Hybridizable Discontinuous Galerkin Method Research Articles

A Unified Error Analysis of Hybridizable Discontinuous Galerkin Methods for the Static Maxwell Equations

We propose a framework that allows us to analyze different variants of hybridizable discontinuous Galerkin (HDG) methods for the static Maxwell equations using one simple analysis. It reduces all t...

New analytical tools for HDG in elasticity, with applications to elastodynamics

We present some new analytical tools for the error analysis of hybridizable discontinuous Galerkin (HDG) method for linear elasticity. These tools allow us to analyze more variants of HDG method using the projection-based approach, which renders the error analysis simple and concise. The key result is a tailored projection for the Lehrenfeld-Schoberl type HDG (HDG+ for simplicity) methods. By using the projection we recover the error estimates of HDG+ for steady-state and time-harmonic elasticity in a simpler analysis. We also present a semi-discrete (in space) HDG+ method for transient elastic waves and prove it is uniformly-in-time optimal convergent by using the projection-based error analysis. Numerical experiments supporting our analysis are presented at the end.

Conservative super-convergent and hybrid discontinuous Galerkin methods applied to nonlinear Schrödinger equations

Using a unified framework, the formulation of a super-convergent discontinuous Galerkin (SDG) method and a hybridized discontinuous Galerkin (HDG) version, both applied to a general nonlinear Schrödinger equation is presented. Conservation of the mass and the energy is studied, theoretically for the semi-discrete formulation; and, for the fully discrete method using the Modified Crank–Nicolson time scheme. Conservation of both quantities is numerically validated on two dimensional problems and high order approximations. A numerical study of convergence illustrates the advantages of the new formulations over the traditional Local Discontinuous Galerkin (LDG) method. Numerical experiments show that the approximation of the initial discrete energy converges with order 2k+1, which is better than that obtained by the standard (continuous) finite element, which is only of order 2k when polynomials of degree k are used.

A superconvergent hybridizable discontinuous Galerkin method for Dirichlet boundary control of elliptic PDEs

We begin an investigation of hybridizable discontinuous Galerkin (HDG) methods for approximating the solution of Dirichlet boundary control problems governed by elliptic PDEs. These problems can involve atypical variational formulations, and often have solutions with low regularity on polyhedral domains. These issues can provide challenges for numerical methods and the associated numerical analysis. We propose an HDG method for a Dirichlet boundary control problem for the Poisson equation, and obtain optimal a priori error estimates for the control. Specifically, under certain assumptions, for a 2D convex polygonal domain we show the control converges at a superlinear rate. We present 2D and 3D numerical experiments to demonstrate our theoretical results.

A hybridizable discontinuous Galerkin phase‐field model for brittle fracture with adaptive refinement

SummaryIn this paper, we propose an adaptive refinement strategy for phase‐field models of brittle fracture, which is based on a novel hybridizable discontinuous Galerkin (HDG) formulation of the problem. The adaptive procedure considers standard elements and only one type of h‐refined elements, dynamically located along the propagating cracks. Thanks to the weak imposition of interelement continuity in HDG methods, and in contrast with other existing adaptive approaches, hanging nodes or special transition elements are not needed, which simplifies the implementation. Various numerical experiments, including one branching test, show the accuracy, robustness, and applicability of the presented approach to quasistatic phase‐field simulations.

Implicit hybridized discontinuous Galerkin methods for compressible magnetohydrodynamics

We present hybridized discontinuous Galerkin (HDG) methods for ideal and resistive compressible magnetohydrodynamics (MHD). The HDG methods are fully implicit, high-order accurate and endowed with a unique feature which distinguishes themselves from other discontinuous Galerkin (DG) methods. In particular, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby resulting in considerably smaller global degrees of freedom than other DG methods. Furthermore, we develop a shock capturing method to deal with shocks by appropriately adding artificial bulk viscosity, molecular viscosity, thermal conductivity, and electric resistivity to the physical viscosities in the MHD equations. We show the optimal convergence of the HDG methods for ideal MHD problems and validate our resistive implementation for a magnetic reconnection problem. For smooth problems, we observe that employing a generalized Lagrange multiplier (GLM) formulation can reduce the errors in the divergence of the magnetic field by two orders of magnitude. We demonstrate the robustness of our shock capturing method on a number of test cases and compare our results, both qualitatively and quantitatively, with other MHD solvers. For shock problems, we observe that an effective treatment of both the shock wave and the divergence-free constraint is crucial to ensuring numerical stability.

Superconvergent Interpolatory HDG Methods for Reaction Diffusion Equations I: An HDG$$_{k}$$ Method

In our earlier work [8], 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.

