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Stability of a weighted L2 projection in a Sobolev space

by M. Averseng

(Report number 2022-33)

Abstract
In this paper, we prove the stability of a weighted \(L^2\) projection operator onto finite-dimensional subspaces of a weighted Sobolev space. This stability property is needed for the analysis of the preconditioners introduced by Alouges and the author in ``Quasi-local and frequency robust preconditioners for the Helmholtz first-kind integral equations on the disk". Namely, we consider the orthogonal projections \(\pi_{N,\omega}: L^2(\mathbb{D},1/\omega(x)dx) \to \mathcal{X}_N\), where \(\mathbb{D} \subset \mathbb{R}^2\) is the unit disk and \(\omega(x) = \sqrt{1 - |x|^2}\). The spaces \(\mathcal{X}_N\) are finite-dimensional subspaces of a weighted Sobolev-type space \(T^1\), and consist of piecewise linear functions on a family of shape-regular and quasi-uniform triangulations of \(\mathbb{D}\). We show that \(\pi_{N,\omega}\) is continuous from \(T^1\) to \(T^1\) and prove an upper bound on the continuity constant that does not depend on \(N\).

Keywords: Boundary element methods, Poincaré inequality, A priori analysis.

@Techreport{A22_1021,
  author = {M. Averseng},
  title = {Stability of a weighted L2 projection in a Sobolev space},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-33},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-33.pdf },
  year = {2022}
}

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