An ${H^1(\mathcal{P}^{\mathsf{h})}$‐Coercive Discontinuous Galerkin Formulation for the Poisson Problem: 1D Analysis

Abstract

Abstract. Coercivity of the bilinear form in a continuum variational problem is a fundamental property for finite-element discretizations: By the classical Lax-Milgram theorem, any conforming discretization of a coercive variational problem is stable; i.e., discrete approximations are well-posed and possess unique solutions, irrespective of the specifics of the underlying approximation space. Based on the prototypical one-dimensional Poisson problem, we establish in this work that most concurrent discontinuous Galerkin formulations for second-order elliptic problems represent instances of a generic conventional formulation and that this generic formulation is noncoercive. Consequently, all conventional discontinuous Galerkin formulations are a fortiori noncoercive, and typically their well-posedness is contingent on approximation-space-dependent stabilization parameters. Moreover, we present a new symmetric nonconventional discontinuous Galerkin formulation based on element Green's functions and the data local to the edges. We show that the new discontinuous Galerkin formulation is coercive on the broken Sobolev space H 1 (P h ), viz., the space of functions that are elementwise in the H 1 Sobolev space. The coercivity of the new formulation is supported by calculations of discrete inf-sup constants, and numerical results are presented to illustrate the optimal convergence behavior in the energy-norm and in the L 2 (Ω)-norm.Key words. finite element method, discontinuous Galerkin, elliptic problems, coercivity AMS subject classifications. 65N30, 65N12 DOI. 10.1137/05063057X1. Introduction. The recent renewal of interest in discontinuous Galerkin (DG) methods for second-order elliptic boundary value problems can be attributed to twofold reasons. First, DG methods provide robust finite-element discretizations for hyperbolic conservation laws, as the interelement discontinuities enable an extension of Godunov's method for finite-volume methods. However, to extend these techniques to singularly perturbed elliptic problems, an appropriate treatment for the elliptic part of the operator is required. Second, the absence of interelement-continuity constraints renders DG methods ideally suited for hp adaptivity, e.g., based on a posteriori error estimation; see, for instance, [1,8]. A comprehensive overview of the historical development of DG methods is provided in [7].A framework for analyzing DG formulations for elliptic problems has recently been erected in [2]. Although the analysis in [2] clarifies basic properties of the different formulations, it does not seem to warrant a clear preference. The literature on DG methods for elliptic problems is dominated by formulations that possess edge terms composed of linear combinations of the jumps and averages of the test and trial functions and their normal derivatives. That is, denoting by u and v the test and trial functions, and by [[ · ]] and {·} the jump and average of (·) at an interelement edge, these formulations contain terms conforming to {∂ n u} [[v]

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