An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates

Abstract:

A simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two-phase Stefan problems in 2D. A typical triangulation is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas it is locally refined in the vicinity of the discrete interface so that the relation becomes hyperbolic. Several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations. Mesh selection is based upon equidistributing pointwise interpolation errors between consecutive meshes and imposing that discrete interfaces belong to the so-called refined region. Consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated. A general theory for interpolation between noncompatible meshes is set up in ${L^p}$-based norms. The resulting scheme is stable in various Sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods on quasi-uniform meshes, namely $O({\tau ^{ - 3/2}})$ as opposed to $O({\tau ^{ - 2}})$, to achieve the same global asymptotic accuracy; here $\tau > 0$ is the (uniform) time step. A rate of convergence of essentially $O({\tau ^{1/2}})$ is derived in the natural energy spaces provided the total number of mesh changes is restricted to $O({\tau ^{ - 1/2}})$, which in turn is compatible with the mesh selection procedure. An auxiliary quasi-optimal pointwise error estimate for the Laplace operator is proved as well. Numerical results illustrate the scheme’s efficiency in approximating both solutions and interfaces.