A stable and jump-aware projection onto a discrete multi-trace space
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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 Γ be a multi-screen resolved by a triangulation (ℳ_Γ,h), and let 𝕍_h(Γ) be the space of continuous piecewise-linear multi-traces on Γ. We construct a linear operator Π_h: ℍ^1/2(Γ) →𝕍_h(Γ) with the following properties: (i) Π_h u_ℍ^1/2≤ C_h u_ℍ^1/2(Γ) for all u ∈ℍ^1/2(Γ), (ii) Π_h u_h = u_h for u_h ∈𝕍_h(Γ) and, (iii) [Π_h u] = 0 for every single trace u ∈ H^1/2([Γ]). The stability constant C_h only depends on the aspect ratio of the elements of ℳ_Ω,h, where ℳ_Ω,h is a tetrahedralization of ℳ_Γ,h. We deduce uniform bounds for the stability of the discrete harmonic lifting, and the equivalence of the H^1/2 norm with a discrete quotient norm.
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