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Generalised Stefan problems : linear analysis and computation Donaldson, Roger David

Abstract

We consider elliptic problems in which the domain is separated into two regions by a free boundary, on which mixed Dirichlet-Neumann conditions are specified. Led by the classical Stefan condition applied to change of phase models, we consider idealised numerical methods which evolve the interface by a trial method that uses the error in one of the boundary conditions as the normal velocity of the boundary. Using linear perturbation analysis of simple cases, we show exactly which interfacial conditions lead to well-posed problems and which choices of velocities lead to convergent methods. Moreover, some velocities lead to methods having superior numerical properties. Analysis of numerics representing the free boundary by a cubic spline fit is presented, followed by an example computation. Related ongoing work is introduced.

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