High-order Hybridizable Discontinuous Galerkin Research Articles

An unfitted HDG method for a distributed optimal control problem

We analyze a high order hybridizable discontinuous Galerkin (HDG) method for an optimal control problem where the computational mesh does not necessarily fit the domain. The method is based on transferring the boundary data to the computational boundary by integrating the approximation of the gradient. We prove optimal order of convergence in the L2-norm for all the variables of the state and adjoint problems, and the control variable as well. More precisely, order hk+1 if the local discrete spaces are constructed using polynomials of degree at most k on a triangulation of meshsize h. We present numerical experiments illustrating the performance of method.

A high order unfitted hybridizable discontinuous Galerkin method for linear elasticity

Abstract This work analyses a high-order hybridizable discontinuous Galerkin (HDG) method for the linear elasticity problem in a domain not necessarily polyhedral. The domain is approximated by a polyhedral computational domain where the HDG solution can be computed. The introduction of the rotation as one of the unknowns allows us to use the gradient of the displacements to obtain an explicit representation of the boundary data in the computational domain. The boundary data is transferred from the true boundary to the computational boundary by line integrals, where the integrand depends on the Cauchy stress tensor and the rotation. Under closeness assumptions between the computational and true boundaries, the scheme is shown to be well-posed and optimal error estimates are provided even in the nearly incompressible case. Numerical experiments in two dimensions are presented.

A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems

We present a priori and a posteriori error analysis of a high order hybridizable discontinuous Galerkin (HDG) method applied to a semi-linear elliptic problem posed on a piecewise curved, non polygonal domain. We approximate \(\Omega \) by a polygonal subdomain \(\Omega _h\) and propose an HDG discretization, which is shown to be optimal under mild assumptions related to the non-linear source term and the distance between the boundaries of the polygonal subdomain \(\Omega _h\) and the true domain \(\Omega \). Moreover, a local non-linear post-processing of the scalar unknown is proposed and shown to provide an additional order of convergence. A reliable and locally efficient a posteriori error estimator that takes into account the error in the approximation of the boundary data of \(\Omega _h\) is also provided.

High‐order hybridizable discontinuous Galerkin formulation with fully implicit temporal schemes for the simulation of two‐phase flow through porous media

AbstractWe present a memory‐efficient high‐order hybridizable discontinuous Galerkin (HDG) formulation coupled with high‐order fully implicit Runge‐Kutta schemes for immiscible and incompressible two‐phase flow through porous media. To obtain the same high‐order accuracy in space and time, we propose using high‐order temporal schemes that allow using large time steps. Therefore, we require unconditionally stable temporal schemes for any combination of element size, polynomial degree, and time step. Specifically, we use the Radau IIA and Gauss‐Legendre schemes, which are unconditionally stable, achieve high‐order accuracy with few stages, and do not suffer order reduction in this problem. To reduce the memory footprint of coupling these spatial and temporal high‐order schemes, we rewrite the nonlinear system. In this way, we achieve a better sparsity pattern of the Jacobian matrix and less coupling between stages. Furthermore, we propose a fix‐point iterative method to further reduce the memory consumption. The saturation solution may present sharp fronts. Thus, the high‐order approximation may contain spurious oscillations. To reduce them, we introduce artificial viscosity. We detect the elements with high‐oscillations using a computationally efficient shock sensor obtained from the saturation solution and the post‐processed saturation of HDG. Finally, we present several examples to assess the capabilities of our formulation.

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 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.

Combined Hybridizable Discontinuous Galerkin (HDG) and Runge-Kutta Discontinuous Galerkin (RK-DG) formulations for Green-Naghdi equations on unstructured meshes

In this paper, we introduce some new high-order discrete formulations on general unstructured meshes, especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations. Working with a recent family of optimized asymptotically equivalent equations, we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlinear shallow water equations supplemented with several algebraic source terms, which globally account for the non-hydrostatic effects through the introduction of auxiliary coupling variables. High-order discrete approximations of the main flow variables are obtained with a RK-DG method, while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second-order linear elliptic sub-problems with high-order HDG formulations. The combined use of hybrid unknowns and local post-processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches. The proposed formulation is then extended to a more complex family of three parameters enhanced Green-Naghdi equations. The resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies, showing good convergence properties and very good agreements with several sets of experimental data.

A high order hybridizable discontinuous Galerkin method for incompressible miscible displacement in heterogeneous media

An hybridizable discontinuous Galerkin method of arbitrary high order is formulated to solve the miscible displacement problem in porous media. The spatial discretization is combined with a sequential algorithm that decouples the flow and the transport equations. Hybridization produces a linear system for the globally coupled degrees of freedom, that is smaller in size compared to the system resulting from the interior penalty discontinuous Galerkin methods. We study the impact of increasing the polynomial order on the accuracy of the solution. Numerical experiments show that the method converges optimally and that it is robust for highly heterogeneous porous media in two and three dimensions.

