CONDITIONS FOR SUPERCONVERGENCE OF HDG METHODS FOR SECOND-ORDER ELLIPTIC PROBLEMS BERNARDO COCKBURN * AND KE SHI † Abstract. We provide a projection-based analysis of a large class of finite element methods for second order elliptic problems. They include the hybridized version of well known mixed meth- ods as well as old and new hybridizable discontinuous Galerkin methods. The main feature of this unifying approach is that it reduces the main difficulty of the analysis to the verification of some properties of an auxiliary, locally defined projection and of the local spaces defining the methods. Sufficient conditions for the optimal convergence of the approximate flux and the superconvergence of an element-by-element postprocessing of the scalar variable are obtained. New hybridizable dis- continuous Galerkin methods with these properties are devised which are defined on squares and cubes. 1. Introduction. In this paper, we propose a projection-based a priori error analysis of finite element methods for second-order elliptic problems. The analysis is unifying because it applies to a large class of methods including the hybridized version of most well known mixed methods as well as several hybridizable discontin- uous Galerkin (HDG) methods. The novelty of the approach is that it reduces the whole error analysis to the element-by-element construction of an auxiliary projection satisfying certain orthogonality and approximation properties, and to the verification of very simple inclusion properties of the local spaces defining the methods. For the sake of simplicity, we present our approach in the framework of the following diffusion problem: q + ∇u =0 in Ω, (1.1a) ∇· q = f in Ω (1.1b) u = g on ∂ Ω. (1.1c) Here Ω ∈ R n (n =2, 3) is a bounded polyhedral domain, f ∈ L 2 (Ω) and g ∈ H 1 2 (∂ Ω). Two ideas led to this approach. The first is that many mixed methods, including the method of Raviart-Thomas (RT), [15, 11, 1], Brezzi-Douglas-Marini, [5], and Brezzi-Douglas-Fortin-Marini, [4], were analyzed by using suitably defined auxiliary projections; see also [6]. The second is that both mixed and HDG methods can be seen as particular cases of a single, general numerical method uncovered in [9]. This suggested the possibility of using a similar projection-based approach to analyze HDG methods. Recently, this was actually achieved, first for a particular case of the so-called local discontinuous Galerkin hybridizable (LDG-H) methods (defined on simplexes) in [8], and then for the whole family of those methods in [10]. In this paper, we continue this effort and show that a single error analysis of many of the methods fitting in the general framework proposed in [9] can be realized. To better describe our results, let us begin by introducing the general form of the methods under consideration; we follow [9]. Let Ω h := {K} denote a conforming triangulation of Ω, where K is a polyhedral element. Let E h denote the set of all faces * School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA, email: cockburn@math.umn.edu. Supported in part by the National Science Foundation (Grant DMS- 0712955) and by the University of Minnesota Supercomputing Institute. † School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA, email: shixx075@math.umn.edu. 1