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FEEC-A Report Summary

Project ID: 339643
Funded under: FP7-IDEAS-ERC
Country: Norway

Mid-Term Report Summary - FEEC-A (Finite Element Exterior Calculus and Applications)

The present project is built on the earlier development of finite element exterior calculus (FEEC), mostly done by the PI and his coworkers. The FEEC-theory is a foundation for further research in various directions, including fundamental mathematical research, applications in science and engineering, and the study of more algorithmic aspects. The research plan of the present project describes activities in all these directions.

The results obtained so far can naturally be divided into three research areas. A cornerstone of this project is the construction and analysis of a map refereed to “the bubble transform.” The purpose of this map is to analyze all the natural finite element spaces that arise from a common grid or mesh. In fact, the map decomposes a function in a Sobolev space into a sum of functions of local support, and it has the property that it is invariant on each of the finite element subspaces. This property has a number of consequences, both for the theory and the construction of algorithms. So far we have a well-developed theory for standard second order elliptic operators, while work on extensions of the theory to discretizations of the de Rham complex is ongoing.

Another area where we have made substantial progress is in the design of robust numerical methods for complex materials. Our work has, in particular, been targeted to poroelasticity, where we are in close cooperation with a research group doing medical computations, with special focus on models for deformation of the brain and the spinal cord. For such models, the developments of methods that are robust with respect to large variations in the model parameters are essential. Finally, we have constructed a new family of numerical methods for a large class of problems where standard finite element methods give rise to nonlocal approximations of certain constitutive laws. In various models, for example in flow problems, this is considered to be an undesired effect of the discretization. In contrast, our new approach leads to a systematic construction of methods with local approximations of these constitutive laws.

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