This set of codes solves boundary value problems for piecewise constant coefficient linear PDEs using potential theory, ie boundary integral equations (BIE) on curves. Quadratures that are very high-order or spectral are used, allowing accuracies approaching machine precision with small numbers of unknowns. The idea is to provide a simple and uniform interface to the evaluation of layer potentials and filling of Nystrom matrices for Laplace, Helmholtz, and Stokes kernels, including modern quadratures for singular self-evaluation, and close-evaluation quadratures (eg based on the Cauchy kernel). Simple BVP solvers are included, for various geometries including singly and doubly periodic. This provides a sandbox to deliver implementations of various schemes that are under active research by myself and collaborators, as well as by other experts such as R. Kress, J. Helsing. The code is designed to be reasonably efficient, yet tutorial in nature (easy to read and well-documented), and could be used as a template for faster Fortran/C versions. I plan to include fast algorithms and corner quadratures.

Main author: Alex Barnett

Version: 20210915

Based on work from 2008-2016 including subsuming the integral-equation parts of MPSpack, all of LSC2D, and my BIE tutorial.

Also includes the following contributions and influences:

Bowei Wu - Stokes velocity extension from Laplace
Gary Marple - matrix versions of global close evaluation quadratures
Nick Trefethen - Gaussian quadrature
Jun Wang - 2nd derivs of Cauchy, Laplace SLP, and traction of Stokes SLP

As of 2018, David Stein made a python implementation of most of BIE2D, including new fast versions of close evaluations, in his package pyBIE2D.

Download using git, svn, or as a zip (see green button above).

Open MATLAB in the top level (BIE2D) directory, and run bie2dsetup to add all needed directories to your path.

Test by running testall which should produce lots of error outputs close to machine precision, figures, etc, and yet not crash.

Many functions (eg close routines in kernels) have built-in self-tests run by calling without any arguments.

Codes have not been tested on MATLAB versions prior to R2012a.

kernels : Laplace, Stokes, Cauchy, Helmholtz potential evaluation and matrix filling, including close-evaluation
utils : general numerical and plot utilities
test : test codes (other than built-in self-tests), figure-generating codes
solvers : 2D BVP solver example codes, also serve to test kernels
solvers/closetouchingexpts : close-to-touching curve experiments
panels : panel quadratures including close-evaluation
singlyperiodic : codes for Laplace, Stokes flow in periodic pipes, possibly with vesicles (to do)
doublyperiodic : codes for flow (Laplace, Stokes) in doubly-periodic geometries, computation of effective permeability (in progress)

1. Every kernel can be accessed as a dense matrix mapping density to potential (if no density function is given), or as the potential evaluation given a single density (or stack of density columns). These are bundled into the same calling interface. Usually one of these simply calls the other (eg the matrix filling for close-evaluation is currently done by sending in columns of the identity as densities, which is wasteful), but I plan to provide more efficient matrix fillers based on BLAS3 later.

2. For all coordinates in $\mathbb{R}^2$ we use complex numbers in the form $x+iy$, since this is very convenient for geometry. For Cauchy integral interpolation we obviously also use complex numbers.

3. Stokes involves vector-valued densities, velocities, etc. For clarity of coding and visualization we have decided to order the node index fast and the vector index slow, ie to stack all the x-components, followed by all the y-components. Matrices thus have a block structure of the form $[A_{11}, A_{12}; A_{21}, A_{22}]$. The other choice of alternating x and y components would be better for RAM locality, and to feed into direct solvers, but we err instead on the side of simplicity/reability and leave this for a Fortran/C implementation. Likewise, we have avoided the use of complex numbers to represent x and y components for Stokes (apart from in internal routines such as Laplace extension).

4. So far most is based on the global periodic trapezoid rule. There are a couple of panel-based tests, but not a nice general format for panel setup yet.

• decide where to build in switches for close eval - per target? (scf)
• multiple closed curves (islands) helpers, use for dpls figs?
• Green's representation theorem kernel tests for Stokes kernels (easy)
• more BVP solver demos (eg bring over testStokesSDevalclose.m w/ all 4 BVPs)
• FMM MEX interfaces
• [long-term] Convert whole thing to C/Fortran libraries
• [long term] basic fast direct solver examples - jj index list fields in s,t?
• kd-tree for close-evaluation lists (Marple). Need non-Toolbox kd-tree.
• [low-priority] Alpert and other options for log-singular kernels
• corners with panels, bring in from various tests
• [long term] MEX interface to Rachh QBX/FMM ?
• [low-priority] bring in singlyperiodic, get Adrianna codes

• Cau_closeglobal simpler uses "interpolate the derivative", exterior S-W form
• cleaner kernel interface without mat or eval suffices
• many repmats/ones -> bsxfun for speed in kernels
• derivs for 'e' LapSLP_closeglobal corrected for nonzero totchg.
• Stokes mat fills bsxfun, StoSLP, StoSLP_closeglobal debug via vel BVP for now
• Stokes SLP pressure closeglobal eval, upsampling
• StoDLP_cg self-test & pressure close, StointvelBVP all pres plots added
• finish bring over doublyperiodic, pres figs
• srcsum2 for speed (targ sum not src, better for close eval)
• initial close-touching expts
• much accelerated the Cauchy close global matrix fill, using BLAS3 for all the O(N^3) parts, hence accelerating Laplace & Stokes (which call Cauchy in non-sparse way due to CSLP matrix being dense)
• Helmholtz kernels including S and D self-eval from MPSpack, far+derivs
• Green's representation theorem off- and on-surf kernel tests for Lap, Helm
• Reference kernel implementations for Laplace, Helmholtz; tests against them
• Basic Laplace panel GRF test