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Efficient computation of dendritic growth with \(r\)-adaptive finite element methods. (English) Zbl 1148.65052

Summary: This paper deals with the application of a moving grid method to the solution of a phase-field model for dendritic growth in two- and three-dimensions. A mesh is found as the solution of an optimization problem that automatically includes the boundary conditions and is solved using a multi-grid approach. The governing equations are discretized in space by linear finite elements and a split time-level scheme is used to numerically integrate in time. One novel aspect of the method is the choice of a regularized monitor function. The moving grid method enables us to obtain accurate numerical solutions with much less degree of freedoms. It is demonstrated numerically that the tip velocity obtained by our method is in good agreement with previously published results.

MSC:

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
80A22 Stefan problems, phase changes, etc.
80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer

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