UCL Discovery
UCL home » Library Services » Electronic resources » UCL Discovery

Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div)

Ern, A; Gudi, T; Smears, I; Vohralík, M; (2022) Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div). IMA Journal of Numerical Analysis , 42 (2) pp. 1023-1049. 10.1093/imanum/draa103. Green open access

[thumbnail of loc_glob_div_HAL.pdf]
Preview
Text
loc_glob_div_HAL.pdf - Accepted Version

Download (776kB) | Preview

Abstract

Given an arbitrary function in H(div)⁠, we show that the error attained by the global-best approximation by H(div)-conforming piecewise polynomial Raviart–Thomas–Nédélec elements under additional constraints on the divergence and normal flux on the boundary is, up to a generic constant, equivalent to the sum of independent local-best approximation errors over individual mesh elements, without constraints on the divergence or normal fluxes. The generic constant only depends on the shape-regularity of the underlying simplicial mesh, the space dimension, and the polynomial degree of the approximations. The analysis also gives rise to a stable, local, commuting projector in H(div)⁠, delivering an approximation error that is equivalent to the local-best approximation. We next present a variant of the equivalence result, where robustness of the constant with respect to the polynomial degree is attained for unbalanced approximations. These two results together further enable us to derive rates of convergence of global-best approximations that are fully optimal in both the mesh size h and the polynomial degree p⁠, for vector fields that only feature elementwise the minimal necessary Sobolev regularity. We finally show how to apply our findings to derive optimal a priori hp-error estimates for mixed and least-squares finite element methods applied to a model diffusion problem.

Type: Article
Title: Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div)
Open access status: An open access version is available from UCL Discovery
DOI: 10.1093/imanum/draa103
Publisher version: https://doi.org/10.1093/imanum/draa103
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher's terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10119681
Downloads since deposit
1Download
Download activity - last month
Download activity - last 12 months
Downloads by country - last 12 months

Archive Staff Only

View Item View Item