Hybridizable Discontinuous Galerkin Method Research Articles

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.

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.

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.

Symplectic Hamiltonian finite element methods for electromagnetics

We present several high-order accurate finite element methods for the Maxwell’s equations which provide time-invariant, non-drifting approximations to the total electric and magnetic charges, and to the total energy. We devise these methods by taking advantage of the Hamiltonian structures of the Maxwell’s equations as follows. First, we introduce spatial discretizations of the Maxwell’s equations using mixed finite element, discontinuous Galerkin, and hybridizable discontinuous Galerkin methods to obtain a semi-discrete system of equations which display discrete versions of the Hamiltonian structure of the Maxwell’s equations. Then we discretize the resulting semi-discrete system in time by using a symplectic integrator. This ensures the conservation properties of the fully discrete system of equations. For the Symplectic Hamiltonian HDG method, we present numerical experiments which confirm its optimal orders of convergence for all variables and its conservation properties for the total linear and angular momenta, as well as the total energy. Finally, we discuss the extension of our results to other boundary conditions and to numerical schemes defined by different weak formulations.

Analysis of hybrid discontinuous Galerkin methods for linearized Navier–Stokes equations

AbstractIn this article, we introduce and analyze arbitrary‐order, locally conservative hybrid discontinuous Galkerin methods for linearized Navier–Stokes equations. The unknowns of the global system are reduced to trace variables on the skeleton of a triangulation and the average of pressure on each cell via embedded static condensation. We prove that the lifting operator associated with trace variables is injective for any polynomial degree. This generalizes the result in (Y. Jeon and E.‐J. Park, Numerische Mathematik 123 [2013], no. 1, pp. 97–119), where quadratic and cubic rectangular elements are analyzed. Moreover, optimal error estimates in the energy norm are obtained by introducing nonstandard projection operators for the hybrid DG method. Several numerical results are presented to show the performance of the algorithm and to validate the theory developed in the article.

Convergence Analysis of the Fully Discrete Hybridizable Discontinuous Galerkin Method for the Allen–Cahn Equation Based on the Invariant Energy Quadratization Approach

Convergence Analysis of the Fully Discrete Hybridizable Discontinuous Galerkin Method for the Allen–Cahn Equation Based on the Invariant Energy Quadratization Approach

Domain decomposition and upscaling technique for metascreens

PurposeThis paper aims to consider a multiscale electromagnetic wave problem for a housing with a ventilation grill. Using the standard finite element method to discretise the apertures leads to an unduly large number of unknowns. An efficient approach to simulate the multiple scales is introduced. The aim is to significantly reduce the computational costs.Design/methodology/approachA domain decomposition technique with upscaling is applied to cope with the different scales. The idea is to split the domain of computation into an exterior domain and multiple non-overlapping sub-domains. Each sub-domain represents a single aperture and uses the same finite element mesh. The identical mesh of the sub-domains is efficiently exploited by the hybrid discontinuous Galerkin method and a Schur complement which facilitates the transition from fine meshes in the sub-domains to a coarse mesh in the exterior domain. A coarse skeleton grid is used on the interface between the exterior domain and the individual sub-domains to avoid large dense blocks in the finite element discretisation matrix.FindingsApplying a Schur complement to the identical discretisation of the sub-domains leads to a method that scales very well with respect to the number of apertures.Originality/valueThe error compared to the standard finite element method is negligible and the computational costs are significantly reduced.

Partial differential equations on hypergraphs and networks of surfaces: Derivation and hybrid discretizations

We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces – generalized by the term hypergraphs. To this end, we consider PDEs on hypergraphs as singular limits of PDEs in networks of thin domains (such as fault planes, pipes, etc.), and we observe that (mixed) hybrid formulations offer useful tools to formulate such PDEs. Thus, our numerical framework is based on hybrid finite element methods (in particular, the class of hybrid discontinuous Galerkin methods).

An adjoint-based adaptive error approximation of functionals by the hybridizable discontinuous Galerkin method for second-order elliptic equations

This paper presents a novel, fully computable error approximation and mesh adaptation approach for functionals defined by second-order elliptic equations. The functionals are approximated by the hybridizable discontinuous Galerkin (HDG) method and the error approximation is obtained by the adjoint-based method and a local post-processing technique of the HDG method. Unlike most adjoint-based error estimations in the literature, the novelty of our method is that the error approximation is obtained without requiring an auxiliary finer mesh or higher order approximation spaces for solving the adjoint problem. This reduces the computational cost and eases the implementation of the problem. What's more, the local post-processing technique can be carried out in parallel, which speeds up the method even more. We illustrate the method with a second-order elliptic problem and we present examples of three types of functionals: volume integrals; boundary integrals and eigenvalue problems. Numerical tests with adaptive mesh refinements for non-smooth solutions are presented to show that our method is efficient and robust.

A hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac delta source

In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in L2-norm is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in L2-norm and W1,p-seminorm are also obtained. Finally, numerical examples are provided to validate the theoretical analysis.

Recent Upgrades in a 2D Turbulent Transport Solver Based on a Hybrid Discontinuous Galerkin Method for the Simulation of Fusion Plasma in Tokamak

The simulation of fusion plasmas in realistic magnetic configurations and tokamak geometries still requires the development of advanced numerical algorithms owing to the complexity of the problem. In this context, we propose a Hybrid Discontinuous Galerkin (HDG) method to solve 2D transport fluid equations in realistic magnetic and tokamak wall geometries. This high-order solver can handle magnetic equilibrium free structured and unstructured meshes allowing a much more accurate discretization of the plasma facing components than current solvers based on magnetic field aligned methods associated with finite-differences (volumes) discretization. In addition, the method allows for handling realistic magnetic equilibrium, eventually non steady, a critical point in the modeling of full discharges including ramp up and ramp down phases. In this paper, we introduce the HDG algorithm with a special focus on recent developments related to the treatment of the cross-field diffusive terms, and to an adaptive mesh refinement technique improving the numerical efficiency and robustness of the scheme. The updated solver is verified with a manufactured solution method, and numerical tests are provided to illustrate the new capabilities of the code.

