Adaptive finite element methods for parabolic problems IV: nonlinear problems
SIAM Journal on Numerical Analysis
Algorithms for Computing Motion by Mean Curvature
SIAM Journal on Numerical Analysis
Convergence Past Singularities for a Fully Discrete Approximation of Curvature-Drive Interfaces
SIAM Journal on Numerical Analysis
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
SIAM Journal on Numerical Analysis
Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems
SIAM Journal on Numerical Analysis
Error analysis of a mixed finite element method for the Cahn-Hilliard equation
Numerische Mathematik
A posteriori error analysis for time-dependent Ginzburg-Landau type equations
Numerische Mathematik
Numerical approximation of the Cahn-Larché equation
Numerische Mathematik
Journal of Scientific Computing
Numerical Approximation of Partial Differential Equations
Numerical Approximation of Partial Differential Equations
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A priori and a posteriori error estimates are derived for the numerical approximation of scalar and complex valued phase field models. Particular attention is devoted to the dependence of the estimates on a small parameter and to the validity of the estimates in the presence of topological changes in the solution that represents singular points in the evolution. For typical singularities the estimates depend on the inverse of the parameter in a polynomial as opposed to exponential dependence of estimates resulting from a straightforward error analysis. The estimates naturally lead to adaptive mesh refinement and coarsening algorithms. Numerical experiments illustrate the reliability and efficiency of this approach for the evolution of interfaces and vortices that undergo topological changes.