Applied Numerical Mathematics 56 (2006) 1123–1133 www.elsevier.com/locate/apnum An algebraic grid optimization algorithm using condition numbers Costanza Conti a , Rossana Morandi a , Rosa Maria Spitaleri b,∗ a Dipartimento di Energetica “Sergio Stecco”, via C. Lombroso 6/17, 50134 Firenze, Italy b Istituto per le Applicazioni del Calcolo, Viale del Policlinico 137, 00161 Roma, Italy Available online 20 December 2005 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.  2005 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Algebraic grid generation; Algebraic mixed methods; Grid quality measures; Condition numbers; Grid optimization 1. Introduction Often a system of differential equations, depending on spatial physical variables, is not effectively solved on Carte- sian co-ordinates. Thus, to improve the solution computation it requires to be written in curvilinear coordinates, by variable transformation techniques. The accuracy of approximated solutions depends on the quality of the grid over- layed on the definition domain Ω [9,11]. Usually, quality means that the grid should be as regular and orthogonal as possible in order to decrease the truncation errors introduced by nonuniform and skewed grids. Though, for special problems, stretching cells in a selected field direction could result appropriate to carry out successful computational processes. Grid generation methods are not always able to provide grids that satisfy computational needs of the spe- cific problem. Often a posteriori improvements of the computed grids are required by optimization strategies [1,2]. Grid optimization means the improvement of an existing grid to achieve the optimal one respect to given criteria com- ing from the geometry or the physics of the problem to be solved. Basic requested properties for a grid are regularity and orthogonality [2]. Aim of this paper is to propose an a posteriori grid optimization algorithm “working” on a small number of variables to achieve “optimal” algebraic grids. To this purpose, let M be an algebraic method generating a mapping X(ξ,η) * Corresponding author. E-mail addresses: c.conti@ing.unifi.it (C. Conti), morandi@de.unifi.it (R. Morandi), spitaleri@iac.rm.cnr.it (R.M. Spitaleri). 0168-9274/$30.00  2005 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2005.11.001