A new stabilized finite element formulation for scalar convection–diffusion problems: the streamline and approximate upwind/Petrov–Galerkin method

A high-resolution Petrov–Galerkin method for the 1D convection–diffusion–reaction problem

“…This feature does carry over to all the methods that have the shock-capturing term similar to that in the CAU method viz. the methods presented in [21], [25], [26], [24]. Unfortunately as pointed out in [29] and in Section 5.7.1 of this paper, these methods are often over diffusive.…”

“…Perturbation ðp h Þ Remarks [22] [25] s½ku þ ð1 À kÞu r Á $w h k is a smoothness measure Mod.CAU [22] FIC [13] h…”

“…Nevertheless nonregular solutions continue to exhibit the Gibbs and dispersive oscillations. Several shock-capturing nonlinear Petrov-Galerkin methods were proposed to control the Gibbs oscillations observed across characteristic internal/boundary layers for the convection-diffusion problem [17][18][19][20][21][22][23][24][25][26][27][28]. A thorough review, comparison and state of the art of these and several other shock-capturing methods for the convection-diffusion equations, therein named as spurious oscillations at layers diminishing methods, was done in [29].…”

“…Reactive terms were not considered in the design of these methods and hence they fail to control the localized oscillations in the presence of these terms. Exceptions to this are the consistent approximate upwind (CAU) method [19], the methods presented in [24] and those that take the CAU method as the starting point [21,25,26]. Nevertheless the expressions for the stabilization parameters therein were never optimized for reactive instability and often the solutions are over-diffusive in these cases.…”

Solución Numérica De Una Ecuación Del Tipo Advección-Difusión

“…Sobre este tema se ha realizado mucho trabajo de investigación; por ejemplo, se han propuesto métodos de Petrov-Galerkin para el control de las oscilaciones de Gibbs observadas a lo largo de capas límite internas para el problema de advección-difusión. Véase por ejemplo, Mitsukami et al (1985), Codina (1993), De Sampaio et al (2001), Do Camo et al (2003, Knobloch (2006), Oñate et al, (2006). Un estudio sobre la situación actual de métodos para el control de oscilaciones espurias fue hecho en John V. et al (2007).…”