The aim of this paper is to solve an elliptic interface problemwith a discontinuous coefficient and a singular source term by the spectral collocation method. First, we develop an algorithm for the elliptic interface problem defined in a rectangular domain with a line interface. By using theGordon-Hall transformation, we generalize it to a domain with a curve boundary and a curve interface. The spectral element collocation method is then employed to complex geometries; that is,we decompose the domain into some nonoverlaping subdomains and the spectral collocation solution is sought in each subdomain. We give some numerical experiments to show efficiency of our algorithm and its spectral convergence.
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