Locally adaptive bubble function enrichment for multiscale finite element methods: Application to convection-diffusion problems

We develop a new class of the multiscale finite element method (MsFEM) to solve the convection-diffusion problems. In the proposed framework, we decompose the solution function space into two parts in MsFEM with locally adaptive bubble function enrichment (LABFE). The first part is the one that the multiscale basis functions can resolve, and the second part is an unresolved part that is taken care of by a set of bubble functions. These bubble functions are defined similarly to construct multiscale basis functions. We exchange the local-global information through updated local boundary conditions for these bubble functions. The new multiscale solution recovered from the solution of global numerical formulation provides feedback for updating the local boundary conditions on each coarse element. As the approach iterates, the quality of MsFEM-LABFE solutions improves since these multiscale basis functions with bubble function enrichment are expected to capture the multiscale feature of the approximate solution more accurately. However, the most expansive part of the algorithm is reconstructing the bubble functions on each coarse element. To reduce the overhead of the bubble function reconstruction, we update the local bubble function only when the intermediate multiscale solution is not well resolved within the region corresponding to the sharp local gradient or the discontinuity of the solution. We illustrate the effectiveness of the proposed method through some numerical examples for convection-diffusion benchmark problems.