Hybridizable Discontinuous Galerkin Research Articles

Hybrid Discontinuous Galerkin Approach for the Solution of Quantum Liouville-Type Equations

Hybrid Discontinuous Galerkin Approach for the Solution of Quantum Liouville-Type Equations

Hybridizable discontinuous Galerkin methods for space-time fractional advection-dispersion equations

This work focuses on the numerical solution of the initial and boundary value problems for space-time fractional advection-diffusion equations. The well-posedness of the weak solutions is shown by Lax-Milgram lemma. Two fully discrete methods are established. The main idea is based on a hybridizable discontinuous Galerkin approach in spatial direction and two finite difference schemes in temporal direction: L1 formula, the weighted and shifted Grünwald-Letnikov formula. The stability and convergence analyses of the proposed methods are derived in detail. Several numerical experiments are provided to illustrate the theoretical results.

An MLMCE-HDG method for the convection diffusion equation with random diffusivity

In this paper, a multilevel Monte Carlo ensemble hybridizable discontinuous Galerkin (MLMCE-HDG) method is proposed to solve the convection diffusion equation with random diffusion coefficients. This method combines the MLMC method in probability space, an ensemble approach in time space and the HDG method in physical space. The proposed method possesses a second order accuracy in time space and optimal L 2 convergence rate in physical space. The stability analysis, error estimates and computational cost are provided. Finally, to illustrate the theoretical analyses, several numerical experiments are performed.

BDDC Algorithms for Oseen problems with HDG Discretizations

Abstract The balancing domain decomposition by constraints (BDDC) methods are applied to the linear system arising from the hybridizable discontinuous Galerkin (HDG) discretization of the Oseen equation of incompressible flow. The generalized minimal residual method (GMRES) is used to accelerate the convergence. The original system is first reduced to a subdomain interface problem that is asymmetric indefinite, but can be positive definite in a special subspace. Edge/face average constraints can ensure all BDDC-preconditioned GMRES iterates stay in this special subspace. The convergence of the algorithm is analyzed, and additional edge/face constraints are used to improve the convergence. If the subdomain size is small enough, the number of iterations is independent of the number of subdomains, and depends only slightly on the subdomain problem size when the viscosity is large. When the viscosity is small, the convergence deteriorates. However, the numerical examples can give satisfactory results for the linear HDG discretization even with a small viscosity.

Efficiency enhancement of ultrathin CIGS solar cells by optimal bandgap grading. Part II: finite-difference algorithm and double-layer antireflection coatings.

In Part I [Appl. Opt.58, 6067 (2019)APOPAI003-693510.1364/AO.58.006067], we used a coupled optoelectronic model to optimize a thin-film C u I n 1-ξ G a ξ S e 2 (CIGS) solar cell with a graded-bandgap photon-absorbing layer and a periodically corrugated backreflector. The increase in efficiency due to the periodic corrugation was found to be tiny and that, too, only for very thin CIGS layers. Also, it was predicted that linear bandgap-grading enhances the efficiency of the CIGS solar cells. However, a significant improvement in solar cell efficiency was found using a nonlinearly (sinusoidally) graded-bandgap CIGS photon-absorbing layer. The optoelectronic model comprised two submodels: optical and electrical. The electrical submodel applied the hybridizable discontinuous Galerkin (HDG) scheme directly to equations for the drift and diffusion of charge carriers. As our HDG scheme sometimes fails due to negative carrier densities arising during the solution process, we devised a new, to the best of our knowledge, computational scheme using the finite-difference method, which also reduces the overall computational cost of optimization. An unfortunate normalization error in the electrical submodel in Part I came to light. This normalization error did not change the overall conclusions reported in Part I; however, some specifics did change. The new algorithm for the electrical submodel is reported here along with updated numerical results. We re-optimized the solar cells containing a CIGS photon-absorbing layer with either (i)a homogeneous bandgap, (ii)a linearly graded bandgap, or (iii)a nonlinearly graded bandgap. Considering the meager increase in efficiency with the periodic corrugation and additional complexity in the fabrication process, we opted for a flat backreflector. The new algorithm is significantly faster than the previous algorithm. Our new results confirm efficiency enhancement of 84% (resp. 63%) when the thickness of the CIGS layer is 600nm (resp. 2200nm), similarly to Part I. A hundredfold concentration of sunlight can increase the efficiency by an additional 27%. Finally, the currently used 110-nm-thick layer of M g F 2 performs almost as well as optimal single- and double-layer antireflection coatings.

