Numerical Solution of Diffusion Problems with Non-Local Free Boundary Conditions

Among the many numerical methods proposed for elliptic and parabolic free boundary problems only front tracking has sufficient flexibility to tackle boundary value problems which do not necessarily have a natural fixed domain equivalent (such as an enthalpy or variational inequality formulation). A particular front tracking method for problems with regular free boundaries is the sequentially one-dimensional approach based on a method of lines discretization of elliptic and time-discretized parabolic equations and the numerical solution of the resulting system of ordinary differential equations with a line iterative sweep method. As has been demonstrated repeatedly, if the free boundary is geometrically simple then this approach can handle problems of considerable mathematical complexity. A recent exposition of this method with model problems from laminar flame theory and non-equilibrium phase transition (the Stefan problem with a Gibbs-Thomson interface condition) may be found in [3]. It is the purpose of this paper to illustrate that the same formalism can be applied to solve a class of free boundary problems where the boundary condition involves a functional of the whole free boundary. As will be seen, algorithmically, the proposed sequentially one-dimensional method takes little note of globally defined free boundary conditions. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.