BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Construction of bounded cochain projections and their role in the FE exterior calculus Falk, Richard S.


Projection operators which commute with the governing differential operators are key tools for the stability analysis of finite element methods associated to a differential complex. In fact, such projections have been a central feature of the analysis of mixed finite element methods since the beginning of such analysis. However, a key difficulty is that, for most of the standard finite element spaces, the canonical projection operators based on the degrees of freedom require additional smoothness to be well--defined and thus are not bounded on the appropriate function spaces. More recently, bounded commuting projections have been constructed, but these lack a key property of the canonical projections; they are not locally defined. In this talk, we review the ideas behind the construction of bounded projections that commute with the exterior derivative and show how, using local operators defined on overlapping macroelements, it is possible to construct such operators that are also locally defined.

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