| Issue |
ESAIM: M2AN
Volume 59, Number 1, January-February 2025
|
|
|---|---|---|
| Page(s) | 231 - 264 | |
| DOI | https://doi.org/10.1051/m2an/2024069 | |
| Published online | 08 January 2025 | |
On interpolation spaces of piecewise polynomials on mixed meshes
1
Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile
2
Institut für Analysis und Scientific Computing, Technische Universit¨at Wien, Wiedner Hauptstrasse 8-10, Wien, Austria
* Corresponding author: michael.karkulik@usm.cl
Received:
23
June
2023
Accepted:
4
September
2024
We consider fractional Sobolev spaces Hθ, θ ∈ (0, 1), on 2D domains and H1-conforming discretizations by globally continuous piecewise polynomials on a mesh consisting of shape-regular triangles and quadrilaterals. We prove that the norm obtained from interpolating between the discrete space equipped with the L2-norm on the one hand and the H1-norm on the other hand is equivalent to the corresponding continuous interpolation Sobolev norm, and the norm-equivalence constants are independent of meshsize and polynomial degree. This characterization of the Sobolev norm is then used to show an inverse inequality between H1 and Hθ.
Mathematics Subject Classification: 46B70 / 46E35 / 65N30
Key words: Inverse estimates / interpolation spaces / stable localization / domain decomposition
© The authors. Published by EDP Sciences, SMAI 2025
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