A new upwind function in stabilized finite element formulations, using linear and quadratic elements for scalar convection–diffusion problems

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.…”

“…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.…”

Accurate FIC-FEM formulation for the multidimensional steady-state advection–diffusion–absorption equation

“…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 [19,[23][24][25][26][27][28][29]. A state of the art review of these and several other shock-capturing methods for the convection-diffusion equations was done in [30].…”

“…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 [23], the methods presented in [31] and those that take the CAU method as the starting point [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.…”