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UID:news1167@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20210627T202614
DTSTART;TZID=Europe/Zurich:20210423T110000
SUMMARY:Seminar in Numerical Analysis: Markus Melenk (TU Wien)
DESCRIPTION:We consider the Helmholtz equation with piecewise analytic coef
 ficients at large wavenumber k > 0. The interface where the coefficients j
 ump is assumed to be analytic. We develop a k-explicit regularity theory f
 or the solution that takes the form of a decomposition into two components
 : the first component is a piecewise analytic\, but highly oscillatory fun
 ction and the second one has finite regularity but features wavenumber-ind
 ependent bounds. This decomposition generalizes earlier decompositions of 
 [MS10\, MS11\, EM11\, MSP12]\, which considered the Helmholtz equation wit
 h constant coefficients\, to the case of (piecewise) analytic coefficients
 . This regularity theory allows to show for high order Galerkin discretiza
 tions (hp-FEM) of the Helmholtz equation that quasi-optimality is reached 
 if (a) the approximation order p is selected as p = O(log k) and (b) the m
 esh size h is such that kh/p is sufficiently small. This extends the resul
 ts of [MS10\, MS11\, EM11\, MSP12] about the onset of quasi-optimality of 
 hp-FEM for the Helmholtz equation to the case of the heterogeneous Helmhol
 tz equation.\\r\\nJoint work with: Maximilian Bernkopf (TU Wien)\, Théoph
 ile Chaumont-Frelet (Inria).\\r\\nReferences [EM11]    S. Esterhazy and
  J.M. Melenk\, On stability of discretizations of the Helmholtz equation\,
  in: Numerical Analysis of Multiscale Problems\, Graham et al.\, eds\, Sp
 ringer 2012 [MS10]    J.M. Melenk and S. Sauter\, Convergence Analysis f
 or Finite Element Discretizations of the Helmholtz equation with Dirichlet
 -to-Neumann boundary conditions\, Math. Comp. 79:1871–1914\, 2010 [MS11]
     J.M. Melenk and S. Sauter\, Wavenumber explicit convergence analysis
  for finite element discretizations of the Helmholtz equation\, SIAM J. Nu
 mer. Anal.\, 49:1210–1243\, 2011 [MSP12] J.M. Melenk\, S. Sauter\, A. Pa
 rsania\, Generalized DG-methods for highly indefinite Helmholtz problems\,
  J. Sci. Comp. 57:536–581\, 2013\\r\\nFor further information about the 
 seminar\, please visit this webpage [t3://page?uid=1115].
X-ALT-DESC:<p>We consider the Helmholtz equation with piecewise analytic co
 efficients at large wavenumber k &gt\; 0. The interface where the coeffici
 ents jump is assumed to be analytic. We develop a k-explicit regularity th
 eory for the solution that takes the form of a decomposition into two comp
 onents: the first component is a piecewise analytic\, but highly oscillato
 ry function and the second one has finite regularity but features wavenumb
 er-independent bounds. This decomposition generalizes earlier decompositio
 ns of [MS10\, MS11\, EM11\, MSP12]\, which considered the Helmholtz equati
 on with constant coefficients\, to the case of (piecewise) analytic coeffi
 cients. This regularity theory allows to show for high order Galerkin disc
 retizations (hp-FEM) of the Helmholtz equation that quasi-optimality is re
 ached if (a) the approximation order p is selected as p = O(log k) and (b)
  the mesh size h is such that kh/p is sufficiently small. This extends the
  results of [MS10\, MS11\, EM11\, MSP12] about the onset of quasi-optimali
 ty of hp-FEM for the Helmholtz equation to the case of the heterogeneous H
 elmholtz equation.</p>\n<p>Joint work with: Maximilian Bernkopf (TU Wien)\
 , Théophile Chaumont-Frelet (Inria).</p>\n<p><strong>References</strong><
 br /> [EM11]&nbsp\; &nbsp\;&nbsp\;S. Esterhazy and J.M. Melenk\, On stabil
 ity of discretizations of the Helmholtz equation\, in: Numerical&nbsp\;Ana
 lysis of Multiscale Problems\, Graham et al.\, eds\, Springer 2012<br /> [
 MS10] &nbsp\; &nbsp\;J.M. Melenk and S. Sauter\, Convergence Analysis for 
 Finite Element Discretizations of the Helmholtz equation with Dirichlet-to
 -Neumann boundary conditions\, Math. Comp. 79:1871–1914\, 2010<br /> [MS
 11] &nbsp\; &nbsp\;J.M. Melenk and S. Sauter\, Wavenumber explicit converg
 ence analysis for finite element discretizations of the Helmholtz equation
 \, SIAM J. Numer. Anal.\, 49:1210–1243\, 2011<br /> [MSP12] J.M. Melenk\
 , S. Sauter\, A. Parsania\, Generalized DG-methods for highly indefinite H
 elmholtz problems\, J. Sci. Comp. 57:536–581\, 2013</p>\n<p>For further 
 information about the seminar\, please visit this <a href="t3://page?uid=1
 115" title="Opens internal link in current window">webpage</a>.</p>
DTEND;TZID=Europe/Zurich:20210423T120000
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