Interval study of convergence in the solution of 1D Burgers by least squares finite element method (LSFEM) + Newton linearization

The vast majority of physical problems is governed, or may be represented by Partial Differential Equations. Some mathematical methods are capable of producing analytical solutions of physical problems, more precisely of heat and mass transfer problems (Arpaci, 1966; Bejan, 1996; Carslaw and Jaeger, 1986), but only of some, and problems very simplified. So it is taken as a fundamental tool for solving heat and mass transfer problems the numerical methods. For decades, numerical methods have been used to solve such problems, among them stand out the finite difference method (Smith, 1971), finite volume (Chung, 2002) and finite element (Lewis et al., 2004; Donea and Huerta, 2003; Reddy, 1993). Since the start of the 50's with Turner et al. (1956), Clough (1960), Argyris (1963), Zienkiewicz and Cheung (1965), Oden and Wellford (1972), the finite element method has been used with great success in various branches of engineering. In Donea and Quartapelle (1992) the authors present a finite element method to solve transient problems governed by linear or nonlinear equations with dominant advective terms. The first is the Generalized Galerkin Method, which provides excellent results due to a correct relationship between the spatial and temporal variations expressed by the theory of characteristics. Beyond that show that the least square method used shows the simplicity of the Taylor-Galerkin method and the unconditional stability of methods of characteristics, however, its accuracy is committed to numbers Courant larger than unity.