ERROR ESTIMATES FOR A FINITE ELEMENT DISCRETIZATION OF A PHASE FIELD MODEL FOR MIXTURES

• Published in: SIAM journal on numerical analysis Vol. 47; no. 6; pp. 4429 - 4445
• Main Authors: ECK, CH, JADAMBA, B., KNABNER, P.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
• Subjects: Algebra; Applied mathematics; Approximation; Constitutive relationships; Convexity; Discretization; Energy; Entropy; Errors; Estimates; Exact sciences and technology; Finite differences and functional equations; Finite element method; Internal energy; Mathematical analysis; Mathematical constants; Mathematical models; Mathematical vectors; Mathematics; Modeling; Number theory; Numerical analysis; Numerical analysis. Scientific computation; Operator theory; Parametric models; Phase transformations; Phase transitions; Sciences and techniques of general use; Finite field; 80A22; Operator equation; Error estimation; phase field model; Entropy; Phase transitions; Discretization method; Exact solution; thermodynamically consistent model; Finite element method; Numerical analysis; Non linear model; Solution regularity; A priori estimation; Approximation error; 65M60; Convexity; a priori error estimate
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/050637984

We derive error estimates for finite element discretizations of phase field models that describe phase transitions in nonisothermal mixtures. Special attention is paid to the applicability of the result for a large class of models with nonlinear constitutive relations and to an approach that avoids an exponential dependence of the constants in the error estimate on the approximation parameter that models the thickness of the diffuse phase transition region. The main assumptions on the model are a convexity condition for a function that can be interpreted as the negative local part of the entropy of the system, a suitable regularity of the exact solutions, and a spectrum estimate for the operator of the Allen—Cahn equation. The spectrum estimate is crucial to avoid the exponential dependence of error constants on the approximation parameters in the model. This is done by a technique introduced in [X. Feng and A. Prohl, Math. Comp., 73 (2004), pp. 541—567] for phase transitions of pure materials with linear constitutive relations.