On Hanging Node Constraints for Nonconforming Finite Elements using the Douglas-Santos-Sheen-Ye Element as an Example

On adaptively refined quadrilateral or hexahedral meshes, one usually employs constraints on degrees of freedom to deal with hanging nodes. How these constraints are constructed is relatively straightforward for conforming finite element methods: The constraints are used to ensure that the discrete solution space remains a subspace of the continuous space. On the other hand, for nonconforming methods, this guiding principle is not available and one needs other ways of ensuring that the discrete space has desirable properties. In this paper, we investigate how one would construct hanging node constraints for nonconforming elements, using the Douglas–Santos–Sheen–Ye (DSSY) element as a prototypical case. We identify three possible strategies, two of which lead to provably convergent schemes with different properties. For both of these, we show that the structure of the constraints differs qualitatively from the way constraints are usually dealt with in the conforming case.