Improved Multiscale Finite Element Methods for Advection-Dominated Problems

  • Biezemans, Rutger (École Nationale des Ponts et Chaussées)

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The multiscale finite element method (MsFEM) is a finite element (FE) approach that allows to solve partial differential equations (PDEs) posed in heterogeneous media on a coarse mesh, i.e. a mesh with elements of size much larger than the characteristic scale of the heterogeneities. To do so, MsFEMs use pre-computed basis functions, adapted to the differential operator, and that take into account the small scales of the problem. When the PDE contains dominating advection terms, naive finite element approximations lead to spurious oscillations, even in the absence of oscillatory coefficients. Stabilization techniques are to be adopted. In spite of various proposals to combine multiscale and stabilization methods, a universally best method has not yet been identified. The main contribution of this talk will be the addition of suitable bubble functions to the approximation space of previously studied MsFEMs. The extent to which an MsFEM is stabilized by these bubbles turns out to vary greatly from one MsFEM variant to another. In particular, we will build on an adaptation of the Crouzeix-Raviart finite element in the MsFEM-spirit, for which the stabilization by bubble functions is much more effective than for other MsFEM variants. Differences in the performance of various methods will be illustrated by numerical experiments.