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An \(hp\)-adaptive flux-corrected transport algorithm for continuous finite elements

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This paper presents an \(hp\)-adaptive flux-corrected transport algorithm for continuous finite elements. The proposed approach is based on a continuous Galerkin approximation with unconstrained higher-order elements in smooth regions and constrained \(P_1/Q_1\) elements in the neighborhood of steep fronts. Smooth elements are found using a hierarchical smoothness indicator based on discontinuous higher-order reconstructions. A gradient-based error indicator determines the local mesh size \(h\) and polynomial degree \(p\). The discrete maximum principle for linear/bilinear finite elements is enforced using a linearized flux-corrected transport (FCT) algorithm. The same limiting strategy is employed when it comes to constraining the \(L^2\) projection of data from one finite-dimensional space into another. The new algorithm is implemented in the open-source software package Hermes. The use of hierarchical data structures that support arbitrary-level hanging nodes makes the extension of FCT to \(hp\)-FEM relatively straightforward. The accuracy of the proposed method is illustrated by a numerical study for a two-dimensional benchmark problem with a known exact solution.

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Acknowledgments

This research was supported by the German Research Association (DFG) under Grant KU 1530/6-1.

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  1. Applied Mathematics III, University Erlangen-Nuremberg, Cauerstr. 11, 91058 , Erlangen, Germany

    Melanie Bittl & Dmitri Kuzmin

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  1. Melanie Bittl
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  2. Dmitri Kuzmin
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Correspondence to Melanie Bittl.

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Bittl, M., Kuzmin, D. An \(hp\)-adaptive flux-corrected transport algorithm for continuous finite elements. Computing 95 (Suppl 1), 27–48 (2013). https://doi.org/10.1007/s00607-012-0223-y

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