Abstract
We consider the finite element method on locally damaged meshes allowing for some distorted cells which are isolated from one another. In the case of the Poisson equation and piecewise linear Lagrange finite elements, we show that the usual a priori error estimates remain valid on such meshes. We also propose an alternative finite element scheme which is optimally convergent and, moreover, well conditioned, i.e. the conditioning number of the associated finite element matrix is of the same order as that of a standard finite element method on a regular mesh of comparable size.
Highlights
We are interested in the finite element method on meshes containing some isolated degenerate cells
The first goal of the present work is to highlight that we can recover the optimal convergence of the finite element method even if the mesh contains several isolated almost degenerate simplexes
We are able to prove that such a scheme is optimally convergent and well conditioned, i.e. its conditioning is of the same order as that of a standard finite element method on a usual regular mesh of comparable size, provided the number of degenerate cells remains uniformly bounded
Summary
We are interested in the finite element method on meshes containing some isolated degenerate cells. The first goal of the present work is to highlight that we can recover the optimal convergence of the finite element method even if the mesh contains several isolated almost degenerate simplexes. We are able to prove that such a scheme is optimally convergent and well conditioned, i.e. its conditioning is of the same order as that of a standard finite element method on a usual regular mesh of comparable size, provided the number of degenerate cells remains uniformly bounded. The optimal H1-convergence has been proved in [22] for second order elliptic equation and in [21] for linear elasticity equations under the minimum angle condition in 2D: there exists α0 ∈ (0, π) such that for each considered mesh Th and any mesh cell K ∈ Th,.
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