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A class of mixed finite element methods based on the Helmholtz decomposition
Mira Schedensack (HU Berlin and Universität Bonn)
Sala conferenze IMATI-CNR - Lunedì 11 Maggio 2015 h.15:00
Abstract. The construction of non-conforming finite element methods (FEMs) is motivated by robust discretizations and mass conservation properties in the simulation of solid and fluid mechanics and by low-order ansatz spaces for higher-order problems as the biharmonic problem for the Kirchhoff plate in structural mechanics. A natural generalization to higher polynomial degrees which preserves the inherent properties of the discretizations is not known so far.
This talk generalizes the non-conforming FEMs of Morley and Crouzeix and Raviart by novel mixed formulations for mth-Laplace equations of the form (−1)m∆mu=f for arbitrary m = 1, 2, 3, . . . These formulations are based on a new Helmholtz decomposition which decomposes an unstructured tensor field into a higher-order derivative and a curl. The new formulations allow for ansatz spaces of arbitrary polynomial degree and its discretizations coincide with the mentioned non-conforming FEMs for the lowest polynomial degree. The discretizations presented in this talk allow not only for a uniform implementation for arbitrary m, but they also allow for lowest-order ansatz spaces, e.g., piecewise affine polynomials for arbitrary m. Besides the a priori and a posteriori analysis, the talk presents optimal convergence rates for adaptive algorithms for the new discretizations.
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