This code solves the Laplace equation ∇²u = 0 for electrostatic potentials u bounded in concentric regions using the Gauss–Seidel algorithm with successive over-relaxation.

Clone and make && make clean.

If building on macOS, first brew install argp-standalone and

export CFLAGS="$CFLAGS -I/usr/local/include/"
export LDFLAGS="$LDFLAGS -L/usr/local/lib/ -largp"

to make GLIBC argument parsing available.

./seidel with no arguments will print the (automatically chosen) parameters and the value of the double integral of E² over the region, a measure of the total electric field.

Run with --help to expose the optional flags.

Usage: seidel [-cpqrt?] [-a a] [-b b] [-n n] [-w w] [--xboundary=a]
        [--yboundary=b] [--clean] [--size=n] [--print] [--norelax]
        [--relax] [--translate] [--omega=w] [--help] [--usage]

The boundary point (a, b) defines the north-west point of the inner boundary.

./seidel --print produces data files of the form

x   y   f(x,y)

for the potential (u.dat), the x- and y- components of the electric field (ex.dat, ey.dat) and a file cs.dat of the form

y   E_x   E_y

which is a cross-section of the electric field at x = 0.5 * n. These files can be removed with --clean (and also with make delete). Run the .plt scripts to plot the results.

The --translate option outputs the value of the integral L (above) as the inner boundary is translated vertically through the region.

The program over-relaxes by default (using successive over-relaxation parameter ω = 1.92, the optimum value for the default configuration). This can be disabled with --norelax (which sets ω = 1), assigned with --omega or calculated with --relax, which prints the value of ω and the number of iterations required for the solution to converge until it hits the minimum.