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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:We consider the Helmholtz equation with piecewise analytic co
efficients at large wavenumber k >\; 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.

\nJoint work with: Maximilian Bernkopf (TU Wien)\
, Théophile Chaumont-Frelet (Inria).

\n**References**<
br /> [EM11] \; \; \;S. Esterhazy and J.M. Melenk\, On stabil
ity of discretizations of the Helmholtz equation\, in: Numerical \;Ana
lysis of Multiscale Problems\, Graham et al.\, eds\, Springer 2012

[
MS10] \; \;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

[MS
11] \; \;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

[MSP12] J.M. Melenk\
, S. Sauter\, A. Parsania\, Generalized DG-methods for highly indefinite H
elmholtz problems\, J. Sci. Comp. 57:536–581\, 2013

\nFor further
information about the seminar\, please visit this webpage.

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