Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 49, Issue 1, Pages 110-134Publisher
SIAM PUBLICATIONS
DOI: 10.1137/090751530
Keywords
Allen-Cahn equation; Ginzburg-Landau equation; mean curvature flow; finite element method; error analysis; adaptive methods
Categories
Funding
- DFG [SFB 611]
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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.
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