Hybrid discontinuous Galerkin-finite volume techniques for compressible flows on unstructured meshes

In this paper we develop a family of arbitrarily high-order non-oscillatory hybrid Discontinuous Galerkin(DG)-Finite Volume(FV) schemes for mixed-element unstructured meshes. Their key ingredient is a switch between a DG method and a FV method based on the CWENOZ scheme when invalid solutions are detected by a troubled cell indicator checking the unlimited DG solution. Therefore, the high order of accuracy offered by DG is preserved in smooth regions of the computational domain, while the robustness of FV is utilized in regions with strong gradients. The high-order CWENOZ variant used has the same spatial order of accuracy as the DG variant, while representing one of the most compact applications on unstructured meshes, therefore simplifying the implementation, reducing the computational overhead associated with large stencils of the original WENO reconstruction without sacrificing the desirable non-oscillatory properties of the schemes. We carefully investigate several parameters associated with the switching between DG and FV methods including the troubled cell indicators in a priori fashion. For the first time in the literature, we investigate the definition of the bounds for an admissible solution, the frequency by which we use the troubled cell indicators, and the evolution of the percentage of troubled cells for unsteady test problems. The 2D and 3D Euler equations are solved for well established test problems and compared with computational or experimental reference solutions. All the methods have been implemented and deployed within the UCNS3D open-source high-order unstructured Computational Fluid Dynamics (CFD) solver. The present coupling has the potential to improve the shortcomings of both FV-DG in a computational efficient manner. The improved accuracy and robustness provided is a characteristic of paramount importance for industrial-scale CFD applications, and favours the extension to other systems of governing equations.

A hybridizable discontinuous Galerkin method for the coupled Navier–Stokes and Darcy problem

We present and analyze a strongly conservative hybridizable discontinuous Galerkin finite element method for the coupled incompressible Navier–Stokes and Darcy problem with Beavers–Joseph–Saffman interface condition. An a priori error analysis shows that the velocity error does not depend on the pressure, and that velocity and pressure converge with optimal rates. These results are confirmed by numerical examples.

A New Explicit–Implicit Hybridizable Discontinuous Galerkin Time-Domain Method for Electromagnetics

For time-domain electromagnetics, the choice of time scheme is very pivotal. Usually, an explicit time scheme has the stability constraint on small grid size. Though the implicit time scheme is unconditionally stable, it has a necessary expense of computing the global system at each time iteration. To alleviate the expensive computational cost, a new explicit-implicit hybridizable discontinuous Galerkin time-domain method that combines the explicit DGTD (exDGTD) method and our former imHDGTD method will be proposed for the first time. Here let’s call it exDGTD-imHDGTD (ex-imHDGTD). This letter gives its implementation, including a new transmission condition. Unlike the traditional explicit–implicit methods, imHDGTD is used to replace traditional implicit algorithms for the refined region, which leads to a remarkable reduction of degrees of freedom (DOFs) and a better matrix solving ability. Numerical results show that the proposed ex–imHDGTD is very effective both on accuracy and performance.

An ensemble Monte Carlo HDG method for parabolic PDEs with random coefficients

We use Monte Carlo, ensemble and hybrid discontinuous Galerkin method (EMC-HDG) to numerically solve parabolic partial differential equations (PDEs) with random coefficients. The proposed method reduces the computational cost and the storage requirement by solving multiple linear systems with a common coefficient matrix. Error analysis shows the proposed method is first-order accurate in time and optimal convergence order in physical space. In the end, several numerical experiments are presented to verify the theoretical results.

