 
Summary: An Error Analysis and the Mesh Independence
Principle for a Nonlinear Collocation Problem
Rakhim Aitbayev
Department of Mathematics, New Mexico Institute of Mining and Technology,
Socorro, New Mexico 87801
Received 31 July 2005; accepted 8 December 2005
Published online in Wiley InterScience (www.interscience.wiley.com).
DOI 10.1002/num.20152
A nonlinear Dirichlet boundary value problem is approximated by an orthogonal spline collocation scheme
using piecewise Hermite bicubic functions. Existence, local uniqueness, and error analysis of the collocation
solution and convergence of Newton's method are studied. The mesh independence principle for the collo
cation problem is proved and used to develop an efficient multilevel solution method. Simple techniques are
applied for estimating certain discretization and iteration constants that are used in the formulation of a mesh
refinement strategy and an efficient multilevel method. Several mesh refinement strategies for solving a test
problem are compared numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 22:
000000, 2006
Keywords: orthogonal spline collocation; nonlinear boundary value problem; Newton's method; mesh
independence principle
I. INTRODUCTION
In this work, we study an orthogonal spline collocation (OSC) approximation of a Dirichlet
