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Definition of Different Types of Finite Elements

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Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations

Part of the book series: Scientific Computation ((SCIENTCOMP))

Abstract

This chapter presents a wide variety of finite elements of different shapes (quadrilaterals, hexahedra, triangles, tetrahedra, pyramids and wedges) useful for the numerical resolution of wave equations. More precisely, the H1, H(curl) and H(div) conforming finite elements are described in details by focusing on their spectral version which induces the important concept of mass-lumping for quadrilaterals and hexahedra. This concept enables us to construct performant algorithms.

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Notes

  1. 1.

    The inverse of a n-diagonal matrix is not n-diagonal in general, which avoids the storage of \(M_{1,r}^{-1}\). The correct approach would be to invert \(M_{1,r}\) at each time-step by an iterative method. Since \(\varDelta t\) is generally small, moving from \(t_n\) to \(t_{n+1}\) requires few iterations. However, each iteration is expensive compared to the product by a diagonal matrix.

  2. 2.

    But, of course, not plane faces for hexahedra, unless the summits of a face are coplanar.

  3. 3.

    Unless all the faces are planes, which provides a linear mapping.

  4. 4.

    This difficulty is also troublesome for time-harmonic problems.

  5. 5.

    \(P_{1}\) is naturally mass-lumped by using the Trapezoidal rule for triangles and tetrahedra.

  6. 6.

    Unless by splitting tetrahedra into four hexahedra, which provides a very distorted mesh. This distortion leads to very bad performance in terms of accuracy and stability.

  7. 7.

    One must keep in mind that, besides the fact that the faces of tetrahedra lean on three edges and those of hexahedra on four, the first faces are plane while the second ones are not always plane. Thereby, even by sticking the edges of two triangular faces of two tetrahedra to the four edges of the quadrangular face of a an hexahedron, we get overlapping or underlapping of the elements. For this reason, pyramids and wedge are useful interfaces between tetrahedra and hexahedra.

  8. 8.

    More precisely, the use of polynomial functions does not keep the plane character of the triangular faces when one transforms a reference pyramid into a pyramid of any shape.

  9. 9.

    The (not obvious) H(curl)-conform character of this mapping for hexahedra is proven in the Annex of [8].

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Authors and Affiliations

  1. Inria, Saclay, France

    Gary Cohen

  2. The French Aerospace Lab, ONERA, Toulouse, France

    Sébastien Pernet

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Cohen, G., Pernet, S. (2017). Definition of Different Types of Finite Elements. In: Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7761-2_2

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  • DOI: https://doi.org/10.1007/978-94-017-7761-2_2

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