Iterative solution of elliptic problems by approximate factorization

An iterative method for the numerical solution of singularly perturbed second-order linear elliptic problems is presented. It is a defect correction iteration in which the approximate operator is the product of two first-order operators, which is readily inverted numerically. The approximate operator is generated by formal asymptotic factorization of the original operator. Hence this is a QUasi Analytic Defect correction iteration (QUAD). Both its continuous and discrete versions are analyzed in one dimension. The scheme is extended to a variety of two dimensional operators and it is analyzed for a model advection-diffusion equation. Numerical calculations show the effectiveness of the scheme over a wide range of values of the small parameter.