A Review of some Developments in Collocation Algorithms

This paper reviews some developments in error estimation and mesh selection for collocation methods for ordinary differential equations. The basic idea of collocation has great generality and simplicity. Given some operator equation we look for an approximate solution in the form of a linear combination of some fixed basis functions. The coefficients in the linear combination are found by substitution into the equation, then by satisfying any boundary conditions and the differential equation at certain ‘collocation’ points. The number of points is chosen so that the number of equations is equal to the number of unknowns. The idea applies to a wide range of operator equations, but the discussion here will be confined to ordinary differential equations, mainly higher order boundary value problems. One particular attraction is the ease with which boundary conditions involving higher derivatives can be dealt with. For differential equations, unlike integral equations, collocation leads directly to a set of algebraic equations for the parameters. The most common sets of functions used are global or piecewise polynomials. These are easily differentiated so that substitution in the equations is easy. Some care is needed in the choice of collocation points if an effective algorithm is to be obtained. The aim of this paper is to review error estimation for these methods and mesh selection for piecewise polynomial collocation. Some mesh selection criteria which take into account the way the error arises are introduced. These criteria are, however, expensive, and in spite of their good motivation they do not seem to produce significantly better distributions than simpler criteria, so