On interpolation spaces of piecewise polynomials on mixed meshes

Abstract

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.