Evaluating Local Approximations of the L 2 -Orthogonal Projection Between Non-Nested Finite Element Spaces

Abstract

We present quantitative studies of transfer operators between finite element spaces associated with unrelated meshes. Several local approximations of the global L2-orthogonal projection are reviewed and evaluated computationally. The numerical studies in 3D provide the first estimates of the quantitative differences between a range of transfer operators between non-nested finite element spaces. We consider the standard finite element interpolation, Clement’s quasi-interpolation with different local polynomial degrees, the global L2-orthogonal projection, a local L2-quasi-projection via a discrete inner product, and a pseudo-L2-projection defined by a Petrov-Galerkin variational equation with a discontinuous test space. Understanding their qualitative and quantitative behaviors in this computational way is interesting per se; it could also be relevant in the context of discretization and solution techniques which make use of different non-nested meshes. It turns out that the pseudo-L2-projection approximates the actual L2-orthogonal projection best. The obtained results seem to be largely independent of the underlying computational domain; this is demonstrated by four examples (ball, cylinder, half torus and Stanford Bunny).