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Spectral Element Discretization of Optimal Control Problems

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Spectral and High Order Methods for Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 76))

Abstract

In this work we consider the numerical solution of a distributed optimal control problem associated with an elliptic partial differential equation. We approximate the optimality system by the spectral element method and derive a posteriori error estimates with respect to the cost functional. Then we use an hN adaptive refinement technique to reduce this error: the error indicator is used to mark what elements must be refined. The choice between an h or N refinement is based on the use of a predicted error reduction algorithm. Numerical results show the way this algorithm works.

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Acknowledgements

We thank Dr. Luca Dedè for helpful discussions and comments. We acknowledge the support of the Italian MIUR through the project COFIN07 “Mathematical and numerical modelling for cardiovascular and fluid dynamics applications.”

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Authors and Affiliations

  1. MOX-Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Piazza L. da Vinci 32, 20133, Milano, Italy

    Loredana Gaudio & Alfio Quarteroni

  2. Chaire de Modelisation et Calcul Scientifique (CMCS), École Polytechnique Fédérale de Lausanne (EPFL), FSB, Station 8, 1015, Lausanne, Switzerland

    Alfio Quarteroni

Authors
  1. Loredana Gaudio
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  2. Alfio Quarteroni
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Correspondence to Loredana Gaudio .

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Editors and Affiliations

  1. , Division of Applied Mathematics, Brown University, George Street 182, Providence, 02912, Rhode Island, USA

    Jan S. Hesthaven

  2. , Department of Mathematical Sciences, Norwegian University of Science and Tech, Trondheim, 7491, Norway

    Einar M. Rønquist

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Gaudio, L., Quarteroni, A. (2011). Spectral Element Discretization of Optimal Control Problems. In: Hesthaven, J., Rønquist, E. (eds) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 76. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15337-2_37

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