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Temporal derivatives in the finite-element method on continuously deforming grids. (English) Zbl 0747.65083
Using deforming grids when solving time-dependent partial differential equations leads sometimes to the necessity to differentiate the trial solution with respect to the movable grid points. In a straightforward investigation the authors give a result which enables these “nonstandard derivatives” to be expressed in terms of the spatial derivatives. The main idea of the article lies in the use of barycentric coordinates and the application of implicit differentiation.
The authors give examples ( Lagrange finite elements) and counterexamples ( Hermite finite elements) and obtain useful results, indeed, but the term “nonstandard derivative” is more meaningful in the field of differential geometry and nonstandard analysis and should better be omitted to avoid misunderstanding.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
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