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For the “Grandes Écoles” competitive entrance exams in 2011, I wrote a small program to find minimal surfaces with fixed boundary.
It’s not very mature, it doesn’t work all the time and it’s not well-documented, but it is available here.
To compile it, install OCaml, OCamlBuild, and the OpenGL libraries for OCaml.
Controls are rudimentary: SPC to start/stop, left-click and scroll to move around, right-click to choose one of the four predefined boundaries, s to swap between full and hollow.
You then get images like this:
I absolutely do not guarantee that the program is correct. The surfaces found do resemble the theoretical ones, but I do not have a mathematical proof that they are the right ones.