A Quick Tutorial on DG Methods for Elliptic Problems
In this paper we recall a few basic definitions and results concerning the use of DG methods for elliptic problems. As examples we consider the Poisson problem and the linear elasticity problem. A hint on the nearly incompressible case is given, just to s
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Classification 65N12, 65N15, 65N30 Key words. Discontinuous Galerkin, Elliptic problems, Linear elasticity
1. Introduction. The main purpose of this paper is to present the basic features of Discontinuous Galerkin Methods for elliptic problems. We will give some hints on the basic mathematical tools typically used to study and analyze them and on their capability to avoid some common troubles (as the discretization of nearly incompressible materials). We will state approximation properties and show how to derive a-priori estimates. We will also present some possible variants and relationships with other approaches (as mixed or hybrid methods for linear elasticity) that possibly deserve a deeper analysis. The paper is addressed to readers with a more engineering oriented background, and an interest in continuum mechanics, with the idea to help them in getting more familiar with the basic concepts and features of DG methods, that indeed, according to the latest developments, show an interesting potential also in structural problems. Actually, applications of DG methods to other problems, and in particular to hyperbolic problems, conservation laws and the like, started already forty years ago, and are fully developed nowadays (see, e.g., [19, 35]). In this book these applications are discussed at a much higher level (starting from the “parallel” contribution of Chi Wang Shu [36]); this is quite natural, since the interested people are (in general) already acquainted with all the basic instruments and with the applications to the more common problems. Instead, most practitioners in structural engineering and continuum mechanics, so far, are not yet familiar with the use of DG methods, that have been pushed forward mainly by applied mathematicians and more ∗ Istituto Universitario di Studi Superiori (IUSS), Piazza della Vittoria 15, 27100 Pavia, Italy and Department of Mathematics, King Abulaziz University, PO Box 80203, Jeddah 21589, Saudi Arabia, [email protected] † Dipartimento di Matematica, Universit` a di Pavia, Via Ferrata 1, 27100 Pavia, Italy, [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 1, © Springer International Publishing Switzerland 2014
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Franco Brezzi and L. Donatella Marini
“mathematically oriented” engineers. Hence the idea of addressing people who are less familiar with the DG machinery but are interested in trying them on their problems. We will not discuss issues related to a-posteriori estimates and meshadaptivity, a very interesting subject that, however, goes beyond the scope of this paper. For this we refer, for instance, to [30–32]. For the same reason, we will not discuss matters related to the solution of the final linear systems, such as the construction of efficient solvers and preconditioners (see, e.g., [9, 25]). The paper is organized as follows. In Sect. 2 we recall some basic instruments, such a
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