ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
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

Session Overview 
Session  
MS 33a: fast solvers for wave problems
 
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, highorder 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.  
Presentations  
4:00pm  4:30pm
Comparison of optimized highorder domain decomposition methods with crosspoint treatment for Helmholtz problems ^{1}Institut Montefiore, Université de Liège, Belgium; ^{2}POEMS, CNRS, Inria, ENSTA Paris, Institut Polytechnique de Paris, France; ^{3}Département Aérospatial et Mécanique, Université de Liège, Belgium; ^{4}IECL, Université de Lorraine, Inria, EPI SPHINX, France The parallel finiteelement solution of largescale timeharmonic problems is addressed with a nonoverlapping substructuring domain decomposition method (DDM) using highorder 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 highorder absorbing boundary conditions (HABCs) or perfectly matched layers (PML) have proved to be well suited, as a good compromise between basic impedance conditions and nonlocal conditions. Recently, a numerical treatment of interior crosspoints (where more than two subdomains meet) and boundary crosspoints (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 highfrequency scattering problems, with both methods implemented in the same highorder 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 finiteelement matrices. References [1] Modave A, Royer A, Antoine X and Geuzaine C. An optimized Schwarz domain decomposition method with crosspoint treatment for timeharmonic acoustic scattering, Comput. Methods Appl. Mech. Engrg. (2020) 368 :113162. [2] Modave A, Geuzaine C and Antoine X. Corner treatments for highorder local absorbing boundary conditions in highfrequency acoustic scattering, J. Comput. Phys. (2020) 401 :109029. [3] Royer A, Modave A, Béchet E, and Geuzaine C. A nonoverlapping domain decomposition method with perfectly matched layer transmission conditions applied to Helmholtz equation, in progress. [4] Royer A, Béchet E, Geuzaine C. GmshFem : An Efficient Finite Element Library Based On Gmsh, 14th WCCMECCOMAS Congress (2021) 4:30pm  5:00pm
A new approach to spacetime boundary integral equations for the wave equation ^{1}TU Graz, Austria; ^{2}University of Vienna; ^{3}TU 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 infsup stable spacetime variational formulation for the wave equation we derive infsup stability conditions for all boundary integral operators for the wave equation in suitable Sobolev spaces. 5:00pm  5:30pm
Fast Direct SpaceTime Finite Element Solvers for the SecondOrder Wave Equation University of Vienna, Austria For the discretisation of timedependent 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 spacetime methods, where the spacetime 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 spacetime variational formulation in spacetime 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 spacetime variational formulation leads to a spacetime Galerkin finite element method, which is unconditionally stable, i.e. no CFL condition is required. However, this spacetime 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 tensorproduct approach with piecewise polynomial, globally continuous ansatz and test functions is used. The developed solvers are based on the BartelsStewart 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 highorder polynomials with the Lukacs Theorem 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) CamposPinto/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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