International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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: 8th Dec 2022, 11:28:32pm CET
MS 33c: fast solvers for wave problems
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
4:00pm - 4:30pm
A stable boundary integral formulation of an acoustic wave transmission problem with mixed boundary conditions
University Zurich, Switzerland
In this talk, we consider an acoustic wave transmission problem with mixed boundary conditions of Dirichlet, Neumann, and impedance type. We will derive a formulation as a direct, space-time retarded boundary integral equation, where both Cauchy data are kept as unknowns on the impedance part of the boundary. This requires the definition of single-trace spaces which incorporate homogeneous Dirichlet and Neumann conditions on the corresponding parts on the boundary.
We prove the continuity and coercivity of the formulation by employing the technique of operational calculus in the Laplace domain.
This talk comprises joined work with S. Eberle, F. Florian, and R. Hiptmair
4:30pm - 5:00pm
Efficient solution of 2D wave propagation problems by CQ-wavelet BEM
1University of Parma, Italy; 2Politecnico di Torino, Italy
We consider wave propagation problems in 2D unbounded domains, including dissipative effects, reformulated in terms of space-time boundary integral equations. For their solution, we employ a Convolution Quadrature (CQ)  for the temporal and a Galerkin Boundary Element Method (BEM) for the spatial discretization. It is known that one of the main advantages of the CQ-BEMs is the use of the Fast Fourier Transform (FFT) algorithm to retrieve the discrete time integral operators with an optimal linear complexity in time, up to a logarithmic term. Besides, it is also known that a key ingredient for the success of such methods is the efficient and accurate evaluation of all the integrals that define the matrix entries associated to the full space-time discretization. This topic has nowadays been successfully addressed when standard Lagrangian basis functions are considered for the space discretization. However, it results that, for such a choice of the basis, the BEM matrices are in general fully populated, a drawback that prevents the application of CQ-BEMs to large scale problems. As a possible remedy to reduce the global complexity of the method, we consider approximant functions of wavelet type . In particular, we propose a numerical procedure that, by taking advantage of the fast wavelet transform, allows on the one hand to compute the matrix entries associated to the choice of wavelet basis functions by maintaining the accuracy of those associated to the Lagrangian basis ones and, on the other hand, to generate sparse matrices without the need of storing a priori the fully populated ones . Such an approach allows in principle the use of wavelet basis of any type and order, combined with CQ based on any stable ordinary differential equations solver . Several numerical results, showing the accuracy of the solution and the gain in terms of computer memory saving, are presented and discussed.
 Lubich C., On the multistep time discretization of linear initial boundary value problems and their boundary integral equations, Numerische Matematik, 67, (1994), 365–389.
 Daubechies I., Ten Lectures on Wavelets, SIAM (2004).
 Bertoluzza S., Falletta S., Scuderi L., Wavelets and convolution quadrature for the efficient solution of a 2D space-time BIE for the wave equation, Applied Mathematics and Computation, 366, (2020), 124726
 Desiderio L., Falletta S., Efficient solution of 2D wave propagation problems by CQ-wavelet BEM: Algorithm and Applications, SIAM Journal on Scientific Computing, 42(4), (2020), B894-B920
5:00pm - 5:30pm
Galerkin spectral method for Helmholtz boundary integral equations on screens
Universidad Adolfo Ibañez, Chile
This work is concerned with the solution of boundary integral formulations for the Helmholtz equations, with boundary conditions (Dirichlet or Neumann) imposed on an open regular surface embeeded in R3.
The problems are transformed from a PDE model to two different first-kind integral equations (depending on the boundary condition) utilizing the standard representation formulas. We then employ a Galerkin method to form the corresponding discretization.
The bases for the Galerkin method are constructed from the projection of the spherical harmonics into the unitary disk. In particular, these bases correspond to the eigenvalues of the Laplace integral equation with Dirichlet condition on the unitary disk as it was proved by Wolfe.
From classical results on differential problems with boundary conditions on open manifolds, we found that the solution for both of the present cases (Dirichlet and Neumann) exhibits singular behavior, that prevents optimal convergence rate for traditional discretizations. In our case, this issue is solved by explicitly including the singular behavior on the spectral bases.
Under the assumption of arbitrary smooth geometry, we show by using scale of functional spaces, that the spectral methods converge super-algebraically.
We give details on the computation of the integral operators. We found that, in contrast to classical boundary integral implementations, the explicit singularities included on the spectral bases prevent the fast convergence of the computation of the discretization of the integral operators. To remedy these we employ a sequence of polar change of variables combined with the Gauss-Jacobi rule to extract the singularities.
Finally, we analyze the effect of the quadrature procedure on the convergence rate. This is done through a complex extension of the spectral basis and subsequent bounds of the derivatives by the Cauchy integral formula.
5:30pm - 6:00pm
Higher-order boundary elements for elastodynamics
Heriot-Watt University, United Kingdom
We discuss h, p and hp-versions of the time domain boundary element method for time dependent linear elasticity. We particularly discuss scattering problems outside an open curve or a polygonal scatterer, where the solution exhibits singularities from the edges and corners. Numerical experiments illustrate the theory in 2d. (joint with A. Aimi, G. Di Credico and E. P. Stephan)
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