Error Control and Andaptivity for a Phase Relaxation Model

¹ Institute of Mathematics, Academia Sinica, Beijing 100080, PR China. The first author was partially supported by the National Natural Science Foundation of China under the grant No. 19771080 and China National Key Project "Large Scale Scientific and Engineering Computing" .
² Department of Mathematics, University of Maryland, College Park, MD 20742, USA. Partially supported by NSF Grant DMS-9623394 and NSF SCREMS 9628467.
³ Institut für Angewandte Mathematik, Universität Freiburg, 79104 Freiburg, Germany. Partially supported by DFG and EU Grant HCM "Phase Transitions and Surface Tension" . (alfred@mathematik.uni-freiburg.de)

Received: 2 July 1999

Abstract

The phase relaxation model is a diffuse interface model with small parameter ε which consists of a parabolic PDE for temperature θ and an ODE with double obstacles for phase variable χ. To decouple the system a semi-explicit Euler method with variable step-size τ is used for time discretization, which requires the stability constraint τ ≤ ε. Conforming piecewise linear finite elements over highly graded simplicial meshes with parameter h are further employed for space discretization. A posteriori error estimates are derived for both unknowns θ and χ, which exhibit the correct asymptotic order in terms of ε, h and τ. This result circumvents the use of duality, which does not even apply in this context. Several numerical experiments illustrate the reliability of the estimators and document the excellent performance of the ensuing adaptive method.