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Computing (2008) 82: 1130
DOI 10.1007/s00607-008-0260-8
Printed in The Netherlands
A Newton method for Bernoullis free boundary problem in three dimensions
H. Harbrecht
Institut fr Numerische Simulation, Universitat Bonn, Bonn, Germany
Received 31 January 2007; Accepted 28 February 2008; Published online 18 March 2008 Springer-Verlag 2008
Summary
The present paper is dedicated to the numerical solution of Bernoullis free boundary problem in three dimensions. We reformulate the given free boundary problem as a shape optimization problem and compute the shape gradient and Hessian of the given shape functional. To approximate the shape problem we apply a RitzGalerkin discretization. The necessary optimality condition is resolved by Newtons method. All information of the state equation, required for the optimization algorithm, are derived by boundary integral equations which we solve numerically by a fast wavelet Galerkin scheme. Numerical results conrm that the proposed Newton method yields an efcient algorithm to treat the considered class of problems.
AMS Subject Classications: 49Q10; 49K20; 49M15; 65K10.
Keywords: free boundary problem; shape calculus; Newton method; boundary integral equations; wavelets.
1. Introduction
The present paper is devoted to the efcient numerical solution of Bernoullis free boundary problem in three dimensions by means of shape optimization. We reformulate the free boundary problem as a shape optimization problem which we are going to solve iteratively by a Newton method. The particular problem under consideration is given as follows.
Let T R3 denote a bounded domain with boundary T = . Inside the domain T we assume the existence of a simply connected subdomain S T with boundary S = . The resulting annular domain T \S is denoted by . The topological situation is visualized in Fig. 1.
We consider the following free boundary problem: seek the free boundary such that the overdetermined boundary value problem
Correspondence: Helmut Harbrecht, Institut fr Numerische Simulation, Universitat Bonn, Wegelerstr. 6, 53115 Bonn, Germany (E-mail: [email protected])
12 H. Harbrecht
Fig. 1. The domain and its boundaries and
u = f in , u = 0,
u
n =g on , u = h on ,
(1.1)
is satised. Herein, we suppose that g, h > 0 and f 0 are sufciently smooth functions on R3 such that u C2,( U) for a xed > 0 and U denoting a...