MPSpack : 2D Helmholtz scattering and eigenvalue problems via particular solutions and integral equations

Alex Barnett 5/31/23. Version 1.41 [legacy code; barely supported]

MPSpack is a user-friendly and fully object-oriented MATLAB toolbox that implements the method of particular solutions (aka Trefftz or nonpolynomial FEM, including the method of fundamental solutions, Fourier-Bessel local expansions, singular corner expansions), and integral equation methods (including some basic corner handling), for the efficient and often spectrally-accurate solution of Laplace eigenvalue problems, interior/exterior Helmholtz boundary-value problems (e.g. wave scattering), periodic diffraction problems, and related PDE problems, on piecewise-homogeneous 2D domains.

Version 1.0 was released in 2009, and co-authored with Timo Betcke. Since then I have been the main developer; it has settled into a repository for a variety of new numerical methods developed for corner domains, layer potentials, periodic problems, and high-frequency Dirichlet and Neumann eigenvalue problems, enabling this research to be reproducible. It is stable and will not have much future development. Instead I and colleagues expect to release a replacement package for integral equations which will include close-evaluation quadratures (Helsing, QBX, etc).

I am grateful for the support of the National Science Foundation under grants DMS-0811005 and DMS-1216656; and Betcke for support of the Engineering and Physical Sciences Research Council Grant EP/H00409/1. We also are thankful for the inclusion of codes by V. Rokhlin, L. N. Trefethen, A. Pataki, Z. Gimbutas, D. M. Schwarz, S. Hawkins, B. Gustavsson, and several others.

Requirements

1. MATLAB 2008a or newer (in particular, no toolboxes needed)

2. Optional requirements for tweaks and fast algorithms (see manual):

• C and Fortran compilers such as gcc and gfortran.

• GNU Scientific Library GSL

• FMMLIB2D for Helmholtz fast multipole method

• LP2D for Alpert quadrature correction to FMM on smooth curves

MPSpack is released under GPL v.3; please contact me for other license options.

Installation

Install git (eg on an ubuntu/debian linux system use sudo apt-get install git). Then as usual do git clone https://github.com/ahbarnett/mpspack to download and create the directory mpspack containing the package.

In MATLAB, type addpath /path/to/mpspack. See Usage below to test your installation.

Add the above addpath command to your MATLAB startup.m file if you want the MPSpack toolbox available by default.

To install tweaks (MEX interfaces to Bessel/Hankel/inpoly), which are tested only in a linux environment, from a shell in the directory mpspack type make. If you have trouble, edit the library locations in make.inc.

See Sec. 4.1 of the manual for linking to fast algorithms (FMM).

Note an obsolete snapshot of version 1.33 from 2014 sits on the sadly-defunct googlecode. Please use the github version.

Usage

To test your basic installation: in MATLAB make sure you're in the mpspack top directory and type run test/testdielscatrokh which should take about 1 second to run and produce a wave scattering figure from a smooth dielectric domain, along with a pointwise error, which should be small (ie around 1e-14).

To test whether your tweaks installation worked:

• run test/testbasis which should give around 0.2 us per eval for MFS, SLP, and DLP bases. Older versions of MATLAB will give only 2 us per eval.

• run test/testinpolywrapper which should compare the slow MATLAB against the fast Luong code.

See tutorial and manual for detailed examples and usage.

Examples

1. Frequency-domain scattering from a square, accurate to 10 digits, computed in a few seconds on a laptop. Spectral convergence is achieved using the following ingredients: decomposition into subdomains (nonpolynomial FEM), fractional-order Fourier-Bessel expansions around corners, and an exterior fundamental solutions representation. In MPSpack this only 20 lines of code for Dirichlet or Neumann cases (see examples/tut_square.m):

   

2. The first 45 Dirichlet eigenmodes of a smooth planar domain, computed to 12 digit accuracy and evaluated on a grid of 3600 points, in around 1 second per mode. Convergence is again spectral, using a layer potential, Kress quadratures, and analytic root-finding on a Fredholm determinant. 9 lines of MPSpack code (see examples/tut_evp.m):

   

3. Finally, an entertaining example of acoustic (sound-hard) scattering from some smoothly digitized letter shapes, computed to 10 digit accuracy by Perrin Meyer (contact him for code). The wave is incident from about 4 o'clock:

   

There are other pictures in the gallery and plenty in the tutorial.

References

Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues, Alex Barnett, Andrew Hassell, and Melissa Tacy, Duke Math. J. 167(16), 3059-3114 (2018).

An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons, Alex H. Barnett and Timo Betcke, SIAM J. Sci. Comp., 32(3), 1417-1441 (2010).

Fast computation of high frequency Dirichlet eigenmodes via the spectral flow of the interior Neumann-to-Dirichlet map, Alex Barnett and Andrew Hassell, Comm. Pure Appl. Math., 67(3) 351-407 (2014).

Robust and efficient solution of the drum problem via Nyström approximation of the Fredholm determinant, L. Zhao and A. H. Barnett, SIAM J. Numer. Anal., 53(4) 1984-2007 (2015).

A new integral representation for quasi-periodic scattering problems in two dimensions, Alex Barnett and Leslie Greengard, BIT Numer. Math. 51, 67-90 (2011)

To do list

• Extract the best quadrature schemes for a new BIE2D package

• Interpolation to replace Zp, Zpp for convenience but losing digits