An adaptive variational multiscale element free Galerkin method based on the residual-based a posteriori error estimators for convection–diffusion–reaction problems

Abstract

As we know, the variational multiscale element free Galerkin (VMEFG) method may still suffer from non-physical oscillations near the boundary or interior layers when solving the convection–diffusion–reaction problems with strong convection-dominated. To overcome this shortcoming, we consider incorporating an adaptive algorithm based on the residual error estimations into the VMEFG, which formed the adaptive VMEFG (AVMEFG) method. With respect to the adaptive technique adopted, two residual-based a posteriori error estimators in the H1-semi norm and energy norm are respectively used to locate and remark the high-gradient numerical solution regions. Several stationary convection–diffusion–reaction problems are solved to verify the effectiveness of the proposed method. Among them, the first example makes a comparison between the proposed method and the adaptive element free Galerkin method, while the other remaining examples compare two different residual-based a posteriori error estimators for the proposed method. Numerical examples illustrate that the proposed method is effective and efficient in solving the convection-dominated problem involving various layers. Moreover, the AVMEFG methods with the two residual-based a posteriori error estimators are generally equivalent except in the case that the solution has an exponential boundary layer, where the energy norm error estimator can achieve significantly better results than the other one.