Projection in negative norms and the regularization of rough linear functionals
F. Millar, I. Muga, S. Rojas, K. G. Van der Zee
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.
Numerische Mathematik
https://doi.org/10.1007/s00211-022-01278-z
Publicada - abr. 2022
10.1007/s00211-022-01278-z
Profundice en los temas de investigación de 'Projection in negative norms and the regularization of rough linear functionals'. En conjunto forman una huella única.
Sobolev spaces Ingeniería y ciencia de los materiales 100%
Polynomial approximation Ingeniería y ciencia de los materiales 75%
Linear Functional Matemáticas 56%
Partial differential equations Ingeniería y ciencia de los materiales 56%
Rough Matemáticas 55%
Regularization Matemáticas 48%
Projection Matemáticas 41%
Norm Matemáticas 34%
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
Millar, F. ; Muga, I. ; Rojas, S. et al. / Projection in negative norms and the regularization of rough linear functionals. En: Numerische Mathematik. 2022 ; Vol. 150, N.º 4. pp. 1087-1121.
@article{34075da8ac954e02953b1309b8136475,
title = "Projection in negative norms and the regularization of rough linear functionals",
abstract = "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{\textquoteright}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.",
author = "F. Millar and I. Muga and S. Rojas and {Van der Zee}, {K. G.}",
note = "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{\textquoteright}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{\'o}n 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: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",
doi = "10.1007/s00211-022-01278-z",
journal = "Numerische Mathematik",
Millar, F, Muga, I, Rojas, S & Van der Zee, KG 2022, 'Projection in negative norms and the regularization of rough linear functionals', Numerische Mathematik, vol. 150, n.º 4, pp. 1087-1121. https://doi.org/10.1007/s00211-022-01278-z
Projection in negative norms and the regularization of rough linear functionals. / Millar, F.; Muga, I.; Rojas, S. et al.
En: Numerische Mathematik, Vol. 150, N.º 4, 04.2022, p. 1087-1121.
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.
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.
U2 - 10.1007/s00211-022-01278-z
DO - 10.1007/s00211-022-01278-z
JO - Numerische Mathematik
JF - Numerische Mathematik
Millar F, Muga I, Rojas S, Van der Zee KG. Projection in negative norms and the regularization of rough linear functionals. Numerische Mathematik. 2022 abr.;150(4):1087-1121. doi: 10.1007/s00211-022-01278-z