Hybridization and postprocessing in finite element exterior calculus

We hybridize the methods of finite element exterior calculus for the Hodge–Laplace problem on differential k k -forms in R n \mathbb {R}^n . In the cases k = 0 k=0 and k = n k=n , we recover well-known primal and mixed hybrid methods for the scalar Poisson equation, while for 0 > k > n 0>k>n , we obtain new hybrid finite element methods, including methods for the vector Poisson equation in n = 2 n=2 and n = 3 n=3 dimensions. We also generalize Stenberg postprocessing [RAIRO Modél. Math. Anal. Numér. 25 (1991), pp. 151–167] from k = n k=n to arbitrary k k , proving new superconvergence estimates. Finally, we discuss how this hybridization framework may be extended to include nonconforming and hybridizable discontinuous Galerkin methods.

Robust BDDC algorithms for the Brinkman problem with HDG discretizations

The balancing domain decomposition by constraints methods (BDDC) have been applied to solve the saddle point problem arising from a hybridizable discontinuous Galerkin (HDG) discretization for the Brinkman equations. Edge/face average constraints are enforced across the subdomain interface for each velocity component to ensure that the BDDC preconditioned conjugate gradient (CG) iterations stay in a special subspace, where the reduced system from the original saddle point problem is positive definite. The condition number is proved to be uniformly bounded under both Stokes and Darcy dominant cases with a deluxe scaling for the parameters with jumps only across the subdomain interface. When the parameters are highly discontinuous with large jumps inside each subdomains, additional adaptively chosen primal constraints, obtained by solving local generalized eigenvalue problems, are introduced to control the condition numbers. Numerical experiments confirm the theory.

An interface-tracking space–time hybridizable/embedded discontinuous Galerkin method for nonlinear free-surface flows

We present a compatible space–time hybridizable/embedded discontinuous Galerkin discretization for nonlinear free-surface waves. We pose this problem in a two-fluid (liquid and gas) domain and use a time-dependent level-set function to identify the sharp interface between the two fluids. The incompressible two-fluid equations are discretized by an exactly mass conserving space–time hybridizable discontinuous Galerkin method while the level-set equation is discretized by a space–time embedded discontinuous Galerkin method. Different from alternative discontinuous Galerkin methods is that the embedded discontinuous Galerkin method results in a continuous approximation of the interface. This, in combination with the space–time framework, results in an interface-tracking method without resorting to smoothing techniques or additional mesh stabilization terms.

Afternote to “Coupling at a Distance”: Convergence Analysis and A Priori Error Estimates

Abstract In their article “Coupling at a distance HDG and BEM”, Cockburn, Sayas and Solano proposed an iterative coupling of the hybridizable discontinuous Galerkin method (HDG) and the boundary element method (BEM) to solve an exterior Dirichlet problem. The novelty of the numerical scheme consisted of using a computational domain for the HDG discretization whose boundary did not coincide with the coupling interface. In their article, the authors provided extensive numerical evidence for convergence, but the proof of convergence and the error analysis remained elusive at that time. In this article we fill the gap by proving the convergence of a relaxation of the algorithm and providing a priori error estimates for the numerical solution.

Low-order Prandtl-Glauert-Lorentz based Absorbing Boundary Conditions for solving the convected Helmholtz equation with Discontinuous Galerkin methods

We construct Absorbing Boundary Conditions (ABCs) for the convected Helmholtz equation that are easy to implement in a Hybridizable Discontinuous Galerkin (HDG) formulation. The construction is based upon the Prandtl-Glauert-Lorentz map which transforms the convected Helmholtz equation into the regular Helmholtz equation. The new ABCs are thus issued from classical Bayliss-Gunztburger-Turkel ABCs and are valid for carrier flows that vary inside the computational domain but become uniform far away from the source. They lead to accurate numerical results for low and intermediate Mach numbers using a HDG formulation with the acoustic potential and the total flux as unknowns.

Analysis of Pressure-Robust Embedded-Hybridized Discontinuous Galerkin Methods for the Stokes Problem Under Minimal Regularity

We present analysis of two lowest-order hybridizable discontinuous Galerkin methods for the Stokes problem, while making only minimal regularity assumptions on the exact solution. The methods under consideration have previously been shown to produce \(H(\text {div})\)-conforming and divergence-free approximate velocities. Using these properties, we derive a priori error estimates for the velocity that are independent of the pressure. These error estimates, which assume only \(H^{1+s}\)-regularity of the exact velocity fields for any \(s \in [0, 1]\), are optimal in a discrete energy norm. Error estimates for the velocity and pressure in the \(L^2\)-norm are also derived in this minimal regularity setting. Our theoretical findings are supported by numerical computations.

