Stabilization of advection dominated problems through a generalized finite element method

Traditional finite element approaches are well-known to introduce spurious oscillations when applied to advection dominated problems. We explore alleviation of this issue from the perspective of a generalized finite element formulation, which enables stabilization of a linear differential operator through enrichments based on fundamental solutions. This is demonstrated through application to steady/unsteady one- and two-dimensional advection-diffusion problems. Here, boundary layer development is efficiently captured using a set of exponential functions derived from fundamental solutions to the problems considered. Results demonstrate smooth, numerical solutions with convergence observed to be in relative agreement with expected convergence rates for elliptic problems. Furthermore, significantly improved error levels are observed compared to traditional finite element formulations. Additional insights in improvements offered by the generalized finite element method are illuminated using a consistent decomposition of the variational multiscale method, enabling comparison with classical stabilized methods.