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Higher order convergent trial methods for Bernoulli's free boundary problem

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Free Boundary Problems and Related Topics

Co-author: Helmut Harbrecht (University of Basel)

Free boundary problem is a partial differential equation to be solved in a domain, a part of whose boundary is unknown the so-called free boundary. Beside the standard boundary conditions that are needed in order to solve the partial differential equation, an additional boundary condition is imposed at the free boundary. One aims thus to determine both, the free boundary and the solution of the partial differential equation.

This work is dedicated to the solution of the generalized exterior Bernoulli free boundary problem which is an important model problem for developing algorithms in a broad band of applications such as optimal design, fluid dynamics, electromagnentic shaping etc. For its solution the trial method, which is a fixed-point type iteration method, has been chosen.

The iterative scheme starts with an initial guess of the free boundary. Given one boundary condition at the free boundary, the boundary element method is applied to compute an approximation of the violated boundary data. The free boundary is then updated such that the violated boundary condition is satisfied at the new boundary. Taylor’s expansion of the violated boundary data around the actual boundary yields the underlying equation, which is formulated as an optimization problem for the sought update function. When a target tolerance is achieved, the iterative procedure stops and the approximate solution of the free boundary problem is detected.

The efficiency of the trial method as well as its speed of convergence depends significantly on the update rule for the free boundary, and thus on the violated boundary condition. This talk focuses on the trial method with violated Dirichlet boundary data and on the development of higher order convergent versions of the trial method with the help of shape sensitivity analysis.

This talk is part of the Isaac Newton Institute Seminar Series series.

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