Abstract. We present the first a priori error analysis of the h-version of the hybridizable discontinuous Galkerin (HDG) methods applied to convection-dominated diffusion problems. We show that, when using polynomials of degree no greater than k, the L 2 -error of the scalar variable converges with order k + 1/2 on general conforming quasi-uniform simplicial meshes, just as for conventional DG methods. We also show that the method achieves the optimal L 2 -convergence order of k + 1 on special meshes. Moreover, we discuss a new way of implementing the HDG methods for which the spectral condition number of the global matrix is independent of the diffusion coefficient. Numerical experiments are presented which verify our theoretical results.

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SUMMARYWe present the first a priori error analysis for the first hybridizable discontinuous Galerkin method for linear elasticity proposed in Internat. J. Numer. Methods Engrg. 80 (2009), no. 8, 1058-1092. We consider meshes made of polyhedral, shape-regular elements of arbitrary shape and show that, whenever piecewisepolynomial approximations of degree k > 0 are used and the exact solution is smooth enough, the antisymmetric part of the gradient of the displacement converges with order k, the stress and the symmetric part of the gradient of the displacement converge with order k C 1=2, and the displacement converges with order k C 1. We also provide numerical results showing that the orders of convergence are actually sharp.

We develop exact piecewise polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree. An Alfeld split of a simplex is obtained by connecting the vertices of an n-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.Hence, κω ∈ P r+1 Λ k−1 (T ).Suppose that ω = σ a σ dλ σ ∈ P r Λ k (T ) and suppose that f = [τ (0), τ (1), τ (2), . . . , τ (s)] is an ssimplex of T . Then the trace of ω on f is given bywhere tr f a σ := a σ | f is simply the restriction of a σ to f . We say that σ ⊂ τ if {σ(1), . . . , σ(k)} ⊂ {τ (0), τ (1), . . . , τ (s)}.We define the spaceThe following result is contained in [3, Theorem 4.8].We also need a result in the case r = 0. To do this, we first state a result from Arnold et al. [3, Lemma 4.6].Proposition 2.2. Let ω ∈ P r Λ k (T ). Suppose that tr Fi ω = 0, for 1 ≤ i ≤ n and T ω ∧ η, for all η ∈ P r−n+k Λ n−k (T ).Then, ω = 0. In particular, if ω ∈ P 0 Λ k (T ) with k ≤ n − 1 satisfies tr Fi ω = 0 for 1 ≤ i ≤ n, then ω = 0.Lemma 2.3. Define the set of k-simplices that have x 0 as a vertex:

We propose a new tool, which we call M M M-decompositions, for devising superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized-mixed methods for linear elasticity with strongly symmetric approximate stresses on unstructured polygonal/polyhedral meshes. We show that for an HDG method, when its local approximation space admits an M M M-decomposition, optimal convergence of the approximate stress and superconvergence of an element-by-element postprocessing of the displacement field are obtained. The resulting methods are locking-free. Moreover, we explicitly construct approximation spaces that admit M M M-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings.

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