On interpolation spaces of piecewise polynomials on mixed meshes

TL;DR

This paper establishes a norm equivalence for fractional Sobolev spaces on mixed meshes using piecewise polynomials, which aids in deriving inverse inequalities independent of mesh size and polynomial degree.

Contribution

It proves that the interpolated discrete norm is equivalent to the continuous Sobolev norm on mixed meshes, with constants independent of mesh parameters.

Findings

Norm equivalence between interpolated discrete and continuous Sobolev norms

Constants are independent of mesh size and polynomial degree

Application to inverse inequalities between H^1 and H^θ

Abstract

We consider fractional Sobolev spaces $H^\theta$, $\theta\in (0,1)$, on 2D domains and $H^1$-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 $L^2$-norm on the one hand and the $H^1$-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 $H^1$ and $H^\theta$.