Sine methods for domain decomposition

Sinc basis functions form a desirable basis for solving singular problems via domain decomposition. This is because both the Sinc-Galerkin and sinc-collocation methods converge exponentially, even in the presence of boundary singularities. This suggests the future possibility of combining sinc methods used in the vicinity of singularities with other methods, such as finite difference methods, used in the remainder of the original domain. For these reasons, a thorough investigation of the implementation of sine methods in the context of domain decomposition is the necessary first step. This work deals with sine methods for second-order ordinary differential equations with homogeneous Dirichlet boundary conditions. Both sine-collocation and Sinc-Galerkin methods are presented. The two traditional methods of domain decomposition, overlapping and patching, are described. Thus all the groundwork is laid to readily determine which method is most suited to any given problem. Numerical results are presented for both decomposition methods that exhibit the nearly identical errors achieved whether one uses the sine-collocation method or the Sinc-Galerkin method.