Exponentially accurate approximations for Dirichlet problems with discontinuous data (Q1276071)

A method to obtain exponentially accurate approximations for Dirichlet problems with discontinuous boundary data for Laplace's equation in two dimensions is presented and discussed. Model problems with circular or rectangular boundaries, whose solutions can be obtained by separation of variables involving Fourier series, are discussed in detail. The boundary data \(g\) is expressed as the sum of a singular function \(\widetilde S_M\), which is a linear combination of ``singular basis functions'' \(\{S_n\}\), and \(g-\widetilde S_M\), which is much smoother than the original data. The solution \(u\) of the boundary-value problem is then expressed as the sum of a linear combination of the harmonic extensions \(\{\varphi_n\}\) of \(\{S_n\}\), and a function \(v\), which satisfies the boundary condition \(v=g- \widetilde S_M\).