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, 11:06:51pm CET
MS 27a: fast and high order solution techniques for boundary integral equations
Boundary integral equations for the solution of boundary value problems for partial differential
equations such as Stokes, Helmholtz, and (frequency-domain) Maxwell’s equations have a computa-
tional cost that scales linearly with respect to the number of boundary degrees of freedom. Due the
complexity of accurately discretizing and dealing with the resulting dense linear system, boundary
integral equations have not been computational tractable until recently. Key contributions to mak-
ing integral equations useful for large scale applications include the development of representations
that yield second-kind equations, techniques for handling complex geometries, fast algorithms, fast
direct solvers, and high order singular quadrature methods. The contributions highlighted in this
minisymposium represent some of the significant advances towards this effort.
As these methods are utilize high order discretizations and have the corresponding accuracy, cost
profile and capabilities, this session naturally fits into the ICOSAHOM program. Based on a small
but significant presence of integral equation researchers at the past ICOSAHOM in London, we hope
to deepen and grow the connection between the two communities. To this end, this minisymposium
brings together a highly international and predominantly junior group of researchers working at the
forefront of research on integral equation methods.
1:50pm - 2:20pm
A Fast Integral Equation-Method for the Two-Dimensional Advection-Diffusion Equation on Time-Dependent Domains
KTH Royal Institute of Technology, Sweden
Boundary integral methods are attractive for solving homogeneous linear elliptic partial differential equations in complex geometry. However, these numerical methods are not straightforward to apply to parabolic equations, which often arise in science and engineering. We address this problem by presenting an integral equation-based solver for the advection-diffusion equation in complicated moving and deformable geometries in two space dimensions. Our method can be explained as applying a high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction. Here, one time-step involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching, which requires several recently developed methods. These include special purpose quadrature, the function extension technique partition of unity extension (PUX), and Ewald summation for the modified Helmholtz equation. Special care is also taken to handle the time-dependent geometries with PUX, which is applicable to other numerical methods than boundary integral-methods.
2:20pm - 2:50pm
Accurate trapezoidal rule-based quadrature for non-parametrized Boundary Integral Methods
1KTH Royal Institute of Technology, Sweden; 2University of Texas at Austin, USA
Boundary integral methods are well renowned for their usefulness with homogeneous elliptic equations such as Laplace or Helmholtz, especially in unbounded domains. With these methods, the solution is built by solving a boundary integral equation (BIE). The integrands are singular in a point, but there are many techniques to accurately discretize the integrals using the explicit parametrization of the surface.
Instead of using an explicit parametrization, we consider the case where the surface is described by a closest point mapping, or equivalently a distance function. Avoiding explicit parametrization can be advantageous when dealing with surfaces with complicated shapes or available in the form of point cloud data.
Using the technique from , we rewrite the surface integral as an integral over a small volume encompassing the surface using the closest-point mapping to the surface, without explicit parametrization. The integrands in the non-parametrized version of the BIEs are singular not in a point but along a line, and few methods have been developed to accurately evaluate these integrals.
In this work, we present new quadrature methods based on corrections to trapezoidal rule, which accurately integrate functions singular along straight lines in 3D. The methods developed are specific for the kind of singularities encountered in the Single-Layer and Double-Layer potentials for Laplace and Helmholtz boundary integral formulations in the non-parametric setting.
 Kublik, C., and Tsai, R. - Integration over curves and surfaces defined by the closest point mapping. Research in the mathematical sciences 3.1 (2016): 3.
2:50pm - 3:20pm
Quadrature error estimates for layer potentials near curved surfaces
1KTH Royal Institute of Technology; 2La Sapienza, University of Rome, Italy
Boundary integral methods are a powerful tool for the simulations of fluid mechanics at the micro scale such as in droplet-based microfluidics, with tiny drops interacting in Stokes flow. We have developed highly accurate numerical methods for drops covered by insoluble surfactants and placed in electric fields. This involves the accurate evaluation of layer potentials on and close to the drop surfaces. In this talk, we will focus on quadrature error estimates for the evaluation of nearly singular layer potentials in two and three dimensions when using two quadrature methods that are commonly used: the panel based Gauss-Legendre quadrature rule and the global trapezoidal rule.
3:20pm - 3:50pm
Quadrature by Matched Asymptotic Expansions (QBMAX) for the Evaluation of Layer Potentials with Boundary Layers
Department of Computer Science, University of Illinois Urbana-Champaign, United States of America
Many quadrature schemes for the evaluation of layer potentials become costly when the kernel decays rapidly, such as for the Helmholtz kernel with large imaginary wave number. Such kernels (and the associated computational cost) are particularly prevalent in the numerical modeling of boundary layers. In this work, we introduce the quadrature-by-matched-asymptotic-expansion (QBMAX) method. The method can greatly reduce the cost of high-order accurate layer potential evaluation in the presence of boundary layers. Based on quadrature by expansion (QBX), QBMAX employs a local change of variables using information derived from the matched asymptotic expansions of the underlying PDE near the boundary. We show illustrative accuracy results as well as empirical data supporting the accuracy of the scheme. We also show the efficiency improvements when solving the Morse-Ingard equations for thermoacoustic scattering in three dimensions when using QBMAX for the near-field interactions coupled with QBX-based fast algorithm for far-field evaluation.
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