High order HDG method and domain decomposition solvers for frequency‐domain electromagnetics

AbstractThis work is concerned with the numerical treatment of the system of three‐dimensional frequency‐domain (or time‐harmonic) Maxwell equations using a high order hybridizable discontinuous Galerkin (HDG) approximation method combined with domain decomposition (DD) on the basis of hybrid iterative‐direct parallel solution strategies. The proposed HDG method preserves the advantages of classical DG methods previously introduced for the time‐domain Maxwell equations, in particular, in terms of accuracy and flexibility with regards to the discretization of complex geometrical features, while keeping the computational efficiency at the level of the reference edge element‐based finite element formulation widely adopted for the considered PDE system. We study in details the computational performances of the resulting DD solvers in particular in terms of scalability metrics by considering both a model test problem and more realistic large‐scale simulations performed on high performance computing systems consisting of networked multicore nodes.

Multiscale Hybridizable Discontinuous Galerkin Method for Flow Simulations in Highly Heterogeneous Media

We propose a multiscale hybridizable discontinuous Galerkin method for Darcy flow and two phase flow simulations in highly heterogeneous media. The multiscale space consists of offline and online multiscale basis functions. The offline basis functions are constructed by solving appropriate local spectral problem, and thus contain important local media information. The online basis functions are computed iteratively with the residuals of previous multiscale solution on selected local regions. Typically, the offline basis provides initial multiscale solution for constructing online basis. For the two phase flow simulations, we only compute the basis space for the initial permeability field and keep it fixed as time advancing. Numerical experiments show the multiscale solution can approximate the fine scale solution accurately for both types of flow simulations.

Modeling and observation of mid-infrared nonlocality in effective epsilon-near-zero ultranarrow coaxial apertures

With advances in nanofabrication techniques, extreme-scale nanophotonic devices with critical gap dimensions of just 1–2 nm have been realized. Plasmons in such ultranarrow gaps can exhibit nonlocal response, which was previously shown to limit the field enhancement and cause optical properties to deviate from the local description. Using atomic layer lithography, we create mid-infrared-resonant coaxial apertures with gap sizes as small as 1 nm and observe strong evidence of nonlocality, including spectral shifts and boosted transmittance of the cutoff epsilon-near-zero mode. Experiments are supported by full-wave 3-D nonlocal simulations performed with the hybridizable discontinuous Galerkin method. This numerical method captures atomic-scale variations of the electromagnetic fields while efficiently handling extreme-scale size mismatch. Combining atomic-layer-based fabrication techniques with fast and accurate numerical simulations provides practical routes to design and fabricate highly-efficient large-area mid-infrared sensors, antennas, and metasurfaces.

High-order hybridisable discontinuous Galerkin method for the gas kinetic equation

ABSTRACT The high-order hybridisable discontinuous Galerkin (HDG) method is used to find steady-state solution of gas kinetic equations on two-dimensional geometry. The velocity distribution function and its traces are approximated in piecewise polynomial space on triangular mesh and mesh skeleton, respectively. By employing a numerical flux derived from the upwind scheme and imposing its continuity on mesh skeleton, the global system for unknown traces is obtained with fewer coupled degrees of freedom, compared to the original DG method. The solutions of model equation for the Poiseuille flow through square channel show the higher order solver is faster than the lower order one. Moreover, the HDG scheme is more efficient than the original DG method when the degree of approximating polynomial is larger than 2. Finally, the developed scheme is extended to solve the Boltzmann equation with full collision operator, which can produce accurate results for shear-driven and thermally induced flows.

A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs

We investigated a hybridizable discontinuous Galerkin (HDG) method for a convection diffusion Dirichlet boundary control problem in our earlier work (Gong et al. SIAM J Numer Anal 56(4):2262–2287, 2018) and obtained an optimal convergence rate for the control under some assumptions on the desired state and the domain. In this work, we obtain the same convergence rate for the control using a class of embedded DG methods proposed by Nguyen et al. (J Comput Phys 302:674–692, 2015) for simulating fluid flows. Since the global system for embedded DG methods uses continuous elements, the number of degrees of freedom for the embedded DG methods are smaller than the HDG method, which uses discontinuous elements for the global system. Moreover, we introduce a new simpler numerical analysis technique to handle low regularity solutions of the boundary control problem. We present some numerical experiments to confirm our theoretical results.

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).

Solvability of the fractional Volterra‐Fredholm integro differential equation by hybridizable discontinuous Galerkin method

This article proves the existence and uniqueness of the solution obtained by the hybridizable discontinuous Galerkin (HDG) method of the fractional Volterra‐Fredholm integro differential equation. The method based on local solvers and transmission condition is applied to the equation using two auxiliary variables. The form of the equation is amenable for achieving the solvability criteria of the problem according to the HDG method. We also calculate a numerical solution of the problem whose exact solution is taken as a smooth or fractional function. This results in a tridiagonal, symmetric, and positive definite stiffness matrix.

