Volume 12, Number 2, Summer 2004
A SEQUENTIAL SPECTRAL METHOD FOR
NONLINEAR ELLIPTIC
INTEGRODIFFERENTIAL EQUATIONS
M. J. GANDER, A. MANSOORA AND K. K. TAM
Abstract. A sequential spectral method (SSM) is a
spectral method in which the unknown coefficients of the approximate
spectral expansion are computed sequentially, one
at a time, instead of simultaneously, like in a classical spectral method (CSM).
These methods are especially suited for
approximating difficult nonlinear integrodifferential equations
where the solution behavior is largely unknown and the problem
lies in finding a solution of the large scale nonlinear system
of equations for the unknown expansion coefficients. The SSM
leads by iteration to the same solution as the underlying CSM
and thus inherits its spectral accuracy, but it avoids the
simultaneous solution of a large system of nonlinear equations. We
prove convergence of the SSM to the spectral approximation
of the underlying spectral method for a general class of
nonlinear elliptic integrodifferential equations, and illustrate the
effectiveness of the method on two applications.
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