Page 457 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 457
NALYSIS, CONTROL AND INVERSE PROBLEMS FOR PARTIAL DIFFERENTIAL
EQUATIONS (MS-22)
Boundedness in Total Variation regularization
Gwenael Mercier, gwenael.mercier@univie.ac.at
University of Vienna, Austria
In this talk, we investigate boundedness and convergence of total variation regularized linear
inverse problems. We present a simple proof of boundedness of the minimizer for fixed regular-
ization parameter, and derive the existence of uniform bounds for small enough noise under a
source condition and adequate a priori parameter choices. We present a few (counter)examples.
This is a joint work wirth K. Bredies (Graz) and José A. Iglesias (RICAM, Linz).
Existence and regularity of weak solutions for a fluid interacting with a
non-linear shell in 3D
Boris Muha, borism@math.hr
Faculty of Science, University of Zagreb, Croatia
Coauthor: Sebastian Schwarzacher
We study the unsteady incompressible Navier-Stokes equations in three dimensions interacting
with a non-linear flexible shell of Koiter type. We study weak solutions to the corresponding
fluid-structure interaction (FSI) problem. The known existence theory for weak solutions is
extended to non-linear Koiter shell models. We introduce a-priori estimates that reveal higher
regularity of the shell displacement beyond energy estimates. These are essential for non-linear
Koiter shell models, since such shell models are non-convex (w.r.t. terms of highest order).
The estimates are obtained by introducing new analytical tools that allow to exploit dissipative
effects of the fluid for the (non-dissipative) solid. The regularity result depends on the geo-
metric constitution alone and is independent of the approximation procedure; hence it holds
for arbitrary weak solutions. The developed tools are further used to introduce a generalized
Aubin-Lions type compactness result suitable for fluid-structure interactions.
Analysis of finite-element based discretizations in nonlinear acoustics
Vanja Nikolic´, vanja.nikolic@ru.nl
Radboud University, Netherlands
Nonlinear effects can be easily observed in sound waves with sufficiently large amplitudes.
The nonlinearity will be apparent even sooner in high-frequency waves because these effects
accumulate over the distance measured in wavelengths. This makes high-intensity ultrasonic
waves inherently nonlinear. Their many applications range from non-invasive surgery to indus-
trial welding and motivate the mathematical investigation into nonlinear acoustics.
In this talk, we will discuss the a priori analysis of finite-element-based discretizations
of nonlinear acoustic equations. In particular, we will focus on the conforming and (hybrid)
discontinuous Galerkin discretizations in space for acoustic equations with nonlinearities of
quadratic type, such as the Westervelt and Kuznetsov equations. These are quasilinear strongly
damped wave equations that serve as classical models of sound propagation through thermovis-
cous fluids and gases. The general approach in the a priori error analysis combines the stability
and convergence analysis of their linearizations with the Banach fixed-point theorem. Numeri-
455
EQUATIONS (MS-22)
Boundedness in Total Variation regularization
Gwenael Mercier, gwenael.mercier@univie.ac.at
University of Vienna, Austria
In this talk, we investigate boundedness and convergence of total variation regularized linear
inverse problems. We present a simple proof of boundedness of the minimizer for fixed regular-
ization parameter, and derive the existence of uniform bounds for small enough noise under a
source condition and adequate a priori parameter choices. We present a few (counter)examples.
This is a joint work wirth K. Bredies (Graz) and José A. Iglesias (RICAM, Linz).
Existence and regularity of weak solutions for a fluid interacting with a
non-linear shell in 3D
Boris Muha, borism@math.hr
Faculty of Science, University of Zagreb, Croatia
Coauthor: Sebastian Schwarzacher
We study the unsteady incompressible Navier-Stokes equations in three dimensions interacting
with a non-linear flexible shell of Koiter type. We study weak solutions to the corresponding
fluid-structure interaction (FSI) problem. The known existence theory for weak solutions is
extended to non-linear Koiter shell models. We introduce a-priori estimates that reveal higher
regularity of the shell displacement beyond energy estimates. These are essential for non-linear
Koiter shell models, since such shell models are non-convex (w.r.t. terms of highest order).
The estimates are obtained by introducing new analytical tools that allow to exploit dissipative
effects of the fluid for the (non-dissipative) solid. The regularity result depends on the geo-
metric constitution alone and is independent of the approximation procedure; hence it holds
for arbitrary weak solutions. The developed tools are further used to introduce a generalized
Aubin-Lions type compactness result suitable for fluid-structure interactions.
Analysis of finite-element based discretizations in nonlinear acoustics
Vanja Nikolic´, vanja.nikolic@ru.nl
Radboud University, Netherlands
Nonlinear effects can be easily observed in sound waves with sufficiently large amplitudes.
The nonlinearity will be apparent even sooner in high-frequency waves because these effects
accumulate over the distance measured in wavelengths. This makes high-intensity ultrasonic
waves inherently nonlinear. Their many applications range from non-invasive surgery to indus-
trial welding and motivate the mathematical investigation into nonlinear acoustics.
In this talk, we will discuss the a priori analysis of finite-element-based discretizations
of nonlinear acoustic equations. In particular, we will focus on the conforming and (hybrid)
discontinuous Galerkin discretizations in space for acoustic equations with nonlinearities of
quadratic type, such as the Westervelt and Kuznetsov equations. These are quasilinear strongly
damped wave equations that serve as classical models of sound propagation through thermovis-
cous fluids and gases. The general approach in the a priori error analysis combines the stability
and convergence analysis of their linearizations with the Banach fixed-point theorem. Numeri-
455
