Interpolation operator on negative Sobolev spaces

Diening, Lars; Storn, Johannes; Tscherpel, Tabea

doi:10.1090/mcom/3824

Interpolation operator on negative Sobolev spaces
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by Lars Diening, Johannes Storn and Tabea Tscherpel;

Math. Comp. 92 (2023), 1511-1541

DOI: https://doi.org/10.1090/mcom/3824

Published electronically: February 7, 2023

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Abstract:

We introduce a Scott–Zhang type projection operator mapping to Lagrange elements for arbitrary polynomial order. In addition to the usual properties, this operator is compatible with duals of first order Sobolev spaces. More specifically, it is stable in the corresponding negative norms and allows for optimal rates of convergence. We discuss alternative operators with similar properties. As applications of the operator we prove interpolation error estimates for parabolic problems and smoothen rough right-hand sides in a least squares finite element method.

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