My physical plane has 3 straight lines on one side and a discontinuous mathematical function on the fourth side. I wouls like to map this onto a rectangle/square on the computational plane. Can anyone please help me with this?

It sounds like you want to use a structured rectangular mesh. I suggest you partition your problem space into blocks. The region either side of the discontinuity would lie in different blocks. You could then partition judiciously-chosen blocks by logically-rectangular (structured) meshes compatible at the block interfaces. If using FEM, you could assemble each rectangular block separately, and couple the blocks with constraints before solving. If your graphics program for plotting results is limited to rectangular meshes, then plot each block separately on the same graph.