Continuous Artificial-Viscosity Shock Capturing for Hybrid Discontinuous Galerkin on Adapted Meshes

One technique for capturing shocks with high-order methods is through artificial viscosity. The key considerations of this approach are 1) deciding the amount of artificial viscosity to add; 2) maintaining stability and efficiency of the nonlinear solver; and 3) ensuring accuracy of the resulting solutions, particularly in the presence of strong shocks. To address consideration 1, we test a switch based on intraelement solution variation as well as one based on the difference between the solution and its low-order projection. To address consideration 2, we forego a complete linearization of the artificial-viscosity contribution to the residual in order to keep the residual Jacobian stencil compact. To address consideration 3, we introduce the viscosity in a piecewise-continuous fashion to avoid spurious entropy production. Furthermore, we use output-based error estimation and mesh optimization on the drag and the total enthalpy error, as well as with entropy variables, to minimize the output error. We test the shock capturing method coupled with mesh optimization on aerodynamic flow applications ranging from transonic to supersonic, which are discretized using the standard discontinuous Galerkin (DG) and hybridized DG methods. One of our findings is that the mesh optimization through error sampling and synthesis algorithm does not always generate the ideal mesh in the presence of strong shocks.

Uniform block-diagonal preconditioners for divergence-conforming HDG Methods for the generalized Stokes equations and the linear elasticity equations

Abstract We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming hybridizable discontinuous Galerkin schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning technique (Xu (1994) The Auxiliary Space Method and Optimal Multigrid Preconditioning Techniques for Unstructured Grids, vol. 56. International GAMM-Workshop on Multi-level Methods (Meisdorf), pp. 215–235). A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.

Time- and frequency-domain hybridizable discontinuous Galerkin solvers for the calculation of the Cherenkov radiation

This work proposes novel hybridizable discontinuous Galerkin (HDG) methods, both in the time and in the frequency domain, to accurately compute the Cherenkov radiation emitted by a charged particle travelling in a uniform medium at superluminal speed. The adopted formulations enrich existing HDG approaches for the solution of Maxwell’s equations by including perfectly matched layers (PMLs) to effectively absorb the outgoing waves and Floquet-periodic boundary conditions to connect the boundaries of the computational domain in the direction of the moving charge. A wave propagation problem with smooth solution is used to show the optimal convergence of the HDG variables and the superconvergence of the postprocessed electric field and a second example examines the role of the PML parameters on the absorption of the electromagnetic field. A series of numerical experiments both in 3D and 2D-axisymmetric components show the capability of the proposed methods to faithfully reproduce Cherenkovian effects in different conditions and their high accuracy is confirmed by comparing the numerical results with the Frank–Tamm formula.

A Hybridizable Discontinuous Galerkin and Boundary Element coupling method for electromagnetic simulations

In this paper, the hybridizable discontinuous Galerkin (HDG) method is combined with the electric field integral equation (EFIE) for the numerical simulation of time-harmonic Maxwell’s equation. The key feature of HDG-EFIE is that the HDG method and the boundary element method (BEM) can be coupled naturally through the extra hybrid variable on the faces of elements, the combined method produces a linear system in term of the degrees of freedom of the extra hybrid variable only. A numerical solution was compared to the analytical solution and found to be in excellent agreement.

Generalized multiscale hybridizable discontinuous Galerkin (GMsHDG) method for flows in nonlinear porous media

In this paper, we present a generalized multiscale hybridizable discontinuous Galerkin (GMsHDG) method for nonlinear porous media. We modified the spectral multiscale HDG framework introduced in Efendiev et al. (2015), to solve nonlinear problems. Also, we give projection-based error analysis on the GMsHDG method to numerically solve a nonlinear parabolic problem. The proposed method has two main ingredients: linearization and generating reduced dimensional multiscale spaces. The GMsHDG method yields that the error decreases when the eigenvalue of the local eigenvalue problem for generating a multiscale space increases, as demonstrated in the mathematical error analysis. Through representative numerical experiments, we confirm the reliability of the error estimations and show that the proposed method is practical and efficient.