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Volume 7, Issue 3
Evaluating Local Approximations of the L2-Orthogonal Projection Between Non-Nested Finite Element Spaces

Thomas Dickopf & Rolf Krause

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 288-316.

Published online: 2014-07

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  • 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).

  • AMS Subject Headings

65D05, 65F10, 65N30, 65N50, 65N55

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COPYRIGHT: © Global Science Press

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@Article{NMTMA-7-288, author = {}, title = {Evaluating Local Approximations of the L2-Orthogonal Projection Between Non-Nested Finite Element Spaces}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {3}, pages = {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).

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.1218nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5876.html} }
TY - JOUR T1 - Evaluating Local Approximations of the L2-Orthogonal Projection Between Non-Nested Finite Element Spaces JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 288 EP - 316 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.1218nm UR - https://global-sci.org/intro/article_detail/nmtma/5876.html KW - Finite elements, unstructured meshes, non-nested spaces, transfer operators, interpolation, projection. AB -

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).

Thomas Dickopf & Rolf Krause. (2020). Evaluating Local Approximations of the L2-Orthogonal Projection Between Non-Nested Finite Element Spaces. Numerical Mathematics: Theory, Methods and Applications. 7 (3). 288-316. doi:10.4208/nmtma.2014.1218nm
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