Stabilized finite element methods with shock capturing for advection–diffusion problems

Combined finite volume and finite element method for convection-diffusion-reaction equation

“…Such equations represent the hyperbolic conservation law for which their solutions always contain discontinuity and high gradient; thus accurate numerical solutions are very difficult to obtain. Special treatment must be applied to suppress spurious oscillations of the computed solutions for both the pure convection and convection-dominated problems [5][6][7][8]. At present, better ways to approximate the convection term are still needed, and thus development of accurate numerical modeling for the convection-diffusion equations remains a challenging task in computational fluid dynamics [1][2][3][4][5][6][7][8][9][10][11].…”

An adaptive stabilized finite element scheme for a water quality model

“…Using these results, Sangalli has analyzed a residual a posteriori error estimate for the residual-free bubbles scheme (see [15]). On the other hand, Knop et al have developed some a posteriori error estimates using a stabilized scheme combined with a shock capture technique to control the local oscillations in the crosswind direction (see [13]). Finally, Wang has introduced an error estimate for the advection-diffusion equation based on the solution of local problems on each element of the triangulation (see [19]).…”

An adaptive stabilized finite element scheme for the advection–reaction–diffusion equation

“…Using these results Sangalli has analyzed a residual a posteriori error estimate for the residual-free bubbles scheme (see [10]). On the other hand, Knopp et al have developed some a posteriori error estimates using a stabilized scheme combined with a shock-capturing technique to control the local oscillations in the crosswind direction (see [8]). Finally, Wang has introduced an error estimate for the advection-diffusion equation based on the solution of local problems on each element of the triangulation (see [15]).…”