Abstract
We design adaptive high-order Galerkin methods for the solution of linear elliptic problems and study their performance. We first consider adaptive Fourier-Galerkin methods and Legendre-Galerkin methods, which offer unlimited approximation power only restricted by solution and data regularity. Their analysis of convergence and optimality properties reveals a sparsity degradation for Gevrey classes. We next turn our attention to the h p-version of the finite element method, design an adaptive scheme which hinges on a recent algorithm by P. Binev for adaptive h p-approximation, and discuss its optimality properties.
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Acknowledgements
The authors “Claudio Canuto” and “Marco Verani” were partially supported by the Italian national grant PRIN 2012HBLYE4. The author “Ricardo H. Nochetto” was partially supported by NSF grants DMS-1109325 and DMS-1411808.
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Canuto, C., Nochetto, R.H., Stevenson, R., Verani, M. (2015). High-Order Adaptive Galerkin Methods. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_4
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DOI: https://doi.org/10.1007/978-3-319-19800-2_4
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