Error Control and Andaptivity for a Phase Relaxation Model

Abstract

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.Mathematics Subject Classification. 65N15, 65N30, 65N50, 80A22, 35M05, 35R35.

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.…”

“…This problem was also solved in [3] and [4]. The computational domain is the square ½À1; 1  ½À1; 1.…”

Lattice Boltzmann method for binary fluids based on mass-conserving quasi-incompressible phase-field theory

“…A number of numerical schemes have been developed based on phase-field models including spectral methods [4,12,13], finite element methods [14][15][16] and LBE methods [9][10][11].…”

An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems

“…In this paper, without these two conditions we derive an a posteriori error estimate for the total energy error based on a direct energy estimate argument which has been used in Chen, Nochetto and Schmidt [8] for the phase relaxation model, a system of one parabolic equation coupled with one variational inequality. This energy estimate argument is slightly different from that of [17].…”