SIAM CSE 2025 Minisymposium on Singular Functions for PDEs

Organizers: Simon Dirckx, Heather Wilber, and Alex Barnett

March 3 and 4, 2025, Forth Worth Convention Center, Fort Worth, TX

Here we gather a permanent record of this eight-speaker minisymposium, including titles, abstracts, slides and software links. Each speaker is shown in bold.

Overview: High-order or spectrally accurate numerical methods for PDEs generally demand the manipulation of singular functions, be they fundamental solutions, geometric singularities, boundary value singularities, or singularities in parameters (Wood anomalies, complexified parameterizations such as boundaries or Fourier variables). Examples include methods of fundamental solutions, Lightning Laplace/Stokes, enriched spectral/Galerkin solvers, extension/image methods. The efficiency and conditioning of such schemes often relies fundamentally on understanding the behavior of the singular functions in question. Numerical techniques to handle such singularities often involve approximation theory in the complex plane, or complex contour deformations.

MS52 (Part I)

• Adaptive Green's Function Integration over Bloch Wavevectors with Explicit Handling of Complex Singularities. abstract. Alex H. Barnett and Jason Kaye, Simons Foundation and Flatiron Institute, U.S.; Lorenzo X. Van Muñoz, Massachusetts Institute of Technology, U.S.

  Software links: AutoBZ.jl

• Integration of Singular Functions over Deformable Surfaces – Corrected Quadratures, Regularized Quadratures, and Rational Approximation Quadratures. abstract. Joar Bagge, University of Texas at Austin, U.S.; Bowei Wu, University of Massachusetts, Lowell, U.S.; George Biros and Per-Gunnar Martinsson, University of Texas at Austin, U.S.; Anna-Karin Tornberg and Pritpal Matharu, KTH Royal Institute of Technology, Sweden.

  Video: vesicle in pipe.

• Trapped Acoustic Waves and Raindrops: High-Order Accurate Integral Equation Method for Localized Excitation of a Periodic Staircase. abstract. Fruzsina Agocs, University of Colorado Boulder, U.S.; Alex H. Barnett, Simons Foundation and Flatiron Institute.

• The Resolution of Singularities by Rational Functions and Mesh Refinement. abstract. Daan Huybrechs, Katholieke Universiteit Leuven, Belgium.

MS80 (Part II)

• Parallel Set-Valued Rational Approximation for PDEs. abstract. Simon Dirckx, University of Texas at Austin, U.S.

  preprint.

• An Accurate Method of Fundamental Solutions for Large and Dense Particle Systems in Stokes Flow. abstract. Anna Broms, KTH Royal Institute of Technology, Sweden; Alex H. Barnett, Simons Foundation and Flatiron Institute, U.S.; Anna-Karin Tornberg, KTH Royal Institute of Technology, Sweden

  JCP paper on resistance for spheres, preprint on mobility for spheres and ellipsoids.

• Computing Green's Functions Using Recurrences . abstract Sheehan Olver, Imperial College London, United Kingdom

• Recursive Reduction Quadrature for the Evaluation of Laplace Layer Potentials in Three Dimensions. abstract. Hai Zhu, Flatiron Institute, U.S.; Shidong Jiang, Simons Foundation and Flatiron Institute, U.S.