Hybridizable Discontinuous Galerkin Research Articles

Residual-type a posteriori error analysis of HDG methods for Neumann boundary control problems

We study a posteriori error analysis of linear-quadratic boundary control problems under bilateral box constraints on the control which acts through a Neumann-type boundary condition. We adopt the hybridizable discontinuous Galerkin method as the discretization technique, and the flux variables, the scalar variables, and the boundary trace variables are all approximated by polynomials of degree k. As for the control variable, it is discretized by the variational discretization concept. Then, an efficient and reliable a posteriori error estimator is introduced, and we prove that the error estimator provides an upper bound and a lower bound for the errors. Finally, numerical results are presented to illustrate the performance of the obtained a posteriori error estimator.

Preasymptotic Error Analysis of the HDG Method for Helmholtz Equation with Large Wave Number

This paper addresses several aspects of the linear hybridizable discontinuous Galerkin method (HDG) for the Helmholtz equation with impedance boundary condition at high frequency. First, error estimates with explicit dependence on the wave number k for the HDG approximations to the exact solution u and its negative gradient $${\mathbf {q}}=-\nabla u$$ are derived. It is shown that $$k\Vert u-u_h\Vert _{L^2(\varOmega )}+ \Vert {\mathbf {q}}-{\mathbf {q}}_h\Vert _{L^2(\varOmega )}=O(k^2h^2+k^4h^3)$$ under the conditions that $$k^3h^2$$ is sufficiently small and that the penalty parameter $$\tau \eqsim k$$ , where h is the mesh size. Note that the convergence order in $${\mathbf {q}}_h$$ is full and the pollution error is $$O(k^4h^3)$$ , which improve the existent results. Secondly, by using a standard postprocessing procedure from the HDG method for elliptic problems, a piecewise quadratic function $$u_h^*$$ is obtained so that $$k\Vert u-u_h^*\Vert _{L^2(\varOmega )}=O(k^3h^3+k^4h^3)$$ . Note that the postprocessing procedure improves only the interpolation error (from $$O(k^2h^2)$$ to $$O(k^3h^3)$$ ) but leaves the pollution error $$O(k^4h^3)$$ unchanged. Thirdly, a dispersion analysis and extensive numerical tests show that the pollution effect can be reduced greatly in 2D case by selecting appropriate penalty parameters.

Implementation of hybridizable discontinuous Galerkin method for time‐harmonic anisotropic poroelasticity in two dimensions

AbstractWe apply a hybridizable discontinuous Galerkin (HDG) method to numerically solve two‐dimensional anisotropic poroelastic wave equations in the frequency domain given by Biot theory. The motivation for choosing HDG method comes from the complexity of the considered equations and the high number of unknowns. The HDG method possesses all the advantages of discontinuous Galerkin method (hp‐adaptivity, accuracy, ability to model complex tectonics, etc.) without a drastic increase in the number of degrees of freedom. After a description of its implementation, we illustrate the accuracy of the proposed method by comparisons with analytical solutions. We then perform a sensitivity analysis of the method as a function of stabilization parameters and frequency, and show in particular that there exists optimal values of these parameters in order to obtain a very good level of accuracy. We also show the ability of the method to reproduce the different types of poroelastic waves including the slow Biot wave.

An Exactly Mass Conserving and Pointwise Divergence Free Velocity Method: Application to Compositional Buoyancy Driven Flow Problems in Geodynamics

AbstractTracer methods are widespread in computational geodynamics for modeling the advection of chemical data. However, they present certain numerical challenges, especially when used over long periods of simulation time. We address two of these in this work: the necessity for mass conservation of chemical composition fields and the need for the velocity field to be pointwise divergence free to avoid gaps in tracer coverage. We do this by implementing the hybrid discontinuous Galerkin (HDG) finite element (FE) method combined with a mass conserving constrained projection of the tracer data. To demonstrate the efficacy of this system, we compare it to other common FE formulations of the Stokes system and projections of the chemical composition. We provide a reference of the numerical properties and error convergence rates which should be observed by using these various discretization schemes. This serves as a tool for verification of existing or new implementations. We summarize these data in a reproduction of a published Rayleigh‐Taylor instability benchmark, demonstrating the importance of careful choices of appropriate and compatible discretization methods for all aspects of geodynamics simulations.

Adaptive HDG Methods for the Brinkman Equations with Application to Optimal Control

This paper investigates adaptive hybridizable discontinuous Galerkin methods for the gradient-velocity–pressure formulation of Brinkman equations. We use piecewise polynomials of degree k to approximate the velocity, the velocity gradient, the pressure and the boundary traces. By the $$L^2$$ -projection and inf-sup condition, a residual type a posteriori error estimator is introduced for the error measured by an energy norm. Moreover, we prove that the a posteriori error estimator is robust in the sense that the ratio of the upper and lower bounds is independent of the dynamic viscosity and permeability. Then the idea and the result, which have been used and obtained for Brinkman equations, are extended to solve the Brinkman optimal control problem. Based on the variational discretization concept, we also derive an efficient and reliable a posteriori error estimator for the error measured by an energy norm. Finally, several numerical results are provided to validate the theoretical analysis.

