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A Stable and Jump-Aware Projection onto a Discrete Multi-Trace Space

by M. Averseng

(Report number 2022-48)

Abstract
This work is concerned with boundary element methods on singular geometries, specifically, those falling in the framework of “multi-screens” by Claeys and Hiptmair. We construct a stable quasi-interpolant which preserves piecewise linear jumps on the multi-trace space. This operator is the boundary element analog of the Scott-Zhang quasi-interpolant used in the analysis of finite-element methods. More precisely, let \(\Gamma\) be a multi-screen resolved by a triangulation \((\mathcal{M}_{\Gamma,h})\), and let \(\mathbb{V}_h(\Gamma)\) be the space of continuous piecewise-linear multi-traces on \(\Gamma\). We construct a linear operator \(\Pi_h:\mathbb{H}^{1/2}(\Gamma)\to\mathbb{V}_h(\Gamma)\) with the following properties: (i) \(\left\|\Pi_h u\right\|_{\mathbf{H}^1/2(\Gamma)} \leq C_h \left\|u\right\|_{\mathbf{H}^{1/2}(\Gamma)}\) for all \(u\in\mathbf{H}^{1/2}(\Gamma)\), (ii) \(\Pi_h u_h = u_h\) for \(u_h\in \mathbf{V}_h(\Gamma)\) and, (iii) \([\Pi_h u]=0\) for every single trace \(u\in H^{1/2}(\Gamma)\). The stability constant \(C_h\) only depends on the aspect ratio of the elements of \(\mathcal{M}_{\Omega,h}\), where \(\mathcal{M}_{\Omega,h}\) is a tetrahedral mesh of \(\Omega\) extending \(\mathcal{M}_{\Gamma,h}\). We deduce uniform bounds for the stability of the discrete jump lifting, and the equivalence of the \(\widetilde{H}^{1/2}\) norm with a discrete quotient norm.

Keywords: Boundary Element Methods, Singularities, Interpolation

@Techreport{A22_1036,
  author = {M. Averseng},
  title = {A Stable and Jump-Aware Projection onto a Discrete Multi-Trace Space},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-48},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-48.pdf },
  year = {2022}
}

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