Turbulence, Coherent Structures, Dynamical Systems and SymmetryP. Holmes, J. L. Lumley, G. Berkooz, and C. W. Rowley, 2nd ed., Cambridge University Press, Cambridge, England, U.K., 2012, 386 pp., $90

High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows

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.

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Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods

In this paper, we review and refine the main ideas for devising the so-called hybridizable discontinuous Galerkin (HDG) methods; we do that in the framework of steady-state diffusion problems. We begin by revisiting the classic techniques of static condensation of continuous finite element methods and that of hybridization of mixed methods, and show that they can be reinterpreted as discrete versions of a characterization of the associated exact solution in terms of solutions of Dirichlet boundary-value problems on each element of the mesh which are then patched together by transmission conditions across interelement boundaries. We then define the HDG methods associated to this characterization as those using discontinuous Galerkin (DG) methods to approximate the local Dirichlet boundary-value problems, and using weak impositions of the transmission conditions. We give simple conditions guaranteeing the existence and uniqueness of their approximate solutions, and show that, by their very construction, the HDG methods are amenable to static condensation. We do this assuming that the diffusivity tensor can be inverted; we also briefly discuss the case in which it cannot. We then show how a different characterization of the exact solution, gives rise to a different way of statically condensing an already known HDG method. We devote the rest of the paper to establishing bridges between the HDG methods and other methods (the old DG methods, the mixed methods, the staggered DG method and the so-called Weak Galerkin method) and to describing recent efforts for the construction of HDG methods (one for systematically obtaining superconvergent methods and another, quite different, which gives rise to optimally convergent methods). We end by providing a few bibliographical notes and by briefly describing ongoing work.

Static Condensation, Hybridization, and the Devising of the HDG Methods

We present a superconvergent hybridizable discontinuous Galerkin (HDG) method for the steady-state incompressible Navier-Stokes equations on general polyhedral meshes. For arbitrary conforming polyhedral mesh, we use polynomials of degree k+1, k, k to approximate the velocity, velocity gradient and pressure, respectively. In contrast, we only use polynomials of degree k to approximate the numerical trace of the velocity on the interfaces. Since the numerical trace of the velocity field is the only globally coupled unknown, this scheme allows a very efficient implementation of the method. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L2-norm of the error in each of the above-mentioned variables and the discrete H1-norm of the error in the velocity converge with the order of k+1 for k>=0. We also show that for k>=1, the global L2-norm of the error in velocity converges with the order of k+2. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this method achieves optimal convergence for all the above-mentioned variables in L2-norm for k>=0, superconvergence for the velocity in the discrete H1-norm without postprocessing for k>=0, and superconvergence for the velocity in L2-norm without postprocessing for k>=1.

A superconvergent HDG method for the Incompressible Navier-Stokes Equations on general polyhedral meshes

. This paper presents a new hybridizable discontinuous Galerkin (HDG) method forlinear elasticity, on tetrahedral meshes, based on a strong symmetric stress formulation. The keyfeature of this new HDG method is the use of a special form of the numerical trace of the stresses,which makes the error analysis diﬀerent from the projection-based error analyzes used for most otherHDG methods. On each element, we approximate the stress by using polynomials of degree k ≥ 1and the displacement by using polynomials of degree k+1. In contrast, to approximate the numericaltrace of the displacement on the faces, we use polynomials of degree k only. This allows for a veryeﬃcient implementation of the method, since the numerical trace of the displacement is the onlyglobally-coupled unknown, but does not degrade the convergence properties of the method. Indeed,we prove optimal orders of convergence for both the stresses and displacements on the elements.These optimal results are possible thanks to a special superconvergence property of the numericaltraces of the displacement, and thanks to the use of a crucial elementwise Korn’s inequality.Key words. hybridizable; discontinuous Galerkin; superconvergence; linear elasticity.AMS subject classiﬁcations. 65N30, 65L12, 35L15

An HDG method for linear elasticity with strong symmetric stresses

We propose the first hybridizable discontinuous Galerkin method for the $p$-Laplacian equation. When using polynomials of degree $k \geq 0$ for the approximation spaces of $u$, $\boldsymbol{\nabla} u$, and $|\boldsymbol{\nabla} u|^{p-2}\boldsymbol{\nabla} u$, the method exhibits optimal $k+1$ order of convergence for all variables in $L^1$- and $L^p$-norms in our numerical experiments. For $k \geq 1$, an elementwise computation allows us to obtain a new approximation $u_h^*$ that converges to $u$ with order $k+2$. We rewrite the scheme as discrete minimization problems in order to solve them with nonlinear minimization algorithms. The unknown of the first problem is the approximation of $u$ on the skeleton of the mesh but requires solving nonlinear local problems. The second problem has the approximation on the elements as an additional unknown but it only requires solving linear local problems. We present numerical results displaying the convergence properties of the methods, demonstrating the utility of...

A hybridizable discontinuous Galerkin method for the p-Laplacian

An HDG method for distributed control of convection diffusion PDEs

This paper presents a new hybridizable discontinuous Galerkin (HDG) method for linear elasticity on general polyhedral meshes, based on a strong symmetric stress formulation The key feature of this new HDG method is the use of a special form of the numerical trace of the stresses, which makes the error analysis different from the projection-based error analyzes used for most other HDG methods For arbitrary polyhedral elements, we approximate the stress by using polynomials of degree k>=1 and the displacement by using polynomials of degree k+1 In contrast, to approximate the numerical trace of the displacement on the faces, we use polynomials of degree k only This allows for a very efficient implementation of the method, since the numerical trace of the displacement is the only globally-coupled unknown, but does not degrade the convergence properties of the method Indeed, we prove optimal orders of convergence for both the stresses and displacements on the elements In the almost incompressible case, we show the error of the stress is also optimal in the standard L2-norm These optimal results are possible thanks to a special superconvergence property of the numerical traces of the displacement, and thanks to the use of a crucial elementwise Korn's inequality Several numerical results are presented to support our theoretical findings in the end

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We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. The algorithm initializes and efficiently updates the dominant POD eigenvalues and modes during the time stepping in a PDE solver without storing the simulation data. We prove that the algorithm without truncation updates the POD exactly. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers’ equation and a 2D Navier–Stokes problem.

Jiguang Shen

Papers

An HDG method for linear elasticity with strong symmetric stresses

A hybridizable discontinuous Galerkin method for the p-Laplacian

An HDG method for distributed control of convection diffusion PDEs

An HDG method for linear elasticity with strong symmetric stresses

Incremental proper orthogonal decomposition for PDE simulation data