A Fast, Hybrid, Time-Domain Discontinuous Galerkin-Physical Optics Method for Composite Electromagnetic Scattering Analysis

To accelerate the solution of transient electromagnetic scattering from composite scatters, a novel hybrid discontinuous Galerkin time domain (DGTD) and time-domain physical optics (TDPO) method is proposed. The DGTD method is used to solve the accurate scattering field of the multi-scale objects region, and a hybrid explicit-implicit time integration method is also used to improve the efficiency of multi-scale problems in the time domain. Meanwhile, the TDPO method is used to accelerate the speed of surface current integration in an electrically large region. In addition, the DGTDPO method considers the mutual coupling between two regions, and effectively reduces the number of numerical calculations for the other space of the composite target, thereby significantly reducing the computer memory consumption. Numerical results certified the high efficiency and accuracy of the hybrid DGTDPO. According to the results, in comparison with the DGTD algorithm in the entire computational domain, the DGTDPO method can reduce computing time and memory by 90% and 70% respectively. Meanwhile, the normalized root mean square deviation (NRMSD) of the time-domain, high-frequency approximation method is over 0.2, and that of the DGTDPO method is only 0.0971. That is, compared with the approximation methods, the hybrid method improves the accuracy by more than 64%.

Adaptive HDG Methods for the Steady-State Incompressible Navier–Stokes Equations

We consider a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier–Stokes equations. We use polynomials of degree $$k+1$$ , k, k and k for approximations of the velocity, the velocity gradient, the pressure and the boundary traces. Some stability results for approximate solutions and some relationships between norms are provided. Moreover an a posteriori error estimator is introduced. By $$L^2$$ -projection and inf-sup condition, we prove that the error estimator is robust for the global $$L^2$$ errors in the velocity, the velocity gradient and the pressure. Finally, a Picard iteration method and an adaptive HDG algorithm are presented. Furthermore, several numerical examples are shown to validate the theoretical analysis.

High-Order Hybridizable Discontinuous Galerkin Formulation for One-Phase Flow Through Porous Media

We present a stable high-order hybridizable discontinuous Galerkin (HDG) formulation coupled with high-order diagonal implicit Runge–Kuta (DIRK) schemes to simulate slightly compressible one-phase flow through porous media. The HDG stability depends on the selection of a single parameter and its definition is crucial to ensure the stability and to achieve the high-order properties of the method. Thus, we extend the work of Nguyen et al. in J Comput Phys 228, 8841–8855, 2009 to deduce an analytical expression for the stabilization parameter using the material parameters of the problem and the Engquist-Osher monotone flux scheme. The formulation is high-order accurate for the pressure, the flux and the velocity with the same convergence rate of P+1, being P the polynomial degree of the approximation. This is important because high-order methods have the potential to reduce the computational cost while obtaining more accurate solutions with less dissipation and dispersion errors than low order methods. The formulation can use unstructured meshes to capture the heterogeneous properties of the reservoir. In addition, it is conservative at the element level, which is important when solving PDE’s in conservative form. Moreover, a hybridization procedure can be applied to reduce the size of the global linear system. To keep these advantages, we use DIRK schemes to perform the time integration. DIRK schemes are high-order accurate and have a low memory footprint. We show numerical evidence of the optimal convergence rates obtained with the proposed formulation. Finally, we present several examples to illustrate the capabilities of the formulation.

A Hybridizable Discontinuous Galerkin Method for the Quad-Curl Problem

The quad-curl problem arises in magnetohydrodynamics, inverse electromagnetic scattering problems, and electromagnetic transmission eigenvalue problems. In this paper, we propose a hybridizable discontinuous Galerkin method to solve the quad-curl problem, and perform the analysis with a relatively low regularity assumption. The efficiency and accuracy of the proposed method are illustrated through numerical experiments.

The PML Boundary Application in the Implicit Hybridizable Discontinuous Galerkin Time-Domain Method for Waveguides

The absorbing boundary condition (ABC) is very common to be used to truncate the computational domain in the hybridizable discontinuous Galerkin (HDG) method. However, if solving waveguide transmission problems with multimode scattering, the ABC is not very applicable due to the wave impedances varying with the frequency. Considering the perfectly matched layers (PMLs) boundary can absorb well the outgoing waves, thus this letter presents a new PML application in the implicit HDG time-domain (imHDGTD) method. For PML and non-PML regions, we need to use different Maxwell equations to generate the global linear system, which leads to a completely different assembling process from ABC. For this, we further give specific technologies. Note that this is the first time that the PML is used in HDG. Compared with the explicit discontinuous Galerkin time-domain method, numerical results show its accuracy and efficient performance, which implies that this method would be very useful.

An implicit HDG method for linear convection-diffusion with dual time stepping

This work presents, for the first time, a dual time stepping (DTS) approach to solve the global system of equations that appears in the hybridisable discontinuous Galerkin (HDG) formulation of convection-diffusion problems. A proof of the existence and uniqueness of the steady state solution of the HDG global problem with DTS is presented. The stability limit of the DTS approach is derived using a von Neumann analysis, leading to a closed form expression for the critical dual time step. An optimal choice for the dual time step, producing the maximum damping for all the frequencies, is also derived. Steady and transient convection-diffusion problems are considered to demonstrate the performance of the proposed DTS approach, with particular emphasis on convection dominated problems. Two simple approaches to accelerate the convergence of the DTS approach are also considered and three different time marching approaches for the dual time are compared.