Multiscale hybridizable discontinuous Galerkin method for elliptic problems in perforated domains

We propose a multiscale hybridizable discontinuous Galerkin method in perforated domains. Perforations may have various shapes and sizes. Following Generalized Multiscale Finite Element method, local snapshot spaces are constructed by solving local spectral problems. Then multiscale finite element spaces for numerical traces are built on the coarse edges within the framework of hybridizable discontinuous Galerkin method. Some representative examples are presented to investigate numerical performance of the proposed method.

A posteriori error analysis of an HDG method for the Oseen problem

We propose a residual-type a posteriori error estimator for a hybridizable discontinuous Galerkin method applied to the Oseen problem in a gradient-velocity-pressure formulation. We state reliability and local efficiency results for our estimator with respect to an error measured in the natural norms, with constants depending explicitly on the physical parameters. A weighting function technique to control the L2-error of the velocity, and the approximation properties of the Oswald interpolation operator are the main ingredients in the analysis. Numerical experiments in three dimensions validate our theoretical results.

A Comparison of the Explicit and Implicit Hybridizable Discontinuous Galerkin Methods for Nonlinear Shallow Water Equations

An explicit implementation of the hybridizable discontinuous Galerkin (HDG) method for solving the nonlinear shallow water equations is presented. We follow the common construction of the implicit HDG for nonlinear conservation laws, and then explain the differences between the explicit formulation and the implicit version. For the implicit implementation, we use the approximate traces of the conserved variables ( $${\widehat{\varvec{q}}}$$ ) to express the numerical fluxes in each element. Next, we impose the conservation of the numerical fluxes via a global system of equations. Using the Newton–Raphson method, this global system can be solely expressed in terms of the increments of the approximate traces in each iteration. For the explicit method, having $${\varvec{q}}_h$$ at each time level, we first obtain $${\widehat{\varvec{q}}}_h$$ such that the conservation of the numerical flux is satisfied. This will result in a nonlinear system of equations which is local to each edge of the mesh skeleton. Having the solution ( $${\varvec{q}}_h$$ , $$\widehat{{\varvec{q}}}_h$$ ) for the previous time step, we use the Runge–Kutta time integration method to obtain $${\varvec{q}}_h$$ in the next time step. Hence, the introduced explicit technique is based on local operations over the faces and elements of the mesh. Using different numerical examples, we show the optimal convergence of the solution of the explicit and implicit approach in $$L^2$$ norm. Finally, through numerical experiments, we discuss the advantages of the implicit and explicit techniques from the computational cost point of view.

A divergence‐conforming hybridized discontinuous Galerkin method for the incompressible Reynolds‐averaged Navier‐Stokes equations

SummaryWe present a hybridized discontinuous Galerkin (HDG) method for the incompressible Reynolds‐averaged Navier‐Stokes equations coupled with the Spalart‐Allmaras one‐equation turbulence model. The method extends upon an HDG method recently introduced by Rhebergen and Wells for the incompressible Navier‐Stokes equations. With a special choice of velocity and pressure spaces for both element and trace degrees of freedom (DOFs), the method returns pointwise divergence‐free mean velocity fields and properly balances momentum and energy. We further examine the use of different polynomial degrees and meshes to see how the order of the scalar eddy viscosity affects the convergence of the mean velocity and pressure fields, specifically for the method of manufactured solutions. As is standard with HDG methods, static condensation can be employed to remove the element DOFs and thus dramatically reduce the global number of DOFs. Numerical results illustrate the effectiveness of the proposed methodology.

HDG-NEFEM for two dimensional linear elasticity

This paper proposes a new methodology for the solution of two dimensional linear elastic problems in domains with curved boundaries. The proposed method exploits the advantages of the hybridisable discontinuous Galerkin method to obtain an accurate approximation of both the displacement and the stress fields by solving a global problem that only involves the displacement field on the element edges as unknown. In addition, the methodology incorporates the exact boundary representation of the domain by means of the so-called NURBS-enhanced finite element method. Numerical examples are used to illustrate the three main advantages of the proposed method, namely the reproducibility of polynomials in domains with curved boundaries, the super-convergence of the solution even for linear approximation and the effectiveness and reliability of degree adaptive processes driven by displacement or stresses.

Analysis on an HDG Method for the p-Laplacian Equations

In Cockburn and Shen (SIAM J Sci Comput 38(1):A545–A566, 2016) the authors propose the first hybridizable discontinuous Galerkin method (HDG) for the p-Laplacian equation. Several iterative algorithms are developed and tested. The main purpose of this paper is to provide rigorous error estimates for the proposed HDG method. We first develop the error estimates based on general polyhedral/polygonal triangulations, under standard regularity assumption of the solution, the convergence analysis is presented for $$1<p<2$$ and $$p>2$$ . Nevertheless, when p approaches to the limits ( $$p \rightarrow 1^+$$ or $$p \rightarrow \infty $$ ), the convergence rate shows degeneration for both cases. Finally, this degeneration can be recovered if we use simplicial triangulation of the domain with sufficient large stabilization parameter for the method.