Error Control for Discontinuous Galerkin Methods for First Order Hyperbolic Problems
An a posteriori error bound for a first order linear hyperbolic problem, with constant advection coefficient, discretized by the discontinuous Galerkin method is presented. The bound is derived using a suitable reconstruction framework, but it is essentia
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ction. Discontinuous Galerkin (dG) methods for advection problems have gained considerable popularity in the literature since their introduction in 1971 by Reed and Hill [32] (see, e.g., [5, 8, 11, 13–15, 19], the volume [12] and the references therein.) The a priori error for the dG method has been considered in [5, 22, 26, 29, 33]. In [31], it was shown numerically that the dG method suffers from slightly suboptimal rates of convergence with respect to the mesh parameter when the error is measured in the L2 -norm. Optimal error bounds in the L2 -norm for various classes of structured meshes have been shown in [9, 10, 33]. Recently, Burman [6] proposed an a posteriori bound for the case of constant elements under a saturation assumption. Other works dealing with error control of various types for first order hyperbolic problems include [2, 3, 24, 34]. Overall, it seems that this rather interesting problem requires further study. A posteriori error bounds for dG methods for elliptic problems have been considered in [4, 7, 17, 18, 21, 27, 28]. Such bounds are based on suitable recoveries/post-processing theoretical tools of the dG solution. In this short note, we present some recent results regarding the a posteriori error analysis of the classical dG method of Reed and Hill [32] for a
∗ Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK, [email protected] † School of Mathematical Sciences University of Nottingham, University Park, Nottingham, NG7 2RD, UK, [email protected] ‡ Department of Applied Mathematics, University of Crete, L. Knosou GR 71409, Heraklion-Crete, Greece. Current Address: School of Mathematical and Physical Sciences, University of Sussex, BN1 9QH, UK, [email protected]
X. Feng et al. (eds.), Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, The IMA Volumes in Mathematics and its Applications 157, DOI 10.1007/978-3-319-01818-8 8, © Springer International Publishing Switzerland 2014
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E. H. Georgoulis, E. Hall, and C. Makridakis
scalar first order linear hyperbolic problem. For simplicity, the special case of a two-dimensional computational domain with constant advection fields is considered. We first consider triangular meshes having one characteristic face per element, where the analysis is simpler. The analysis will be extended to the general case in a forthcoming work; here, we briefly discuss the main ideas for this case also. The rest of this work is structured as follows. In Sect. 2, we describe the model problem considered, along with its discretization by the discontinuous Galerkin method. Section 3 is devoted to the derivation of energy norm a posteriori bounds for the discontinuous Galerkin method, under the assumption of a mesh containing one characteristic face per element. A brief discussion on relaxing the characteristic face assumption is given in Sect. 4. We conclude with some numerical experiments in Sect. 5. 2. The Problem and Its Discretization. We start by assuming the
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