Iterative solution of elliptic problems by approximate factorization (Q1375450)

An iterative method for the numerical solution of singularly perturbed linear elliptic problems of second order is developed. The basis of the method is a defect correction iteration, but in the present work the operator, approximating the original differential one, is generated by a formal asymptotic factorization into two first-order operators. The resulting approximate operator is easily inverted numerically by solving a sequence of initial value problems. The algorithm combines the best of the asymptotic and the numerical approaches. It posseses a fast rate of convergence and is a stable one. Convergence analysis of both the continuous and the discrete iteration in one-dimensional case is presented. The scheme is also extended to a class of two-dimensional singularly perturbed elliptic problems. The method is demonstrated numerically on one- and two-dimensional model problems.