In most cases FDMs are used on regular domains. For irregular domains, FEM, FVM, BEM are used. But some books present application of FDM on irregular domains. In that case correction parameters ( α's) are used for step lengths near the boundary domian.
FEM is most direct and simple scheme for PDE,
If you want to tuning your program.
Experimental data is needed.
This is method in industirial reasearch group.
There is no other method to replace experiment.
good luck
I discuss the BEM in four of my books (DiffEq, NumCalc, CFD, and Ptracking), which you can get for free on certain days, as allowed by Amazon. The software is always free. I have attached some examples. The BEM is easy to set up the input files and runs very fast, even for complex domains with several inlets and exits. If you have a problem in mind, I could get it started for you. As this is a boundary problem, my polygon editor (also free) really comes in handy for building the domains.
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