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Extension by zero in discrete trace spaces: Inverse estimates

Authors: Ralf Hiptmair, Carlos Jerez-Hanckes and Shipeng Mao
Journal: Math. Comp. 84 (2015), 2589-2615
MSC (2010): Primary 65N12, 65N15, 65N30
Published electronically: April 21, 2015
MathSciNet review: 3378840
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Abstract: We consider lowest-order $ {\boldsymbol H}^{-\frac {1}{2}}(\operatorname {div}_\Gamma , \Gamma )$- and $ H^{-\frac {1}{2}}(\Gamma )$-conforming boundary element spaces supported on part of the boundary $ \Gamma $ 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) trace norms increase poly-logarithmically with the mesh width. Our approach harnesses multilevel norm equivalences for boundary element spaces, inherited from stable multilevel splittings of finite element spaces.

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Additional Information

Ralf Hiptmair
Affiliation: SAM, ETH Zürich, CH-8092 Zürich, Switzerland

Carlos Jerez-Hanckes
Affiliation: School of Engineering, Pontificia Universidad Católica de Chile, Santiago, Chile

Shipeng Mao
Affiliation: LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

Keywords: Boundary finite element spaces, inverse estimates, multilevel norm equivalences
Received by editor(s): October 22, 2012
Received by editor(s) in revised form: March 18, 2014
Published electronically: April 21, 2015
Additional Notes: 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
Article copyright: © Copyright 2015 American Mathematical Society