The best method to solve Helmholtz equation for a irregular boundary

[wdlang]

i have an almost square region.

By 'almost' i mean the edges are curvy, not completely straight.

i now need to solve the Helmholtz equation with Dirichlet boundary condition

what is the best numerical method?

how is Finite element, though i do not know what Finite element is

 

[Chestermiller]

By proper use of Taylor series expansions at the boundary, you can express a modified boundary condition on a square boundary (taking into account the difference between the actual boundary location and the square boundary location). This will allow you to use finite differences with the square boundary. See Carnahan, Luther, and Wilkes, Applied Numerical Methods.