On Fixed Point (Trial) Methods for Free Boundary Problems

2016

Interfaces Free Bound.

The present article is concerned with the solution of a generalized Bernoulli free boundary problem in three spatial dimensions. We parametrize the free boundary under consideration over the sphere and apply a trial method which updates the free boundary into the normal direction. At the free boundary, we prescribe the Neumann boundary condition and update the free boundary with the help of the remaining Dirichlet boundary condition. An inexact Newton update is employed which leads to a novel second order convergent trial method. Numerical examples show the feasibility of the present approach. In particular, a parametric representation is utilized which imposes no restriction to the free boundary under consideration except for its genus.

“…[12,26,36] and yields a first order convergent trial method (see [36]). Notice that (2.5) is well defined since we assumed h > 0 to ensure the solvability of the free boundary problem.…”

Section: Trial Methods 21 Background and Motivationmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

See 1 more Smart Citation

A second order convergent trial method for a free boundary problem in three dimensions

Bugeanu

2016

Interfaces Free Bound.

“…A superlinearly convergent trial method for Bernoulli's free boundary problem has been proposed in [12], called the implicit Neumann method. Results on a quadratically convergent trial method can be found in [25,26], where the solution of a Robin boundary value problem was suggested. Unfortunately, this Robin boundary value problem is only well-posed if the free boundary is convex.…”

Section: Speeding Up the Convergencementioning

confidence: 99%

“…To the several numerical schemes for the solution of free boundary problems belong parametric trial methods, as proposed in [2,12,25,26], and the pseudo-solid approach, as proposed in [13,19,27]. The level set method for Bernoulli's problem has been used in [6,7,18], enjoying the property of allowing topology changes.…”

mentioning

confidence: 99%

Improved trial methods for a class of generalized Bernoulli problems

Journal of Mathematical Analysis and Applications

Mitrou

2014

The aim of this article is to develop improved trial methods for the solution of a generalized exterior Bernoulli free boundary problem. At the free boundary, we prescribe the Neumann boundary condition and update the free boundary with the help of the remaining Dirichlet boundary condition. Appropriate update rules are obtained by expanding the state's Dirichlet data at the actual boundary via a Taylor expansion of first and second order. The resulting trial methods converge linearly for both cases, although the trial method based on the second order Taylor expansion is much more robust. Nevertheless, via results of shape sensitivity analysis, we are able to modify the update rules such that their convergence is improved. The feasibility of the proposed trial methods and their performance is demonstrated by numerical results.

Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

Math Methods in App Sciences

Mitrou

2014

aWe apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach's fixed-point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical findings.