TY - JOUR
T1 - Projection in negative norms and the regularization of rough linear functionals
AU - Millar, F.
AU - Muga, I.
AU - Rojas, S.
AU - Van der Zee, K. G.
N1 - Funding Information:
The authors want to thank Diego Paredes for helping with preliminary numerical experiments, Michael Karkulik for his pertinent advice, and also want to thank the unknown reviewers for their insightful suggestions. The work by IM and FM was done in the framework of Chilean FONDECYT research project #1160774. The work of SR was supported by the Chilean grant ANID Fondecyt 3210009. IM and SR have also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 777778 (MATHROCKS). The research by KvdZ was supported by the Engineering and Physical Sciences Research Council (EPSRC) under grant EP/T005157/1. IM acknowledges support from the project DI Investigación Innovadora Interdisciplinaria PUCV No 039.409/2021, and project DI PUCV No 039.375/2021.
Funding Information:
The authors want to thank Diego Paredes for helping with preliminary numerical experiments, Michael Karkulik for his pertinent advice, and also want to thank the unknown reviewers for their insightful suggestions. The work by IM and FM was done in the framework of Chilean FONDECYT research project?#1160774. The work of SR was supported by the Chilean grant ANID Fondecyt 3210009. IM and SR have also received funding from the European Union?s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 777778 (MATHROCKS). The research by KvdZ was supported by the Engineering and Physical Sciences Research Council (EPSRC) under grant EP/T005157/1. IM acknowledges support from the project DI Investigaci?n Innovadora Interdisciplinaria PUCV No 039.409/2021, and project DI PUCV No 039.375/2021.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2022/4
Y1 - 2022/4
N2 - In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as W01,q(Ω), where 1 < q< ∞ and Ω is a Lipschitz domain, we propose a projection method in negative Sobolev spaces W-1,p(Ω) , p being 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 of L1(Ω) , but one strives for a regular approximation in L1(Ω). We focus on projections onto discrete finite element spaces Gn, 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 subspace Vm. We show that this idea leads to a fully discrete method given by a mixed problem on Vm× 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.
AB - In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as W01,q(Ω), where 1 < q< ∞ and Ω is a Lipschitz domain, we propose a projection method in negative Sobolev spaces W-1,p(Ω) , p being 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 of L1(Ω) , but one strives for a regular approximation in L1(Ω). We focus on projections onto discrete finite element spaces Gn, 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 subspace Vm. We show that this idea leads to a fully discrete method given by a mixed problem on Vm× 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.
UR - http://www.scopus.com/inward/record.url?scp=85126803897&partnerID=8YFLogxK
U2 - 10.1007/s00211-022-01278-z
DO - 10.1007/s00211-022-01278-z
M3 - Article
AN - SCOPUS:85126803897
VL - 150
SP - 1087
EP - 1121
JO - Numerische Mathematik
JF - Numerische Mathematik
SN - 0029-599X
IS - 4
ER -