Projection in negative norms and the regularization of rough linear functionals

Millar, F.; Muga, I.; Rojas, S.; Van der Zee, K. G.

Abstract

In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as W01,q(Ω), where 1 < q< ∞ and Ωis a Lipschitz domain, we propose a projection method in negative Sobolev spacesW-1,p(Ω) , pbeing the conjugate exponent satisfying p- 1+ q- 1= 1. Our method is particularly useful when one is dealing with a rough (irregular) functional that is a member of W-1,p(Ω) , though not ofL1(Ω) , but one strives for a regular approximation inL1(Ω). We focus on projections onto discrete finite element spacesGn, and consider both discontinuous as well as continuous piecewise-polynomial approximations. While the proposed method aims to compute the best approximation as measured in the negative (dual) norm, for practical reasons, we will employ a computable, discrete dual norm that supremizes over a discrete subspaceVm. We show that this idea leads to a fully discrete method given by a mixed problem onVm× Gn. We propose a discontinuous as well as a continuous lowest-order pair, prove that they are compatible, and therefore obtain quasi-optimally convergent methods. We present numerical experiments that compute finite element approximations to Dirac delta’s and line sources. We also present adaptively generate meshes, obtained from an error representation that comes with the method. Finally, we show how the presented projection method can be used to efficiently compute numerical approximations to partial differential equations with rough data.

Citation

Millar, F., Muga, I., Rojas, S., & Van der Zee, K. G. (2022). Projection in negative norms and the regularization of rough linear functionals. Numerische Mathematik, 150(4), 1087-1121. https://doi.org/10.1007/s00211-022-01278-z

• Journal Article Type: Article
• Acceptance Date: Feb 21, 2022
• Online Publication Date: Mar 21, 2022
• Publication Date: Apr 1, 2022
• Deposit Date: Feb 23, 2022
• Publicly Available Date: Mar 22, 2023
• Journal: Numerische Mathematik
• Print ISSN: 0029-599X
• Electronic ISSN: 0945-3245
• Peer Reviewed: Peer Reviewed
• Volume: 150
• Issue: 4
• Pages: 1087-1121
• DOI: https://doi.org/10.1007/s00211-022-01278-z
• Keywords: Applied Mathematics; Computational Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/7504930
• Publisher URL: https://link.springer.com/article/10.1007/s00211-022-01278-z