The partition-of-unity method for linear diffusion and convection problems: accuracy, stabilization and multiscale interpretation

We investigate the effectiveness of the partition-of-unity method (PUM) for convection-diffusion problems. We show that for the linear diffusion equation, an exponential enrichment function based on an approximation of the analytic solution leads to improved accuracy compared to the standard finite-element method. It is illustrated that this approach can be more efficient than using polynomial enrichment to increase the order of the scheme. We argue that the PUM enrichment, can be interpreted as a subgrid-scale model in a multiscale framework, and that the choice of enrichment function has consequences for the stabilization properties of the method. The exponential enrichment is shown to function as a near optimal subgrid-scale model for linear convection.