An H1(Ph)-coercive discontinuous Galerkin formulation for the Poisson problem: 1D analysis

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 $H^1(\mathcal{P}^{\mathsf{h}})$, 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 $L_2(\Omega)$-norm