The application of sparse matrix methods to the numerical solution of nonlinear elliptic partial differential equations
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We present a new algorithm for solving general semilinear, elliptic partial differential equations. The algorithm is based on Newton's Method but uses an approximate iterative method to solve the linear systems that arise at each step of Newton's Method. We show that the algorithm can maintain the quadratic convergence of Newton's Method and that it may be substantially faster than other available methods for semilinear or nonlinear partial differential equations.
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