Session I.1 - Multiresolution and Adaptivity in Numerical PDEs
Tuesday, June 13, 14:30 ~ 15:00
A priori and a posteriori error analysis in ${\boldsymbol H}(\mathrm{curl})$: localization, minimal regularity, and $p$-optimality
Martin Vohralík
Inria Paris, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
We design a stable local commuting projector from the entire infinite-dimensional Sobolev space ${\boldsymbol H}(\mathrm{curl})$ onto its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh. The projector is defined by simple piecewise polynomial projections and is stable in the $L_2$ norm, up to data oscillation. It in particular allows to establish the equivalence of local-best and global-best approximations in ${\boldsymbol H}(\mathrm{curl})$. This in turn yields to a priori error estimates under minimal Sobolev regularity in ${\boldsymbol H}(\mathrm{curl})$, localized elementwise, optimal both in the mesh size $h$ and in the polynomial degree $p$. In the heart of the projector, there is an ${\boldsymbol H}(\mathrm{curl})$-conforming flux reconstruction procedure. This itself leads to guaranteed, fully computable, constant-free, and $p$-robust a posteriori error estimates in ${\boldsymbol H}(\mathrm{curl})$. Details can be found in [1−3].
[1] Chaumont-Frelet, Théophile and Vohralík, Martin. Equivalence of local-best and global-best approximations in ${\boldsymbol H}(\mathrm{curl})$. Calcolo 58 (2021), 53.
[2] Chaumont-Frelet, Théophile and Vohralík, Martin. $p$-robust equilibrated flux reconstruction in ${\boldsymbol H}(\mathrm{curl})$ based on local minimizations. Application to a posteriori analysis of the curl−curl problem. SIAM Journal on Numerical Analysis (2023), accepted for publication.
[3] Chaumont-Frelet, Théophile and Vohralík, Martin. A stable local commuting projector and optimal $hp$ approximation estimates in ${\boldsymbol H}(\mathrm{curl})$. HAL Preprint 03817302, submitted for publication, 2022.
Joint work with Théophile Chaumont-Frelet (Inria Sophia-Antipolis).