Preconditioning for a Pressure-Robust HDG Discretization of the Stokes Equations

We introduce a new preconditioner for a recently developed pressure-robust hybridized discontinuous Galerkin (HDG) finite element discretization of the Stokes equations. A feature of HDG methods is the straightforward elimination of degrees-of-freedom defined on the interior of an element. In our previous work (J. Sci. Comput., 77(3):1936--1952, 2018) we introduced a preconditioner for the case in which only the degrees-of-freedom associated with the element velocity were eliminated via static condensation. In this work we introduce a preconditioner for the statically condensed system in which the element pressure degrees-of-freedom are also eliminated. In doing so the number of globally coupled degrees-of-freedom are reduced, but at the expense of a more difficult problem to analyse. We will show, however, that the Schur complement of the statically condensed system is spectrally equivalent to a simple trace pressure mass matrix. This result is used to formulate a new, provably optimal preconditioner. Through numerical examples in two- and three-dimensions we show that the new preconditioned iterative method converges in fewer iterations, has superior conservation properties for inexact solves, and is faster in CPU time when compared to our previous preconditioner.

Spatial adaptivity in SOLEDGE3X‐HDG for edge plasma simulations in versatile magnetic and reactor geometries

AbstractWith the ultimate goal to predict plasmas heat and particle fluxes in ITER operation, more efforts are required to deal with realistic magnetic configurations and tokamak geometries. In an attempt to achieve this goal, we propose an adaptive mesh refinement method added to a fluid solver based on a high‐order hybrid discontinuous Galerkin (HDG) method. Based on unstructured meshes, this magnetic equilibrium free numerical scheme has shown promising and encouraging features to solve 2D/3D transport reduced Braginski fluid equations. To improve its numerical efficiency, a mesh refinement based on h‐adpativity is investigated. We describe here an adaptive refinement strategy on a reduced edge particle transport model based on electron density and parallel momentum. This strategy is illustrated in realistic tokamak wall geometry. Computations performed show potential gains in the required number of degrees of freedom against benchmark computations with uniform meshes, along with the potential to give an automated, goal‐oriented, mesh generation technique for edge transport simulations in 2D.

Point-Wise Divergence-Free Extended Hybridizable Discontinuous Galerkin Method for Two-Phase Stokes Flow with Unfitted Meshes and Interfaces

Point-Wise Divergence-Free Extended Hybridizable Discontinuous Galerkin Method for Two-Phase Stokes Flow with Unfitted Meshes and Interfaces

A stabilized hybrid discontinuous Galerkin method for the Cahn–Hilliard equation

In this paper, we present a stabilized mixed hybrid discontinuous Galerkin finite element method for solving fourth-order parabolic problems such as the Cahn–Hilliard equation used to describe the dynamics of avascular tumor growth. The proposed method is compared to other methods previously studied in this context. Several numerical experiments are presented to show convergence, performance and accuracy comparisons with other methods. Besides, we also show the ability of the method to solve a complex case of avascular tumor growth where the behavior of tumor cells towards nutrient gradients drives fingering instabilities. The results show the great potential of the presented method for the efficient and accurate solution of Cahn–Hilliard problems which includes applications on complex tumor growth problems.

A superconvergent hybridizable discontinuous Galerkin method for weakly compressible magnetohydrodynamics

This work proposes a hybridizable discontinuous Galerkin (HDG) method for the solution of magnetohydrodynamic (MHD) problems with weakly compressible flows. A novel fluid formulation that adopts the velocity and the pressure as primal variables is first derived and its superior properties, compared to alternative density–momentum-based approaches, are demonstrated on a simple benchmark. The coupled MHD formulation exhibits superconvergence properties for both the fluid velocity and the magnetic induction, a feature not present in any HDG formulation published in this field. An alternative MHD formulation, adopting a fluid-type solver for the solution of the magnetic subproblem, is also considered and its advantages and disadvantages are discussed. The convergence properties of the proposed formulations for the single physics and for the coupled problem are examined on an extensive set of numerical examples in both two and three dimensions, on structured and unstructured meshes and at low and high Hartmann numbers.

An adjoint-based super-convergent Galerkin approximation of eigenvalues

We present a new method for computing high-order accurate approximations of eigenvalues defined in terms of Galerkin approximations. We consider the eigenvalue as a non-linear functional of its corresponding eigenfunction and show how to extend the adjoint-based approach proposed in Cockburn and Wang (2017) [14], to compute it. We illustrate the method on a second-order elliptic eigenvalue problem. Our extensive numerical results show that the approximate eigenvalues computed by our method converge with a rate of 4k+2 when tensor-product polynomials of degree k are used for the Galerkin approximations. In contrast, eigenvalues obtained by standard finite element methods such as the mixed method or the discontinuous Galerkin method converge with a rate at most of 2k+2. We present numerical results which show the performance of the method for the classic unit square and L-shaped domains, and for the quantum harmonic oscillator. We also present experiments uncovering a new adjoint-corrected approximation of the eigenvalues provided by the hybridizable discontinuous Galerkin method which converges with order 2k+2, as well as results showing the possibilities and limitations of using the adjoint-correction term as an asymptotically exact a posteriori error estimate.