In the case of the L1 norm the interpolation error can be globally minimised without the use of quadrature. A local method using successive over-relaxation is derived and found to be efficient for a small number of points but rather slow for bigger problems. A new shooting method approach starts with the left boundary point and the first interior point and successively calculates points for which the interpolation is optimal. This method is found to be slower than equidistribution, but of the same order of convergence with the additional advantage that the error is globally minimised.

An alternative approach of minimising the error between a higher order representation of the function and the piecewise linear interpolation was also analysed, concentrating on piecewise cubic and quadratic representations. Conditions on the uniqueness of the piecewise linear interpolation require a minimum mesh density. This caused problems for the cubic but not the quadratic representation. These higher order representations were then tested on the numerical solution of the advection diffusion problem and found to work well given a sufficient number of points. However, the uniqueness problems now not only affected the piecewise cubic representation but also the piecewise quadratic.

Generally the methods presented do indeed provide alternative approaches. The shooting method is efficient but in its present form only useful for the L1 norm. The higher order interpolation methods are promising but require further analytical analysis to ensure robustness and to incorporate the effects of the global truncation error from the differential equation solution.

Download
Authors' PDF: rk_phd00.pdf

AMOR home