Iterative solution of elliptic equations

Abstract

We reduce solution of the Dirichlet problem ( x ∈ D ⊂ R m ) Δ u ( x ) + a ( x ) u ( x ) = f ( x ) in  D , u = 0 on  ∂ D to iterative solution of a simpler problem Δ u = f ( x ) in  D , u = 0 on  ∂ D , for which one can use either Fourier series or Green's function method. The method is suitable for numerical computations, particularly when one uses Newton's method for semilinear problems Δ u + g ( x , u ) = 0 in  D , u = 0 on  ∂ D , .