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arXiv · 2105.07925

Quasi-monotonicity and Robust Localization with Continuous Piecewise Polynomials

Francesca Tantardini·Rüdiger Verfürth
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Abstract

We consider the energy norm arising from elliptic problems with discontinuous piecewise constant diffusion. We prove that under the quasi-monotonicity property on the diffusion coefficient, the best approximation error with continuous piecewise polynomials is equivalent to the $\ell_2$-sum of best errors on elements, in the spirit of A. Veeser for the $H^1$-seminorm. If the quasi-monotonicity is violated, counterexamples show that a robust localization does not hold in general, neither on elements, nor on pairs of adjacent elements, nor on stars of elements sharing a common vertex.

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Francesca Tantardini, Rüdiger Verfürth. 2021-05-17. Quasi-monotonicity and Robust Localization with Continuous Piecewise Polynomials. https://arxiv.org/abs/2105.07925

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