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Summary: ORTHOGONAL SPLINE COLLOCATION FOR NONLINEAR
DIRICHLET PROBLEMS
RAKHIM AITBAYEV AND BERNARD BIALECKI
SIAM J. NUMER. ANAL. c 2000 Society for Industrial and Applied Mathematics
Vol. 38, No. 5, pp. 15821602
Abstract. We study the orthogonal spline collocation (OSC) solution of a homogeneous Dirich-
let boundary value problem in a rectangle for a general nonlinear elliptic partial differential equation.
The approximate solution is sought in the space of Hermite bicubic splines. We prove local existence
and uniqueness of the OSC solution, obtain optimal order H1 and H2 error estimates, and prove the
quadratic convergence of Newton's method for solving the OSC problem.
Key words. orthogonal spline collocation, Dirichlet problem, nonlinear, existence, uniqueness,
error estimates, Newton's method
AMS subject classifications. 65N35, 65J15, 65N15
PII. S0036142999354538
1. Introduction. The orthogonal spline collocation (OSC) method for the so-
lution of nonlinear one-dimensional boundary value problems (BVPs) was introduced
by de Boor and Swartz [6]. An extensive survey of spline collocation methods for
solving partial differential equations is given in [5]. In comparison to finite element
Galerkin methods, collocation methods do not involve integral approximations in the
computation of the coefficients of the resulting algebraic equations. Moreover, the
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