Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 10th Dec 2022, 11:05:36am CET

 
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Session Overview
Session
MS 33a: fast solvers for wave problems
Time:
Monday, 12/July/2021:
4:00pm - 6:00pm

Session Chair: Oscar Bruno
Session Chair: Olaf Steinbach
Virtual location: Zoom 5


Session Abstract

Problems in wave propagation have remained some of the most relevant and chal-

lenging in science and engineering. The importance of computational methods in

these areas continues to accelerate, as they span vast areas of application, from

photonics to acoustics, communications, optics, and seismology, among many

others. Significant progress in the field of computational wave scattering has

resulted in recent years from consideration of fast, high-order solvers based on

discretizations of various types, including finite elements, finite differences and

spectral methods, and relying on both differential and integral formulations. The

resulting methods have provided important new mathematical tools applicable to

a wide range of application areas. In many cases, the resulting methods have pro-

vided solution to previously intractable problems. We believe that the proposed

ICOSAHOM minisymposium, which brings together some of the most important

researchers in the area, will help disseminate and advance the field, and it will ul-

timately give rise to significant progress in this important area of computational

science.


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Presentations
4:00pm - 4:30pm

Comparison of optimized high-order domain decomposition methods with cross-point treatment for Helmholtz problems

Anthony Royer1, Axel Modave2, Eric Béchet3, Xavier Antoine4, Christophe Geuzaine1

1Institut Montefiore, Université de Liège, Belgium; 2POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, France; 3Département Aérospatial et Mécanique, Université de Liège, Belgium; 4IECL, Université de Lorraine, Inria, EPI SPHINX, France

The parallel finite-element solution of large-scale time-harmonic problems is addressed with a non-overlapping sub-structuring domain decomposition method (DDM) using high-order elements. It is well known that the efficiency of this method strongly depends on the transmission condition enforced on the interfaces between the subdomains.

Local conditions such as high-order absorbing boundary conditions (HABCs) or perfectly matched layers (PML) have proved to be well suited, as a good compromise between basic impedance conditions and non-local conditions. Recently, a numerical treatment of interior cross-points (where more than two subdomains meet) and boundary cross-points (points that belong to both the exterior boundary and at least two subdomains) has been proposed for both methods [1, 3].

In this work we compare the performance and efficiency of these two methods for high-frequency scattering problems, with both methods implemented in the same high-order finite element framework [4]. In particular we analyse the influence of the parameters of both methods on the computational times, as well as the properties of the resulting finite-element matrices.

References

[1] Modave A, Royer A, Antoine X and Geuzaine C. An optimized Schwarz domain decomposition method with cross-point treatment for time-harmonic acoustic scattering, Comput. Methods Appl. Mech. Engrg. (2020) 368 :113162.

[2] Modave A, Geuzaine C and Antoine X. Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering, J. Comput. Phys. (2020) 401 :109029.

[3] Royer A, Modave A, Béchet E, and Geuzaine C. A non-overlapping domain decomposition method with perfectly matched layer transmission conditions applied to Helmholtz equation, in progress.

[4] Royer A, Béchet E, Geuzaine C. Gmsh-Fem : An Efficient Finite Element Library Based On Gmsh, 14th WCCM-ECCOMAS Congress (2021)



4:30pm - 5:00pm

A new approach to space-time boundary integral equations for the wave equation

Olaf Steinbach1, Marco Zank2, Carolina Urzua-Torres3

1TU Graz, Austria; 2University of Vienna; 3TU Delft

We first discuss the ellipticity of the wave single layer boundary integral operator in one

space dimension where we generalize the energetic boundary element formulaition.

Instead of an additional time derivative of the test function we use a modified

Hilbert transformation which then results in ellipticity and boundedness results in the

energy space $H^{-1/2}(\Sigma)$.

In the general case, and based on a generalied inf-sup stable space-time variational

formulation for the wave equation we derive inf-sup stability conditions for all

boundary integral operators for the wave equation in suitable Sobolev spaces.



5:00pm - 5:30pm

Fast Direct Space-Time Finite Element Solvers for the Second-Order Wave Equation

Marco Zank

University of Vienna, Austria

For the discretisation of time-dependent partial differential equations, the standard approaches are explicit or implicit time stepping schemes together with finite element methods in space. An alternative approach is the usage of space-time methods, where the space-time domain is discretised and the resulting global linear system is solved at once. In this talk, the model problem is the wave equation. First, a space-time variational formulation in space-time Sobolev spaces for the wave equation is discussed, where a modified Hilbert transform is used such that ansatz and test spaces are equal. A conforming discretisation of this space-time variational formulation leads to a space-time Galerkin finite element method, which is unconditionally stable, i.e. no CFL condition is required. However, this space-time Galerkin finite element discretisation leads to a large global linear system of algebraic equations. The main part of this talk investigates new efficient direct solvers for this system. In particular, a tensor-product approach with piecewise polynomial, globally continuous ansatz and test functions is used. The developed solvers are based on the Bartels-Stewart method and on the Fast Diagonalization method, which result in solving a sequence of spatial subproblems. The solver based on the Fast Diagonalization method allows solving these spatial subproblems in parallel, leading to a full parallelization in time. In the last part of the talk, numerical examples are shown and discussed.



5:30pm - 6:00pm

Advances in stability of high-order polynomials with the Lukacs Theorem

Bruno Despres

Sorbonne University, France

The Lukacs Theorem states that bound preserving real polynomials on compact intervals have a (non unique) representation as sum of squares of other real polynomials. It is highly tempting to use this rigorous approach to stabilize high order discretization of partial differential equations. Of course, it has price which is that the structure becomes non linear, however the approximation properties are optimal.

I will review recent advances on the topic with numerical illustration to advection equation. This is based on recent 2 publications:

a) Campos-Pinto/Charles/Després, Bruno Algorithms for positive polynomial approximation. Numer. Anal. 2019

b) Després/Herda, Computation of sum of squares polynomials from data points. Numer. Anal. 2020

Possible connections with WENO techniques will be evoked.



 
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