Sp. Huestis, A FOURIER BOUNDARY INTEGRAL METHOD FOR SOLVING LAPLACES-EQUATION IN 2DIMENSIONS, Communications in numerical methods in engineering, 11(7), 1995, pp. 575-584
Laplace's equation in two dimensions can be solved by a boundary integ
ral method which imposes the required consistency relation between the
boundary function and its outward normal gradient. Rather than incorp
orating this into a standard boundary element approach for the Dirichl
et problem, the known boundary function and the unknown boundary gradi
ent are here expressed as truncated Fourier series expansions. The con
sistency relation then becomes an algebraic relation between expansion
coefficients, whose matrix entries can be found by appropriate applic
ations of the fast Fourier transform. This is solved for the boundary
gradient coefficients; the solution can then be evaluated at any inter
ior point. Numerical experiments explore the effect of the truncation
level, and indicate that reasonable accuracy can be attained without n
eeding a prohibitively large number of Fourier coefficients.