Computational Mathematics, null. https://dx.doi.org/10.1007/s10208-015-9292-6

MLA:

Burger, Martin, Ole Loseth Elvetun, and Matthias Schlottbom. "Analysis of the Diffuse Domain Method for Second Order Elliptic Boundary Value Problems." Foundations of Computational Mathematics null (2015).

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