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hp Adaptive finite elements based on derivative recovery and superconvergence

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Computing and Visualization in Science

Abstract

In this paper, we present a new approach to hp-adaptive finite element methods. Our a posteriori error estimates and hp-refinement indicator are inspired by the work on gradient/derivative recovery of Bank and Xu (SIAM J Numer Anal 41:2294–2312, 2003; SIAM J Numer Anal 41:2313–2332, 2003). For element τ of degree p, R(∂ p u hp ), the (piece-wise linear) recovered function of ∂ p u is used to approximate \({|\varepsilon|_{1,\tau} = |\hat{u}_{p+1} - u_{p}|_{1,\tau}}\) , which serves as our local error indicator. Under sufficient conditions on the smoothness of u, it can be shown that \({\|{\partial^{p}(\hat{u}_{p+1} - u_{p})\|_{0,\Omega}}}\) is a superconvergent approximation of \({\|(I - R){\partial}^p u_{hp}\|_{0,\Omega}}\) . Based on this, we develop a heuristic hp-refinement indicator based on the ratio between the two quantities on each element. Also in this work, we introduce nodal basis functions for special elements where the polynomial degree along edges is allowed to be different from the overall element degree. Several numerical examples are provided to show the effectiveness of our approach.

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Authors and Affiliations

  1. Department of Mathematics, University of California, La Jolla, SanDiego, CA, 92093-0112, USA

    Randolph E. Bank

  2. Department of Computer Science, University of California, Davis, Davis, CA, 95616, USA

    Hieu Nguyen

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  1. Randolph E. Bank
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  2. Hieu Nguyen
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Correspondence to Hieu Nguyen.

Additional information

Communicated by: Gabriel Wittum.

Randolph E. Bank work was supported by the National Science Foundation under contract DMS-0915220.

Hieu Nguyen work was supported by the National Science Foundation under contract DMS-0915220 and a grant from the Vietnam Education Foundation (VEF).

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Bank, R.E., Nguyen, H. hp Adaptive finite elements based on derivative recovery and superconvergence. Comput. Visual Sci. 14, 287–299 (2011). https://doi.org/10.1007/s00791-012-0179-7

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