Claudio Canuto

Adaptivity in High-Order Methods

The design and analysis of adaptive spectral or hp-type finite element discretization methods for
elliptic equations poses formidable challenges, for several reasons.

i) The optimality of the approximation should be assessed with respect to specific sparsity classes in which the best N-term approximation error is allowed to decay exponentially, as opposed to the more familiar classes of algebraic decay which are natural for finite-order methods (we refer to [1] for some representative results in this sense).

ii) The growth of active degrees of freedom during the adaptive iterations may obey a linear (rather than geometric) law, endangering the analysis of optimality (however, see [2] for a more aggressive adaptive strategy).

iii) In hp-FEM, the choice between applying a mesh refinement or a polynomial enrichment is a delicate stage in the adaptive process, since early decisions in one of the two directions should be lately amenable to a correction in order to guarantee the final near-optimality of the adaptive discretization for a prescribed accuracy. The talk will touch these and related issues, building upon recent joint work with R.H. Nochetto, R. Stevenson and M. Verani. In particular, we will describe an abstract framework [3] for adaptive finite element discretizations of hp type (hp-AFEM), which incorporates an hp-near best approximation algorithm recently developed by P. Binev [4]. Several practical realizations of hp-AFEM will be discussed. Particular attention will be devoted to the issue of p-robustness, i.e., the independence from the polynomial degree of the constants involved in the analysis. This requirement suggests, on the one hand, the adoption of appropriate a-posteriori error estimators, such as the so-called equilibrated fluxes considered in [5], and, on the other hand, the use of optimal hp preconditioners, such as the multilevel preconditioners discussed in [6], for the computation of the Galerkin solution at each iteration of the adaptive process.

[1] C. Canuto, R.H. Nochetto and M. Verani, Adaptive Fourier-Galerkin methods, Math. Comput. 83 (2014), 1645–1687.
[2] C. Canuto, R.H. Nochetto, R. Stevenson, and M. Verani, Adaptive spectral Galerkin methods with dynamic marking, SIAM J. Numer. Anal. 54 (2016), 3193–3213.
[3] C. Canuto, R.N. Nochetto, R. Stevenson, and M. Verani, Convergence and optimality of hp-AFEM, Numer. Math. 135 (2017), 1073–1119.
[4] P. Binev, Instance optimality for hp-type approximation. In: Oberwolfach Reports 39 (2013), 14–16.
[5] C. Canuto, R.N. Nochetto, R. Stevenson, M. Verani, On p-robust saturation for hp-AFEM, Comput. Math. Appl. 73 (2017), 2002–2022.
[6] K. Brix, Kolja, M. Campos Pinto, C. Canuto, and W. Dahmen, Multilevel preconditioning of discontinuous Galerkin spectral element methods. Part I: geometrically conforming meshes. IMA J. of Numer. Anal. 35 (2015), 1487–1532.

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