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.

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).

A Particle-in-Cell solver based on a high-order hybridizable discontinuous Galerkin spectral element method on unstructured curved meshes

A high-order hybridizable discontinuous Galerkin spectral element method (HDGSEM) for Particle-In-Cell (PIC) schemes is presented for the simulation of electrostatic applications on three-dimensional unstructured curved meshes. The electrostatic Poisson equation is solved and optionally a Boltzmann relation for the electron species can be used which leads to non-linear source terms. The hybridizable formulation reduces the total number of unknowns of the field solver, allowing the simulation of large problems. The implementation of the HDGSEM solver in a PIC code is described and validated using several test cases with successively increasing complexity. It is shown that the high-order convergence properties are retained on curvilinear meshes, likewise when material jumps are introduced. The simulation of an ion optic illustrates the applicability of the presented method for complex geometries and large problem sizes.

A High Order HDG Method for Stokes Flow in Curved Domains

We propose and analyze a high order hybridizable discontinuous Galerkin (HDG) method for the Stokes equations in a curved domain. It is based on approximating the domain by a polyhedral computational subdomain where an HDG solution is computed. To obtain a high order approximation of the Dirichlet boundary data in the computational domain, we employ a transferring technique based on integrating the approximation of the gradient. In addition, we first seek for a discrete pressure having zero-mean in the computational domain and then the zero-mean condition in the entire domain is recovered by a post-process that involves an extrapolation of the discrete pressure. We prove that the method provides optimal order of convergence for the approximations of the pressure, the velocity and its gradient, that is, order $$h^{k+1}$$ if the local discrete spaces are constructed using polynomials of degree k and the meshsize is h. We present numerical experiments validating the method.

A high-order hybridizable discontinuous Galerkin method with fast convergence to steady-state solutions of the gas kinetic equation

The mass flow rate of Poiseuille flow of rarefied gas through long ducts of two-dimensional cross-sections with arbitrary shape is critical in the pore-network modeling of gas transport in porous media. Here, for the first time, the high-order hybridizable discontinuous Galerkin (HDG) method is used to find the steady-state solution of the linearized Bhatnagar–Gross–Krook equation on two-dimensional triangular meshes. The velocity distribution function and its traces are approximated in piecewise polynomial spaces (of degree up to 4) on the triangular meshes and mesh skeletons, respectively. By employing a numerical flux that is derived from the first-order upwind scheme and imposing its continuity weakly on the mesh skeletons, global systems for unknown traces are obtained with fewer coupled degrees of freedom when compared to the original discontinuous Galerkin formulation. To achieve fast convergence to the steady-state solution, a diffusion-like equation for flow velocity, which is asymptotic-preserving into the fluid dynamic limit, is solved by the HDG simultaneously on the same meshes. The proposed HDG-synthetic iterative scheme is proved to be accurate and efficient. Specifically, for flows in the near-continuum regime, numerical simulations have shown that, to achieve the same level of accuracy, our scheme could be faster than the conventional iterative scheme by two orders of magnitude, also it is faster than the synthetic iterative scheme based on the finite difference discretization in the spatial space by one order of magnitude. In addition, the implicit HDG method is more efficient than an explicit discontinuous Galerkin gas kinetic solver, as well as the implicit discontinuous Galerkin scheme when the degree of approximating polynomial is larger than 2. The HDG-synthetic iterative scheme is ready to be extended to simulate rarefied gas mixtures and the Boltzmann collision operator.

A hybrid-mesh hybridizable discontinuous Galerkin method for solving the time-harmonic Maxwell’s equations

This work presents a high-order hybridizable discontinuous Galerkin (HDG) method based on hybrid mesh to solve the two-dimensional time-harmonic Maxwell’s equations. The hybrid mesh consists of unstructured triangular mesh in the domain with complex geometries and structured quadrilateral mesh in the rest domain. The coupling of different meshes is natural in HDG framework. Numerical simulations show that the HDG method on hybrid mesh converges at the optimal rate. One can save computation costs by employing hybrid mesh due to the reduction in number of degrees of freedom (DOFs).

A High Order HDG Method for Curved-Interface Problems Via Approximations from Straight Triangulations

We propose a novel technique to solve elliptic problems involving a non-polygonal interface/boundary. It is based on a high order hybridizable discontinuous Galerkin method where the mesh does not exactly fit the domain. We first study the case of a curved-boundary value problem with mixed boundary conditions since it is crucial to understand the applicability of the technique to curved interfaces. The Dirichlet data is approximated by using the transferring technique developed in a previous paper. The treatment of the Neumann data is new. We then extend these ideas to curved interfaces. We provide numerical results showing that, in order to obtain optimal high order convergence, it is desirable to construct the computational domain by interpolating the boundary/interface using piecewise linear segments. In this case the distance of the computational domain to the exact boundary is only $$O(h^2)$$O(h2).

