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We consider a bounded Lipschitz polyhedron Ω ⊂ R3 with trivial topology and set Γ:= ∂Ω to be its compact boundary composed of a small number of flat faces

Extension by zero in discrete trace spaces: Inverse estimates

MATHEMATICS OF COMPUTATION, no. 296 (2015): 2589-2615

Cited: 5|Views17
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Abstract

We consider lowest-order H-1/2 (div(Gamma), Gamma)- and H-1/2 (Gamma)-conforming boundary element spaces supported on part of the boundary G of a Lipschitz polyhedron. Assuming families of triangular meshes created by regular refinement, we prove that on these spaces the norms of the extension by zero operators with respect to (localized)...More

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Introduction
  • The authors consider a bounded Lipschitz polyhedron Ω ⊂ R3 with trivial topology and set Γ := ∂Ω to be its compact boundary composed of a small number of flat faces.
  • Inverse estimates, multilevel norm equivalences.
  • On Γh the authors consider the finite element space S1(Γh) ⊂ H1(Γ) of Γh-piecewise linear, globally continuous functions on Γ; see [42, Sect.
  • Inverse inequalities with logarithmic blow-up in terms of the mesh width are typical for such borderline cases and piecewise polynomial finite element spaces of fixed degree [20, Lemma 1.142].
Highlights
  • We consider a bounded Lipschitz polyhedron Ω ⊂ R3 with trivial topology and set Γ := ∂Ω to be its compact boundary composed of a small number of flat faces
  • Inverse inequalities with logarithmic blow-up in terms of the mesh width are typical for such borderline cases and piecewise polynomial finite element spaces of fixed degree [20, Lemma 1.142]
  • We summarize the main results that extend (1.3) to the boundary element spaces RT 0(Γ+l ) and Q0(Γ+) and the corresponding trace norms
  • (4.2) V − ΠRl t V L2(Ω) hl V H (Ω), ∀ V ∈ {V ∈ (H (Ω))3 : div V = 0}. This estimate is instrumental in constructing various discrete extension operators for both boundary element spaces Q0(Γl) and RT 0(Γl)
  • We find c0 ∈ RT 0(Γ+0 ) such that wL · n∂Γ+ dl =
Results
  • The authors summarize the main results that extend (1.3) to the boundary element spaces RT 0(Γ+l ) and Q0(Γ+) and the corresponding trace norms.
  • Similar norm equivalences hold for boundary element spaces and involve the multilevel norms
  • 1 2 (Γ)-conforming boundary element spaces the authors have the following equivalent norms: (3.5) (3.6)
  • The localized trace norm of boundary element functions allows an equivalent characterization through multilevel splittings: (3.9)
  • The stable discrete extension from Lemma 3.1 played a key role in the proof of the multilevel decomposition
  • This estimate is instrumental in constructing various discrete extension operators for both boundary element spaces Q0(Γl) and RT 0(Γl).
  • Similar norm equivalences hold for the edge boundary element spaces RT 0(ΓL) relying on the multilevel norms
  • The authors use the extension operators provided by Lemmas 4.1, 4.5, and 3.2, 3.1 together with Theorem 5.1 in the very same fashion as in the proof of Theorem 3.4; see [29, Sect.
  • Norm equivalences as expressed in Theorem 5.2 hold for spaces of piecewise constant functions on ΓL.
  • In order to prove the “ ” estimate of (5.18), by Lemma 4.4, the trace inequality and Theorem 5.4 the authors have to establish
Conclusion
  • To show the estimate “ ” of (5.18) the authors again apply the trivial local extension operators of Lemma 4.2 to the terms of a multilevel splitting of φL.
  • In analogy to the second part of the proof of Theorem 5.2, the authors use the continuity of the trace operator and the norm equivalence from Theorem 5.4.
  • The authors follow exactly the reasoning of the proof of Theorem 5.3, this time using the discrete extension operator E2l provided by Lemma 4.4.
  • The authors just follow Part ˜ of the proof of Theorem 2.1, as in (6.20) and (6.21) modify the ql in order to enforce zero normal components on ∂Γ+, pay with another factor of L2 in the estimate, as in (6.22), and the authors are done.
Funding
  • The work of the second author was funded by FONDECYT 11121166 and CONICYT project Anillo ACT1118 (ANANUM)
  • The work of the third author was partly supported by Thales SA under contract “Preconditioned Boundary Element Methods for Electromagnetic Scattering at Dielectric Objects” and NSFC 11101414, 11101386, 11471329
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