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Geodesic Finite Elements in Spaces of Zero Curvature

In: Numerical Mathematics and Advanced Applications 2011

Author

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  • O. Sander

    (Freie Universität Berlin)

Abstract

We investigate geodesic finite elements for functions with values in a space of zero curvature, like a torus or the Möbius strip. Unlike in the general case, a closed-form expression for geodesic finite element functions is then available. This simplifies computations, and allows us to prove optimal estimates for the interpolation error in 1d and 2d. We also show the somewhat surprising result that the discretization by Kirchhoff transformation of the Richards equation proposed in Berninger et al. (SIAM J Numer Anal 49(6):2576–2597, 2011) is a discretization by geodesic finite elements in the manifold $$\mathbb{R}$$ with a special metric.

Suggested Citation

  • O. Sander, 2013. "Geodesic Finite Elements in Spaces of Zero Curvature," Springer Books, in: Andrea Cangiani & Ruslan L. Davidchack & Emmanuil Georgoulis & Alexander N. Gorban & Jeremy Levesley (ed.), Numerical Mathematics and Advanced Applications 2011, edition 127, pages 449-457, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-33134-3_48
    DOI: 10.1007/978-3-642-33134-3_48
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