Multiscale methods in computational fluid and solid mechanics

The basic idea of multiscale methods, namely the decomposition of a problem into a coarse scale and a fine scale, has in an intuitive manner been used in engineering for many decades, if not for centuries. Also in computational science, large-scale problems have been solved, and local data, for instance displacements, forces or velocities, have been used as boundary conditions for the resolution of more detail in a part of the problem. Recent years have witnessed the development of multiscale methods in computational science, which strive at coupling fine scales and coarse scales in a more systematic manner. Having made a rigorous decomposition of the problem into fine scales and coarse scales, various approaches exist, which essentially only differ in how to couple the fine scales to the coarse scale. The Variational Multiscale Method is a most promising member of this family, but for instance, multigrid methods can also be classified as multiscale methods. The same conjecture can be substantiated for hp-adaptive methods. In this lecture we will give a succinct taxonomy of various multiscale methods. Next, we will briefly review the Variational Multiscale Method and we will propose a space-time VMS formulation for the compressible Navier-Stokes equations. The spatial discretization corresponds to a high-order continuous Galerkin method, which due to its hierarchical nature provides a natural framework for `a priori' scale separation. The latter property is crucial. The method is formulated to support both continuous and discontinuous discretizations in time. Results will be presented from the application of the method to the computation of turbulent channel flow. Finally, multigrid methods will be applied to fluid-structure interaction problems. The basic iterative method for fluid-structure interaction problems employs defect correction. The latter provides a suitable smoother for a multigrid process, although in itself the associated subiteration process converges slowly. Indeed, the smoothed error can be represented accurately on a coarse mesh, which results in an effective coarse-grid correction. It is noted that an efficient solution strategy is made possible by virtue of the relative compactness of the displacement-to-pressure operator in the fluid-structure interaction problem. This relative compactness manifests the difference in length and time scales in the fluid and the structure and, in this sense, the multigrid method exploits the inherent multiscale character of fluid-structure-interaction problems.