A P1 - P1 Finite Element Method for a Phase Relaxation Model II: Adaptively Refined Meshes

Abstract

We examine the e ect of adaptively generated re ned meshes on the P 1 ?P 1 nite element method with semi-explicit time stepping of Part I, which applies to a phase relaxation model with small parameter " > 0. A typical mesh is highly graded in the so-called re ned region, which exhibits a local meshsize proportional to the time step , and is coarse in the remaining parabolic region where the meshsize is of order p. Three admissibility tests guarantee mesh quality and, upon failure, lead to remeshing and so to incompatible consecutive meshes. The most severe test checks whether the transition region, where phase changes take place, belongs to the re ned region. The other two tests monitor equidistribution of pointwise interpolation errors. The resulting adaptive scheme is shown to be stable in various Sobolev norms and to converge with a rate of order O(= p ") in the natural energy spaces. Several numerical experiments illustrate the scheme's e ciency and enhanced performance as compared with those of Part I.

Anisotropic mesh adaptation for the solution of the Stefan problem

“…In phase-field theory (see [4,6,7,9]), an ordinary differential equation is introduced for the computation of /. In this model it is preferable to use an algebraic equation but this simplification is valid only for the classical Stefan problem.…”

Error Control and Andaptivity for a Phase Relaxation Model

“…We use refining/coarsening procedures based on bisection, which lead to compatible consecutive meshes; this is a major difference with the method proposed in [11]. Given a triangle S ∈ M n , h S stands for its diameter and ρ S for its sphericity and they satisfy h S ≤ 2ρ S / sin(γ S /2), where γ S is the minimum angle of S. Shape regularity of the family of triangulations is equivalent to γ S ≥ γ > 0, with γ independent of n. We denote by B n the collection of interior inter-element boundaries or sides e of M n in Ω; h e stands for the size of e ∈ B n .…”

“…7.3). The transition layer velocity need not be computed explicitly for mesh design, which is a major improvement with respect to [11].…”

“…In contrast to [11], which uses a priori information for mesh design, the present paper is entirely based on a posteriori error estimation. Instrumental to this goal is an interpolation operator which extends a node-wise relation stemming from the discretization of (1.2) to the whole domain, but does not cause additional error contributions in the discrete coincidence region; similar ideas were previously used in [11].…”

“…Instrumental to this goal is an interpolation operator which extends a node-wise relation stemming from the discretization of (1.2) to the whole domain, but does not cause additional error contributions in the discrete coincidence region; similar ideas were previously used in [11]. The importance of this property is evident in practical applications of error control since unnecessary mesh refinement should always be avoided in the discrete coincidence region.…”