An efficient boundary integral method for the Mullins-Sekerka problem
The paper is concerned with the numerical treatment of a free boundary problem which is sometimes called Mullins-Sekerka problem in studying solidification and melting of materials with negligible specific heat. In a different context the considered problem can be regarded as an asymptotic limit of the Cahn-Hilliard equation. The problem is given by the Laplace equation in two space dimensions. On the free boundary one has the Gibbs-Thomson law (surface tension effects) and a condition describing the motion of the interface. This motion is an area preserving and curve shortening motion. After a careful introduction the authors develop an efficient method for the numerical solution which is essentially based on a reformulation of the problem by a boundary integral representation. The second step of the approach consists of the consideration of the evolution of the interface in a \(\theta\)-\(L\) formulation (\(\theta\)-tangent angle, \(L\)-total arclength). This allows to solve the evolution equation implicitly in the Fourier space. Finally, the discretized integral system is solved for the velocity, where the singular part is handled by discrete Fourier transforms. In Section 3 the authors discuss details of the numerical implementation of the proposed method. After deriving some special solutions to be used for the verification of the numerical solution, the authors report in Section 5 on their numerical results obtained for several examples. The paper is concluded by a discussion of possible extensions and applications of this approach to other problems.
scientific article; zbMATH DE number 947699