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: 4th Dec 2022, 07:14:52pm CET
Z3: integral equations, BEM
4:10pm - 4:30pm
On the coupling of Curvilinear Virtual Element and Boundary Element Methods for 2D exterior Helmholtz problems
1Politecnico di Torino; 2Università di Parma
We consider the Helmholtz equation defined in unbounded domains, external to 2D bounded ones having curved shape, endowed with a Dirichlet condition on the boundary and a radiation condition at infinity.
We propose a numerical method that approximates the solution using computations only in an interior finite domain. This is obtained by introducing a curved smooth artificial boundary on which a Non Reflecting Boundary Condition (NRBC) , defined by a Boundary Integral Equation, is imposed.
For the numerical discretization, we propose a Galerkin approach based on Curved Virtual Element Method (CVEM) in the interior of the computational domain.
This choice is based on the fact that the virtual element method allows to broaden the classical family of the finite element method for the discretization of partial differential equations for what concerns both the decomposition of domains with complex geometry and the definition of local high order discrete spaces. Moreover, the use of curvilinear elements instead of polygonal ones allows to avoid the sub-optimal rate of convergence for degrees of accuracy higher than 2. For the discretization of the NRBC, we consider a classical BEM associated to Lagrangian nodal basis functions.
The main challenge in the theoretical analysis, based on the pioneering Johnson-Nédélec paper in which the coupling approach is proposed for the Laplace equation (see ) , is the lack of ellipticity of the associated bilinear form. However, using the Fredholm theory for integral operators, it is possible to prove the well-posedness of the problem in case of computational domains with smooth artificial boundaries. Moreover, the analysis of the Helmholtz problem and of the proposed numerical method for its solution, is carried out by interpreting the new main operators as perturbations of the Laplace ones.
We present the theoretical analysis of the method in a quite general framework, and we provide an optimal error estimate in the energy norm. The numerical tests we present confirm the theoretical results and show the effectiveness of the new proposed approach.
 C. Johnson and J.-C. Nédélec. On the coupling of boundary integral and finite element methods. Math. Comp., 35(152):1063–1079, 1980.
4:30pm - 4:50pm
Computing spectral measures of self-adjoint operators
1University of Cambridge; 2Cornell University
Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonance phenomena, and fluid stability analysis. Despite new results on computing spectra, there has been no method able to compute spectral measures of general infinite-dimensional self-adjoint operators, with previous efforts focused on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators that carry a lot of structure. Unlike its matrix counterpart, the spectral measure of a self-adjoint operator may have an absolutely continuous component and an associated density function, e.g. in applications posed on unbounded domains. In this talk we show how state-of-the-art adaptive spectral methods allow the computation of spectral measures. By dealing with operators directly, opposed to previous "discretise-then-solve" techniques, we construct methods with arbitrarily large rates of convergence that recover the spectral measures of self-adjoint ODEs, PDEs, integral operators and discrete operators. Finally, these algorithms are embarrassingly parallelisable. Numerical examples will be given, demonstrating efficiency, and tackling difficult problems taken from mathematics and other fields such as chemistry and physics. A highlight is the computation of hundreds of eigenvalues of Dirac operators from computational chemistry to machine precision and without spectral pollution. The talk is based on joint work with Andrew Horning (Cornell University) and Alex Townsend (Cornell University).
4:50pm - 5:10pm
Evaluating near-singular integrals with application to axisymmetric Stokes flow
University of New Mexico, United States of America
Boundary integral formulations yield efficient numerical methods to solve elliptic boundary value problems. They are the method of choice for interfacial fluid flow in either the inviscid vortex sheet limit, or the viscous Stokes limit. The fluid velocity at a target point is given by integrals over the interfaces. However, for target points near, but not on the interface, the integrals are near singular and standard quadratures lose accuracy. While several accurate methods exist to resolve the analytic integrals that appear in planar geometries, they dont generally apply to the non-analytic case that arises in axisymmetric geometries. Motivated by the latter, we present a method based on Taylor series expansion of the integrand about basepoints on the interface that accurately resolve a large class of integrals. We describe the method, present analytical convergence results, and apply it to model double emulsion flow through constrictions in axisymmetric Stokes flow.
5:10pm - 5:30pm
Spectral Ewald: a fast Ewald summation method for electrostatics and Stokes flow with arbitrary periodicity
KTH Royal Institute of Technology, Sweden
Large particle systems in e.g. electrostatics or Stokes flow require fast methods that reduce the computational complexity from O(N^2) to O(N) or O(N log N) when computing the interaction of N particles. Ewald-type methods achieve complexity O(N log N) by splitting the potential into a real-space part and a Fourier-space part. The Spectral Ewald method computes the latter, accelerated by the fast Fourier transform (FFT). A unified framework permits treatment of three-dimensional problems with periodicity in three, two, one or none of the spatial directions. Removing periodic boundary conditions only moderately increases run time compared to the triply periodic case. This is accomplished by an adaptive FFT together with a special FFT-based solution technique for the subproblem in the free directions.
In the Spectral Ewald method, the source strengths are spread from the particles onto a uniform grid using a window function. By allowing the support of the window function to vary independently of the size of the uniform grid, the error decays spectrally in both grid size and window support size. We compare different window functions, and in particular consider the Kaiser-Bessel window function which permits a smaller support than e.g. the Gaussian window to reach the same accuracy. To efficiently compute the Kaiser-Bessel function, we approximate it by a piecewise polynomial, with the polynomial degree selected to maintain the accuracy of the method.
5:30pm - 5:50pm
Density Interpolation Methods for Boundary Integral Equations
Institute for Mathematical and Computational Engineering, Pontificia Universidad Catolica de Chile, Santiago, Chile
We will present a class of general regularization techniques for the numerical evaluation of weakly singular, singular, hypersingular, and nearly singular boundary integral operators associated with classical PDEs of the elliptic type. The proposed techniques address longstanding efficiency, accuracy, and practical implementation issues that have hindered the applicability of boundary integral equation methods, such as Nyström and boundary element methods, in science and engineering. Relying on Green’s third identity and local interpolations of density functions in terms of homogeneous solutions of the underlying PDE, these techniques regularize the singularities present in boundary integral operators recasting them in terms of integrands that are bounded or even more regular depending on the density interpolation order. The resulting boundary integrals can then be accurately and efficiently evaluated using elementary quadrature rules. A variety of numerical examples demonstrate the effectiveness of the proposed methodology in the context of Nyström and boundary element methods for the Laplace, Helmholtz, elastostatic, and time-harmonic Maxwell and elastodynamic boundary value problems.
5:50pm - 6:10pm
A Simple Quadrature For Hypersingular Integral Equations
University of Texas at Austin, USA
Hypersingular integral equations are efficient for solving boundary value problems from a wide range of engineering applications, including acoustic and electromagnetic scattering, fracture mechanics, and aerodynamics. A key challenge is to accurately approximate the finite-part integrals in these equations. In this talk, we present a simple trapezoidal quadrature method based on lattice sum theories; the quadrature is high-order accurate for 2D and 3D problems and can be easily combined with existing fast solvers. Theoretically, our quadrature method is connected to a family of zeta functions and can handle integrands with a variety of singularities and in arbitrary dimensions.
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