Variational multiscale element free Galerkin method for convection-diffusion-reaction equation with small diffusion

Variational Multiscale Element Free Galerkin Method Coupled with Low-Pass Filter for Burgers’ Equation with Small Diffusion

“…In the paper, for the 1D problem, the bubble functions fine scale trial solution and weighting function are given as follows [27]: where is the location of the internal virtual node for the piecewise linear bubble and in the paper = −0.5. For the 2D problem, when the triangle background cells are used and in a reference background cell, the bubble for the fine scale trial solution is a quadratic bubble defined as [21] 1 ( , ) = 27 (1 − − ) .…”

“…Although the VMEFG method can play a role in stability for some moderate Re values [21], but for very large Re values, it will still produce spurious oscillations as we will see in the next section. Previous researches [22][23][24]28] indicate that a low-pass filter can suppress the spurious oscillations; thus in the paper we use the high order low-pass filter to smooth the solutions for very large Re values at each time loop.…”

“…The numerical solution of Burgers' equation at a high Reynolds number (Re ≥ 10 5 ) is very difficult due to the presence of shock [21]. A direct application of the VMEFG method for Re = 10 5 leads to highly oscillatory results as shown in Figure 7.…”

“…In order to avoid the selection of the stabilization parameter, Zhang and his coworkers [18] presented variational multiscale element free Galerkin method (VMEFG) which is free of user-defined stability parameters and a consistent stabilized method. Subsequently, they applied the method to solve Stokes problems [18], water wave problems [19], MHD flows problems [20], and convection-diffusion-reaction problem [21] successfully. But, as far as the solution of Burgers' equation is concerned, for the very small diffusivity coefficients, the VMEFG method also produces the numerical pseudo oscillations as we will see in the paper; therefore, we must find another technique to deal with it.…”

A finite element variational multiscale method for incompressible flow

“…An explicit residual-based a posteriori error estimator [21] for linear elasticity, the method is derived from the variational multiscale theory. The variational multiscale element free Galerkin method [22] is presented to solve convection-diffusion-reaction equation. A variational multiscale approach to model blood flow through arteries is proposed.…”