An Analysis of some Properties of Class of Elliptic Partial Differential Operators: a Solution of some Problems: A Fast and Robust Method for Solving Sparse Linear Systems Arising from Elliptic Partial Differential Equations

Abstract

We describe a fast and robust method for solving thelarge sparse linear systems that arise upon the discretization of ellipticpartial differential equations such as Laplace's equation and the Helmholtzequation at low frequencies. While most existing fast schemes for this taskrely on so called \iterative" solvers, the method described here solvesthe linear sys- tem directly We prove Schwarz-Pick type estimates andcoefficient estimates for a class of functions induced by the elliptic partialdifferential operators. Then we apply these results to obtain a Landau type theorem.