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: 8th Dec 2022, 11:06:51pm CET

Session Overview 
Session  
MS 27a: fast and high order solution techniques for boundary integral equations
 
Session Abstract  
Boundary integral equations for the solution of boundary value problems for partial differential equations such as Stokes, Helmholtz, and (frequencydomain) Maxwellâ€™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 secondkind 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.  
Presentations  
1:50pm  2:20pm
A Fast Integral EquationMethod for the TwoDimensional AdvectionDiffusion Equation on TimeDependent 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 equationbased solver for the advectiondiffusion equation in complicated moving and deformable geometries in two space dimensions. Our method can be explained as applying a highorder accurate timestepping scheme based on semiimplicit spectral deferred correction. Here, one timestep involves solving a sequence of nonhomogeneous 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 timedependent geometries with PUX, which is applicable to other numerical methods than boundary integralmethods. 2:20pm  2:50pm
Accurate trapezoidal rulebased quadrature for nonparametrized Boundary Integral Methods ^{1}KTH Royal Institute of Technology, Sweden; ^{2}University 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 [1], we rewrite the surface integral as an integral over a small volume encompassing the surface using the closestpoint mapping to the surface, without explicit parametrization. The integrands in the nonparametrized 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 SingleLayer and DoubleLayer potentials for Laplace and Helmholtz boundary integral formulations in the nonparametric setting. [1] 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 ^{1}KTH Royal Institute of Technology; ^{2}La 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 dropletbased 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 GaussLegendre 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 UrbanaChampaign, 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 quadraturebymatchedasymptoticexpansion (QBMAX) method. The method can greatly reduce the cost of highorder 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 MorseIngard equations for thermoacoustic scattering in three dimensions when using QBMAX for the nearfield interactions coupled with QBXbased fast algorithm for farfield evaluation. 
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