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An H1(Ph)-Coercive Discontinuous Galerkin Formulation for the Poisson Problem: 1D Analysis

Discontinuous Galerkin (DG) methods are finite element techniques for the solution of partial differential equations. They allow shape functions which are discontinuous across inter-element edges. In principle, DG methods are ideally suited for hp-adaptivity, as they handle nonconforming meshes and varying-in-space polynomial-degree approximations with ease. Recently, DG formulations for elliptic problems have been put in a general framework of analysis. Although clarifying basic properties, the analysis does not warrant a clear preference. Specifically, none of the conventional DG formulations possesses a bilinear form that is coercive (and continuous) on an infinite-dimensional broken Sobolev space. Rather, bilinear forms are only weakly coercive or defined on subspaces only and employ stabilization parameters that typically increase unboundedly as the subspace is expanded, e.g., if the polynomial degree is increased. For hp-adaptation, coercivity is a fundamental property: By the classical Lax-Milgram theorem, any conforming discretization of a coercive formulation is stable, i.e., discrete approximations are well-posed and have a unique solution, irrespective of the specifics of the underlying approximation space. In this work we consider the one-dimensional Poisson problem and present a generic consistent conventional DG formulation. We show that conventional DG formulations are necessarily noncoercive. Moreover, we presents a new symmetric DG formulation which contains nonconventional edge terms based on element Green's functions and the data local to the edges. We show that the new DG formulation is coercive on H1(Ph), the space of functions that are piecewise in the H1 Sobolev space. Furthermore, we show that the new DG formulation and the classical Galerkin formulation are equivalent, that is, in the infinite-dimensional case they yield the same solution.