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In Chen et al. (J. Sci. Comput. 81(3): 2188–2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defined on simplicial meshes, for scalar reaction-diffusion equations and showed how to define an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree $$k\geqslant 1$$ . In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3): 635–650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defined on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements affine-equivalent to a finite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for $$k\geqslant 0$$ by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory.

In our earlier work (Cockburn et al. in J Sci Comput 79(3):1777–1800, 2019), we approximated solutions of a general class of scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous Galerkin (interpolatory HDG) method. This method reduces the computational cost compared to standard HDG since the HDG matrices are assembled once before the time integration. Interpolatory HDG also achieves optimal convergence rates; however, we did not observe superconvergence after an element-by-element postprocessing. In this work, we revisit the Interpolatory HDG method for reaction diffusion problems, and use the postprocessed approximate solution to evaluate the nonlinear term. We prove this simple change restores the superconvergence and keeps the computational advantages of the Interpolatory HDG method. We present numerical results to illustrate the convergence theory and the performance of the method.

We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of Δ x \Delta x only. For example, when polynomials of degree k k are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order k + 1 / 2 k+1/2 in the L 2 L^2 -norm, whereas the post-processed approximation is of order 2 k + 1 2k+1 ; if the exact solution is in L 2 L^2 only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order k + 1 / 2 k+1/2 in L 2 ( Ω 0 ) L^2(\Omega _0) , where Ω 0 \Omega _0 is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.

International audience; We build a bridge between the hybrid high-order (HHO) and the hybridizable discontinuous Galerkin (HDG) methods in the setting of a model diffusion problem. First, we briefly recall the construction of HHO methods and derive some new variants. Then, by casting the HHO method in mixed form, we identify the numerical flux so that the HHO method can be compared to HDG methods. In turn, the incorporation of the HHO method into the HDG framework brings up new, efficient choices of the local spaces and a new, delicate construction of the numerical flux ensuring optimal orders of convergence on meshes made of general shape-regular polyhedral elements. Numerical experiments comparing two of these methods are shown.

We present a technique for numerically solving convection-diffusion problems in domains $$\varOmega $$ with curved boundary. The technique consists in approximating the domain $$\varOmega $$ by polyhedral subdomains $$\mathsf{{D}}_h$$ where a finite element method is used to solve for the approximate solution. The approximation is then suitably extended to the remaining part of the domain $$\varOmega $$ . This approach allows for the use of only polyhedral elements; there is no need of fitting the boundary in order to obtain an accurate approximation of the solution. To achieve this, the boundary condition on the border of $$\varOmega $$ is transferred to the border of $$\mathsf{D }_h$$ by using simple line integrals. We apply this technique to the hybridizable discontinuous Galerkin method and provide extensive numerical experiments showing that, whenever the distance of $$\mathsf{{D}}_h$$ to $$\partial \varOmega $$ is of order of the meshsize $$h$$ , the convergence properties of the resulting method are the same as those for the case in which $$\varOmega =\mathsf{{D}}_h$$ . We also show numerical evidence indicating that the ratio of the $$L^2(\varOmega )$$ norm of the error in the scalar variable computed with $$d>0$$ to that of that computed with $$d=0$$ remains constant (and fairly close to one), whenever the distance $$d$$ is proportional to $$\min \{h,Pe^{-1}\}/(k+1)^2$$ , where $$Pe$$ is the so-called Peclet number.

In this paper, we give a simple introduction to the devising of discontinuous Galerkin (DG) methods for nonlinear hyperbolic conservation laws. These methods have recently made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. The DG methods, which are extensions of finite volume methods, incorporate into a finite element framework the notions of approximate Riemann solvers, numerical fluxes and slope limiters coined during the remarkable development of the high-resolution finite difference and finite volume methods for nonlinear hyperbolic conservation laws. We start by stressing the fact that nonlinear hyperbolic conservation laws are usually obtained from well-posed problems by neglecting terms modeling nondominant features of the model which, nevertheless, are essential in crucial, small parts of the domain; as a consequence, the resulting problem becomes ill-posed. The main difficulty in devising numerical schemes for these conservation laws is thus how to re-introduce the neglected physical information in order to approximate the physically relevant solution, usually called the entropy solution. For the classical case of the entropy solution of the nonlinear hyperbolic scalar conservation law, we show how to carry out this process for two prototypical DG methods. The first DG method is the so-called shock-capturing DG method, which does not use slope limiters and is implicit; the second is the Runge–Kutta DG method, which is an explicit method that does not employ a shock-capturing term but uses a slope limiter instead. We then focus on the Runge–Kutta DG methods and show how to obtain a key stability property which holds independently of the accuracy of the scheme and of the nonlinearity of the conservation law; we also show some computational results.