AP¹--P¹Finite Element Method for a Phase Relaxation Model I: Quasi-Uniform Mesh

Abstract

We study a simple model of phase relaxation which consists of a parabolic PDE for temperature and an ODE with a small parameter " and double obstacles for phase variable. The model replaces sharp by di use interfaces and gives rise to superheating e ects. A semi-explicit time discretization with uniform time-step is combined with continuous piecewise linear nite elements for both and , over a xed quasi-uniform mesh of size h. At each time step, an inexpensive nodewise algebraic correction is performed to update , followed by the solution of a linear positive de nite symmetric system for by a preconditioned conjugate gradient method. A priori estimates for both and are derived in L 2-based Sobolev spaces provided the stability constraint " is enforced. Asymptotic behavior of the fully discrete model is examined as "; ; h # 0 independently, which leads to a rate of convergence of order O((+ h)" ?1=2), provided a natural compatibility condition on the initial data is satis ed. Numerical experiments illustrate the performance of the proposed method for the natural choice h ".

Anisotropic mesh adaptation for the solution of the Stefan problem

“…For this reason the presented formulation is called a semi-phase-field model. The proposed formulation presents strong similarities with the phase-field relaxation model introduced in Jiang and Nochetto [6].…”

“…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

“…In Section 4.2 we state a posteriori error estimates for (3.1)-(3.4) along with their optimal asymptotic rate, and prove them in Section 5. In addition to providing computable error estimators, these results simplify and extend the error analysis of [10] to variable step-sizes.…”

“…Both methods are studied in [10] for constant step-size. In Section 4.2 we state a posteriori error estimates for (3.1)-(3.4) along with their optimal asymptotic rate, and prove them in Section 5.…”

Optimal error estimates for semidiscrete phase relaxation models

“…The semidiscrete tra veling waves of [10] reveal the resulting order for the semiexplicit method to be the best possible ; it improves upon those in [15,16] indeed. A new space discretization of this scheme using piecewise linear finite éléments for both <9 n and X n is further analyzed in [6], When x = e and F = 0 the semi-explicit problem coïncides with the so-called nonlinear Chernoffformula studied in [7] ; see also the surveys [9,14]. The somewhat disturbing factor 1/Vë can be eliminated via extrapolation, but at the expense of keeping two consécutive time itérâtes, namely 0 n~l and O nZ , and a more restrictive stability constraint.…”