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We present the first a priori error analysis of a new method proposed in Cockburn & Wang (2017, Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Comput. Sci., 73, 644–666), for computing adjoint-based, super-convergent Galerkin approximations of linear functionals. If $J(u)$ is a smooth linear functional, where $u$ is the solution of a steady-state diffusion problem, the standard approximation $J(u_h)$ converges with order $h^{2k+1}$, where $u_h$ is the Hybridizable Discontinuous Galerkin approximation to $u$ with polynomials of degree $k>0$. In contrast, numerical experiments show that the new method provides an approximation that converges with order $h^{4k}$, and can be computed by only using twice the computational effort needed to compute $J(u_h)$. Here, we put these experimental results in firm mathematical ground. We also display numerical experiments devised to explore the convergence properties of the method in cases not covered by the theory, in particular, when the solution $u$ or the functional $J(\cdot )$ are not very smooth. We end by indicating how to extend these results to the case of general Galerkin methods.

We analyze a technique to improve the spatial accuracy, by the single application at the end of the simulation of a local post-processing, for pure Lagrange–Galerkin (PLG) methods applied to evolutionary convection-diffusion (possibly pure convection/diffusion) equations with time-dependent domains. The post-processing technique is based on a simple convolution that extracts the ‘hidden accuracy’ of Galerkin schemes, and it is used and rigorously analyzed in a fully discrete context. We prove that, when applied to the numerical solution of PLG schemes, it improves the spatial accuracy in the $l^{\infty }(L^2(\varOmega ^0))$-norm from order $k+1$ to at least order $2k$, where $k$ is the degree of the polynomials defining the finite element space and $\varOmega ^0$ any interior region of the computational domain meshed with translation-invariant meshes. For pure convection, a spatial accuracy enhancement in the $l^{\infty }(L^2(\varOmega ^0))$-norm from order $k+1$ to order $2k+2$ is obtained by post-processing the numerical solution of PLG schemes. Numerical tests are presented that confirm these theoretical results.

We propose a new tool, which we call $M$-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 $M$-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 $M$-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings., 45 pages, 2 figures

We analyze the convergence of a multigrid algorithm for the Hybridizable Discontinuous Galerkin (HDG) method for diffusion problems. We prove that a non-nested multigrid V-cycle, with a single smoothing step per level, converges at a mesh independent rate. Along the way, we study conditioning of the HDG method, prove new error estimates for it, and identify an abstract class of problems for which a nonnested two-level multigrid cycle with one smoothing step converges even when the prolongation norm is greater than one. Numerical experiments verifying our theoretical results are presented.

We provide a systematic way of devising superconvergent mixed and hybridizable discontinuous Galerkin (HDG) methods for linear elasticity based on weak stress symmetry formulations. We show that, by suitably modifying the spaces defining superconvergent mixed and HDG methods for diffusion obtained in Cockburn et al. (Conditions for superconvergence of HDG methods for second-order elliptic problems, Math. Comp., 81, 1327–1353.), we obtain optimally convergent approximations for all unknowns, as well as a superconvergent approximation of the displacement. We use a projection-based a priori error analysis to achieve this goal.