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Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations

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Mixed Finite Elements, Compatibility Conditions, and Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1939))

The presented notes review the concept and main results concerning commuting projections and projection-based interpolation operators defined for one-, two- and three-dimensional exact sequences involving the gradient, curl, and divergence operators, and Sobolev spaces. The discrete sequences correspond to polynomial spaces defining the classical, continuous finite elements, the “edge elements” of Nédélec, and “face elements” of Raviart-Thomas. All discussed results extend to the elements of variable order as well as parametric elements. The presentation reproduces results for 2D from [19] and 3D from [17, 25, 20, 13] and attempts to present them in a unified manner for all types of finite elements forming the exact sequences. The idea of the projection-based interpolation for elliptic problems was introduced in [37] and generalized to the exact sequence in [21]. The presented results build on the existence of polynomial preserving extension operators [15, 22].

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Authors and Affiliations

  1. Institute for Computational Engineering and Sciences, The University of Texas at Austin, 78712, Austin, TX, USA

    Leszek F. Demkowicz

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  1. Leszek F. Demkowicz
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Editors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Brescia, Via Valotti 9, 25133, Brescia, Italy

    Lucia Gastaldi

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Demkowicz, L.F. (2008). Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations. In: Boffi, D., Gastaldi, L. (eds) Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics, vol 1939. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78319-0_3

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