An Adaptive hp–DG–FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case

Communications on Applied Mathematics and Computation (2019) 1:309–331 https://doi.org/10.1007/s42967-019-00026-9 ORIGINAL PAPER An Adaptive hp–DG–FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case Paola Antonietti1 · Claudio Canuto2 · Marco Verani1 Received: 31 May 2018 / Revised: 30 December 2018 / Accepted: 14 January 2019 / Published online: 29 July 2019 © Shanghai University 2019 Abstract We propose and analyze an hp-adaptive DG–FEM algorithm, termed hp-ADFEM, and its one-dimensional realization, which is convergent, instance optimal, and h- and p-robust. The procedure consists of iterating two routines: one hinges on Binev’s algorithm for the adaptive hp-approximation of a given function, and finds a near-best hp-approximation of the current discrete solution and data to a desired accuracy; the other one improves the discrete solution to a finer but comparable accuracy, by iteratively applying Dörfler marking and h refinement. Keywords Elliptic problem · Discontinuous Galerkin method · hp-adaptivity · Convergence and optimality Mathematics Subject Classification 65N30 · 65N50 1 Introduction The design and analysis of adaptive hp-type finite-element methods for elliptic problems is significantly more challenging than it is for h-type methods. Indeed, as demonstrated, e.g., by some examples given in [6, Sect.1], one should include in the adaptive procedure the possibility of stepping back from an early choice between h-refinement and p-enrichment: while the true structure of the solution reveals itself along the iterations, one should be able to re-distribute the allocated degrees of freedom between hand p-resolutions. The existence of (rather) pathological situations has not prevented * Claudio Canuto Paola Antonietti Marco Verani 1 MOX‑Dipartimento di Matematica, Politecnico di Milano, P.zza Leonardo Da Vinci 32, 20133 Milan, Italy 2 Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy 13 Vol.:(0123456789) 310 Communications on Applied Mathematics and Computation (2019) 1:309–331 the development of practical hp-adaptive algorithms that work (see, e.g., [9] and the references therein), but in most cases, these procedures are not supported by a sound mathematical theory, which assesses the optimality, and even the convergence, of the method (unless a priori assumptions on the structure of the solution are made). The crucial issue is an approximation problem: how can we build an hp-finite-element space with a minimal dimension in which a given function can be approximated with a prescribed accuracy? A constructive answer to this question has been given by Binev in the past few years (see [5]), who designed a greedy hp-algorithm, which is incremental with respect to the dimension and has instance optimality properties (see Sect. 2.3). With a good answer to such an approximation problem, one may think of recursively applying the hp-adaptive algorithm to a sequence of Galerkin discrete solutions of the elliptic problem, built in a way to get closer and closer to the exact solution. This idea has been implemented in [6], where a general framework for adaptive hp discretizations has been devised, and an adaptive algorithm termed hp-AFEM has been proposed, which guarantees convergence and instance optimality of the sequence of generated Galerkin solutions. The algorithm is both h- and p optimal in one space dimension, whereas in higher dimensions, p-robustness is lost, partly due to the need of going from the non-conforming meshes produced by Binev’s algorithm to the conforming ones needed in a continuous Galerkin method, and partly due to the use of a residual-based error estimator (the latter obstruction may be removed by resorting to equilibrated flux estimators, as in [7]). Since Binev’s algorithm produces non-conforming meshes and discontinuous approximations, it is quite natural to associate with it a discontinuous, rather than a continuous, Galerkin discretization of the elliptic problem. The purpose of this paper is to take a step forward in this direction. In particular, hereafter, we propose an hp-adaptive DG-FEM algorithm, termed hp-ADFEM, and a realization of it in one space dimension which is convergent, instance optimal, and h- and p-robust. We do not require any restrictions on the relative size of neighboring elements, nor on the polynomial degrees used on them. In building a discrete solution that matches a prescribed accuracy, we extend the approach developed in [4] for h-type DG methods to the hp-case, using in the analysis several results on hp-type a posteriori error estimators (see, e.g., [8] and the references therein). The multi-dimensional case is currently under investigation [1]; while our general convergence theorem holds in any dimension, proving p-robustness seems to require a grading property in the distribution of polynomial degrees over the partition, which is not guaranteed by the algorithm proposed in [5]. The remainder of the paper is organized as follows. In Sect. 2, we introduce our general framework for the hp-approximation of a given function, and we present the Binev algorithm. Section 3 describes the hp-DG discretizations that we consider, and collects some of their properties. Section 4 contains the general convergence and instance optimality results, based on the concatenation of Binev’s algorithm and a procedure to compute DG solutions with polynomial data, matching a prescribed tolerance. Finally, in Sect. 5, we illustrate a possible realization of this procedure based on the classical SOLVE → ESTIMATE → MARK → REFINE paradigm. The following notation is used throughout the paper. By A ≲ B , we mean that A can be bounded by a multiple of B, independently of the parameters on which A and B may depend. Similarly, A ≃ B is defined as both A ≲ B and B ≲ A. 13 311 Communications on Applied Mathematics and Computation (2019) 1:309–331 2  hp‑Partitions and hp‑Approximations Let Ω be a bounded open interval of the real line. In view of the hp-adaptive discretization of a boundary-value problem therein, we introduce some notation concerning partitions in Ω and function spaces built on them. 2.1 Partitions of the Domain ̄ into finitely many We assume that we are given an essentially disjoint initial partition K0 of Ω closed subintervals, which are the initial geometric elements; the initial subdivision may depend upon the data of the problem at hand. Then, we apply subsequent dyadic subdivisions, by halving each element K that we encounter into two closed subintervals K ′ and K ′′ of equal size, the “children” of K, such that K = K � ∪ K �� and |K � ∩ K �� | = 0. The set 𝔎 of all these geometric elements forms an infinite binary “master tree”, having as its roots the elements of ̄ . A subtree of the master tree is a finite subset of 𝔎 that contains all the initial partition of Ω roots and for each element in the subset both its parent a (...truncated)