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**A \(P^1-P^1\) finite element method for a phase relaxation model. I: Quasi-uniform mesh.**
*(English)*
Zbl 0972.65067

Summary: We study a simple model of phase relaxation which consists of a parabolic partial differential equation for temperature \(\theta\) and an ordinary differential equation with a small parameter \(\varepsilon\) and double obstacles for phase variable \(\chi\). The model replaces sharp interfaces by diffuse ones and gives rise to superheating effects. A semi-explicit time discretization with uniform time step \(\tau\) is combined with continuous piecewise linear finite elements for both \(\theta\) and \(\chi\), over a fixed quasi-uniform mesh of size h. At each time step, an inexpensive nodewise algebraic correction is performed to update \(\chi\), followed by the solution of a linear positive definite symmetric system for \(\theta\) by a preconditioned conjugate gradient method.

A priori estimates for both \(\theta\) and \(\chi\) are derived in L\(^{2}\)-based Sobolev spaces provided the stability constraint \(\tau\leq\varepsilon\) is enforced. Asymptotic behavior of the fully discrete model is examined as \(\varepsilon,\tau,h\) independently, which leads to a rate of convergence of order \({\O}((\tau+h)\varepsilon^{-1/2})\), provided a natural compatibility condition on the initial data is satisfied. Numerical experiments illustrate the performance of the proposed method for the natural choice \(h\approx\tau\leq\varepsilon\).

A priori estimates for both \(\theta\) and \(\chi\) are derived in L\(^{2}\)-based Sobolev spaces provided the stability constraint \(\tau\leq\varepsilon\) is enforced. Asymptotic behavior of the fully discrete model is examined as \(\varepsilon,\tau,h\) independently, which leads to a rate of convergence of order \({\O}((\tau+h)\varepsilon^{-1/2})\), provided a natural compatibility condition on the initial data is satisfied. Numerical experiments illustrate the performance of the proposed method for the natural choice \(h\approx\tau\leq\varepsilon\).

### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

80A22 | Stefan problems, phase changes, etc. |

80M10 | Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer |

35R35 | Free boundary problems for PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |