Evaluating Local Approximations of the <em>L</em><sup>2</sup>-Orthogonal Projection Between Non-Nested Finite Element Spaces
Year: 2014
Numerical Mathematics: Theory, Methods and Applications, Vol. 7 (2014), Iss. 3 : pp. 288–316
Abstract
We present quantitative studies of transfer operators between finite element spaces associated with unrelated meshes. Several local approximations of the global $L^2$-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, Clément's quasi-interpolation with different local polynomial degrees the global $L^2$-orthogonal projection, a local $L^2$-quasi-projection via a discrete inner product, and a pseudo-$L^2$-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-$L^2$-projection approximates the actual $L^2$-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).
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2014.1218nm
Numerical Mathematics: Theory, Methods and Applications, Vol. 7 (2014), Iss. 3 : pp. 288–316
Published online: 2014-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 29
Keywords: Finite elements unstructured meshes non-nested spaces transfer operators interpolation projection.
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