To run this code, open the script poisson.m in MATLAB and press the RUN button. Change the hfun parameter to control the mesh resolution. The mesh processing can take some time due to the neighbor-finding step, and unfortunately the current implementation does not scale well. Expect to wait a while if you set hfun less than 0.025 :)
This unstructured mesh generation in this code is heavily dependent on the MESH2d library by Darren Engwirda.