Locking-Free and Gradient-Robust $${\varvec{H}}({{\,\mathrm{{\text {div}}}\,}})$$-Conforming HDG Methods for Linear Elasticity

Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new: it assures that dominant gradient fields in the momentum balance do not lead to spurious displacements. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.

An a priori error analysis of adjoint-based super-convergent Galerkin approximations of linear functionals

Abstract 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.

$L^\infty$ Norm Error Estimates for HDG Methods Applied to the Poisson Equation with an Application to the Dirichlet Boundary Control Problem

We prove quasi-optimal $L^\infty$ norm error estimates (up to logarithmic factors) for the solution of Poisson's problem in two dimensional space by the standard hybridizable discontinuous Galerkin (HDG) method. Although such estimates are available for conforming and mixed finite element methods, this is the first proof for HDG. The method of proof is motivated by known $L^\infty$ norm estimates for mixed finite elements. We show two applications: the first is to prove optimal convergence rates for boundary flux estimates, and the second is to prove that numerically observed convergence rates for the solution of a Dirichlet boundary control problem are to be expected theoretically. Numerical examples show that the predicted rates are seen in practice.

Uniform Auxiliary Space Preconditioning for HDG Methods for Elliptic Operators with a Parameter Dependent Low Order Term

The auxiliary space preconditioning (ASP) technique is applied to the hybridizable discontinuous Galerkin (HDG) schemes for three different elliptic problems with a parameter dependent low order te...

A Space-Time Hybridizable Discontinuous Galerkin Method for Linear Free-Surface Waves

We present and analyze a novel space-time hybridizable discontinuous Galerkin (HDG) method for the linear free-surface problem on prismatic space-time meshes. We consider a mixed formulation which immediately allows us to compute the velocity of the fluid. In order to show well-posedness and to obtain a priori error estimates, our space-time HDG formulation makes use of weighted inner products. We perform an a priori error analysis in which the dependence on the time step and spatial mesh size is explicit. The analysis results are supported by numerical examples.

Hybridisable Discontinuous Galerkin Formulation of Compressible Flows

This work presents a review of high-order hybridisable discontinuous Galerkin (HDG) methods in the context of compressible flows. Moreover, an original unified framework for the derivation of Riemann solvers in hybridised formulations is proposed. This framework includes, for the first time in an HDG context, the HLL and HLLEM Riemann solvers as well as the traditional Lax-Friedrichs and Roe solvers. HLL-type Riemann solvers demonstrate their superiority with respect to Roe in supersonic cases due to their positivity preserving properties. In addition, HLLEM specifically outstands in the approximation of boundary layers because of its shear preservation, which confers it an increased accuracy with respect to HLL and Lax-Friedrichs. A comprehensive set of relevant numerical benchmarks of viscous and inviscid compressible flows is presented. The test cases are used to evaluate the competitiveness of the resulting high-order HDG scheme with the aforementioned Riemann solvers and equipped with a shock treatment technique based on artificial viscosity.

A nested hybridizable discontinuous Galerkin method for computing second-harmonic generation in three-dimensional metallic nanostructures

We develop a nested hybridizable discontinuous Galerkin (HDG) method to numerically solve the Maxwell's equations coupled with a hydrodynamic model for the conduction-band electrons in metals. The HDG method leverages static condensation to eliminate the degrees of freedom of the approximate solution defined in the elements, yielding a linear system in terms of the degrees of freedom of the approximate trace defined on the element boundaries. This article presents a computational method that relies on a degree-of-freedom reordering such that the HDG linear system accommodates an additional static condensation step to eliminate a large portion of the degrees of freedom of the approximate trace, thereby yielding a much smaller linear system. For the particular metallic structures considered in this article, the resulting linear system obtained by means of nested static condensations is a block tridiagonal system, which can be solved efficiently. We apply the nested HDG method to compute second harmonic generation on a triangular coaxial periodic nanogap structure. This nonlinear optics phenomenon features rapid field variations and extreme boundary-layer structures that span a wide range of length scales. Numerical results show that the ability to identify structures which exhibit resonances at ω and 2ω is essential to excite the second harmonic response.

A High-Order HDG Method with Dubiner Basis for Elliptic Flow Problems

We propose a standard hybridizable discontinuous Galerkin (HDG) method to solve a classic problem in fluids mechanics: Darcy’s law. This model describes the behavior of a fluid trough a porous medium and it is relevant to the flow patterns on the macro scale. Here we present the theoretical results of existence and uniqueness of the weak and discontinuous solution of the second order elliptic equation, as well as the predicted convergence order of the HDG method. We highlight the use and implementation of Dubiner polynomial basis functions that allow us to develop a general and efficient high order numerical approximation. We also show some numerical examples that validate the theoretical results.

Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system

We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with anL2penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.