Solving steady interface problems using residual velocities
Roger Donaldson, Brian T. R. Wetton · IMA Journal of Applied Mathematics · 2006
We consider elliptic problems in which the domain is separated into two regions by a steady free boundary, on which mixed Dirichlet–Neumann conditions are specified. Led by the classical Stefan condition applied to change of phase models, we consider numerical methods which evolve interfaces to the desired steady shape by using the residual in one of the boundary conditions as a normal velocity. 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, an idea related to early work of Garabedian. Analysis of a discrete scheme in which the free boundary is approximated by a cubic spline fit is presented, followed by an example computation.