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On Trefftz’ integral equation for the Bernoulli free boundary value problem

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Abstract

We propose a new numerical method for the solution of Bernoulli’s free boundary value problem for a harmonic function w in a doubly connected domain D in \(\mathbb {R}^2\) where an unknown free boundary \(\varGamma _0\) is determined by prescribed Cauchy data of w on \(\varGamma _0\) in addition to a Dirichlet condition on the known boundary \(\varGamma _1\). Our method is based on a two-by-two system of boundary integral equations for the unknown boundary \(\varGamma _0\) and the unknown normal derivative \(g=\partial _\nu w\) of w on \(\varGamma _1\). This system is nonlinear with respect to \(\varGamma _0\) and linear with respect to g and we suggest to solve it simultaneously for \(\varGamma _0\) and g by Newton iterations. We establish a local convergence result and exhibit the feasibility of the method by a few numerical examples.

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  1. University of Göttingen, Lotzestrasse 16-18, 37083, Göttingen, Germany

    Rainer Kress

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  1. Rainer Kress
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Kress, R. On Trefftz’ integral equation for the Bernoulli free boundary value problem. Numer. Math. 136, 503–522 (2017). https://doi.org/10.1007/s00211-016-0847-5

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  • DOI: https://doi.org/10.1007/s00211-016-0847-5

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