An algebraic grid optimization algorithm using condition numbers

Abstract

In this paper we present an algorithm able to provide geometrically optimal algebraic grids by using condition numbers as quality measures. In fact, the solution of partial differential equations (PDEs) to model complex problems needs an efficient algorithm to generate a good quality grid since better geometrical grid quality is gained, faster accuracy of the numerical solution can be kept. Moving from classical approaches, we derive new measures based on the condition numbers of appropriate cell matrices to control grid uniformity and orthogonality. We assume condition numbers in appropriate norms as building blocks of objective functions to be minimized for grid optimization. This optimization procedure improves the mixed algebraic grid generation method first discussed in [C. Conti, R. Morandi, D. Scaramelli, Using discrete uniformity property in a mixed algebraic method, Appl. Numer. Math. 49 (4) (2004) 355–366. [3]]. The whole algorithm is able to cheaply generate optimal algebraic grids providing optimal location of the control points defining a small set of free parameters in the tensor product of the mixed algebraic method.