Weak and very weak solutions of the Laplace equation and the Stokes system with prescribed regularity
To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for elliptic boundary value problems.
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| Published in | Examples and counterexamples Vol. 8; p. 100198 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.12.2025
Elsevier |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2666-657X 2666-657X |
| DOI | 10.1016/j.exco.2025.100198 |
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| Abstract | To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for elliptic boundary value problems. |
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| AbstractList | To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for elliptic boundary value problems. |
| ArticleNumber | 100198 |
| Author | Lorenz, Katharina Nicaise, Serge Apel, Thomas |
| Author_xml | – sequence: 1 givenname: Thomas surname: Apel fullname: Apel, Thomas organization: Universität der Bundeswehr München, Institute of Mathematics and Computer-Based Simulation, Werner-Heisenberg-Weg 39, Neubiberg, 85577, Germany – sequence: 2 givenname: Katharina surname: Lorenz fullname: Lorenz, Katharina email: katharina.lorenz@unibw.de organization: Universität der Bundeswehr München, Institute of Mathematics and Computer-Based Simulation, Werner-Heisenberg-Weg 39, Neubiberg, 85577, Germany – sequence: 3 givenname: Serge surname: Nicaise fullname: Nicaise, Serge organization: Université Polytechnique Hauts-de-France, Campus Mont Houy, Valenciennes, 59313, France |
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| Cites_doi | 10.1002/num.22057 10.1090/surv/162 10.1137/0520006 |
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| Keywords | Laplace equation Finite element approximation Stokes system Non-homogeneous boundary conditions |
| Language | English |
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| References | Mazya, Rossmann (b3) 2010 Apel, Nicaise, Pfefferer (b1) 2016; 32 Dauge (b4) 1989; 20 T. Apel, K. Lorenz, J. Pfefferer, Numerical Analysis for the Very Weak Solution of the Stokes Equations in Polygonal and Polyhedral Domains, Technical Report, 2025, in preparation. Kozlov, Maz’ya, Rossmann (b2) 2001 Apel (10.1016/j.exco.2025.100198_b1) 2016; 32 Kozlov (10.1016/j.exco.2025.100198_b2) 2001 Dauge (10.1016/j.exco.2025.100198_b4) 1989; 20 10.1016/j.exco.2025.100198_b5 Mazya (10.1016/j.exco.2025.100198_b3) 2010 |
| References_xml | – year: 2010 ident: b3 article-title: Elliptic equations in polyhedral domains publication-title: Mathematical Surveys and Monographs – year: 2001 ident: b2 article-title: Spectral problems associated with corner singularities of solutions to elliptic equations publication-title: Mathematical Surveys and Monographs – reference: T. Apel, K. Lorenz, J. Pfefferer, Numerical Analysis for the Very Weak Solution of the Stokes Equations in Polygonal and Polyhedral Domains, Technical Report, 2025, in preparation. – volume: 32 start-page: 1433 year: 2016 end-page: 1454 ident: b1 article-title: Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains publication-title: Numer. Methods Partial Differential Equations – volume: 20 start-page: 74 year: 1989 end-page: 97 ident: b4 article-title: Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations publication-title: SIAM J. Math. Anal. – year: 2001 ident: 10.1016/j.exco.2025.100198_b2 article-title: Spectral problems associated with corner singularities of solutions to elliptic equations – volume: 32 start-page: 1433 issue: 5 year: 2016 ident: 10.1016/j.exco.2025.100198_b1 article-title: Discretization of the Poisson equation with non-smooth data and emphasis on non-convex domains publication-title: Numer. Methods Partial Differential Equations doi: 10.1002/num.22057 – year: 2010 ident: 10.1016/j.exco.2025.100198_b3 article-title: Elliptic equations in polyhedral domains doi: 10.1090/surv/162 – ident: 10.1016/j.exco.2025.100198_b5 – volume: 20 start-page: 74 issue: 1 year: 1989 ident: 10.1016/j.exco.2025.100198_b4 article-title: Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations publication-title: SIAM J. Math. Anal. doi: 10.1137/0520006 |
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| Title | Weak and very weak solutions of the Laplace equation and the Stokes system with prescribed regularity |
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