Stable Multiscale Discretizations of L2-Differential Complexes

Global simulations over long time scales in climate sciences often require coarse grids due to computational constraints. This leaves dynamically important smaller scales unresolved. Thus the influence of small scale processes has to be taken care of by different means. State-of-the-art dynamical cores represent the influence of subscale processes typically via subscale parametrizations and often employ heuristic coupling of scales. This, however, unfortunately often lacks mathematical consistency. The aim of this work is to improve mathematical consistency of the upscaling process that transfers information from the subgrid to the coarse scales of the dynamical core and to largely extend the idea of adding subgrid correctors to basis functions for scalar and vector valued elements discretizing various function spaces.

Discussing prototypically the issue of weighted Hodge decompositions I will show that standard techniques on coarse meshes fail to find good projections in all parts of a modified de Rham complex if rough data is involved and discuss an idea of how to construct multiscale finite element (MsFEM) correctors to scalar and vector valued finite elements and, further, how to construct stable multiscale element pairings using the theory of finite element exterior calculus (FEEC). This can be seen as a meta-framework that contains the construction of standard MsFEMs [Efendiev2009, Graham2012]. Application examples here comprise porous media, elasticity, and fluid flow as well as electromagnetism in fine-scale and high-contrast media. I will provide the necessary theoretical background in homological algebra and differential geometry, and discuss a scalable MPI based implementation technique suitable for large clusters. Several computational examples will be shown. I may, if time permits, discuss some ideas from homogenisation theory to attack the problem of a proof of accuracy.