On interpolation spaces of piecewise polynomials on mixed meshes

06/29/2023

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by   Michael Karkulik, et al.

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We consider fractional Sobolev spaces H^θ, θ∈ (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^θ.