Hybridizable discontinuous Galerkin projection methods for Navier–Stokes and Boussinesq equations

Schemes for the incompressible Navier–Stokes and Boussinesq equations are formulated and derived combining the novel Hybridizable Discontinuous Galerkin (HDG) method, a projection method, and Implicit–Explicit Runge–Kutta (IMEX-RK) time-integration schemes. We employ an incremental pressure correction and develop the corresponding HDG finite element discretization including consistent edge-space fluxes for the velocity predictor and pressure correction. We then derive the proper forms of the element-local and HDG edge-space final corrections for both velocity and pressure, including the HDG rotational correction. We also find and explain a consistency relation between the HDG stability parameters of the pressure correction and velocity predictor. We discuss and illustrate the effects of the time-splitting error. We then detail how to incorporate the HDG projection method time-split within standard IMEX-RK time-stepping schemes. Our high-order HDG projection schemes are implemented for arbitrary, mixed-element unstructured grids, with both straight-sided and curved meshes. In particular, we provide a quadrature-free integration method for a nodal basis that is consistent with the HDG method. To prevent numerical oscillations, we develop a selective nodal limiting approach. Its applications show that it can stabilize high-order schemes while retaining high-order accuracy in regions where the solution is sufficiently smooth. We perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented. To complete these studies and to illustrate a range of properties for our new schemes, we employ an unsteady tracer advection benchmark, a manufactured solution for the steady diffusion and Stokes equations, and a standard lock-exchange Boussinesq problem.

Numerical investigation of a high order hybridizable discontinuous Galerkin method for 2d time‐harmonic Maxwell's equations

PurposeThis work is concerned with the development and the numerical investigation of a hybridizable discontinuous Galerkin (HDG) method for the simulation of two‐dimensional time‐harmonic electromagnetic wave propagation problems.Design/methodology/approachThe proposed HDG method for the discretization of the two‐dimensional transverse magnetic Maxwell equations relies on an arbitrary high order nodal interpolation of the electromagnetic field components and is formulated on triangular meshes. In the HDG method, an additional hybrid variable is introduced on the faces of the elements, with which the element‐wise (local) solutions can be defined. A so‐called conservativity condition is imposed on the numerical flux, which can be defined in terms of the hybrid variable, at the interface between neighbouring elements. The linear system of equations for the unknowns associated with the hybrid variable is solved here using a multifrontal sparse LU method. The formulation is given, and the relationship between the considered HDG method and a standard upwind flux‐based DG method is also examined.FindingsThe approximate solutions for both electric and magnetic fields converge with the optimal order of p+1 in L2 norm, when the interpolation order on every element and every interface is p and the sought solution is sufficiently regular. The presented numerical results show the effectiveness of the proposed HDG method, especially when compared with a classical upwind flux‐based DG method.Originality/valueThe work described here is a demonstration of the viability of a HDG formulation for solving the time‐harmonic Maxwell equations through a detailed numerical assessment of accuracy properties and computational performances.

An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations

We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier–Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure–velocity formulation of the incompressible Navier–Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k+1 in the L2-norm, when polynomials of degree k⩾0 are used for all components of the approximate solution. And third, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H(div)-conforming, and converges with order k+2 for k⩾1 and with order 1 for k=0 in the L2-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.

An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations

In this paper, we present hybridizable discontinuous Galerkin methods for the numerical solution of steady and time-dependent nonlinear convection–diffusion equations. The methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary. Applying the Newton–Raphson procedure and the hybridization technique, we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every Newton iteration. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. We then extend the method to time-dependent problems by approximating the time derivative by means of backward difference formulae. When the time-marching method is ( p + 1 ) th order accurate and when polynomials of degree p ⩾ 0 are used to represent the scalar variable, each component of the flux and the approximate trace, we observe that the approximations for the scalar variable and the flux converge with the optimal order of p + 1 in the L 2 -norm. Finally, we apply element-by-element postprocessing schemes to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous interelement normal component, is shown to converge with order p + 1 in the L 2 -norm. The new approximate scalar variable is shown to converge with order p + 2 in the L 2 -norm. The postprocessing is performed at the element level and is thus much less expensive than the solution procedure. For the time-dependent case, the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required. Extensive numerical results are provided to demonstrate the performance of the present method.

An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations

We present a hybridizable discontinuous Galerkin method for the numerical solution of steady and time-dependent linear convection–diffusion equations. We devise the method as follows. First, we express the approximate scalar variable and corresponding flux within each element in terms of an approximate trace of the scalar variable along the element boundary. We then define a unique value for the approximate trace by enforcing the continuity of the normal component of the flux across the element boundary; a global equation system solely in terms of the approximate trace is thus obtained. The high number of globally coupled degrees of freedom in the discontinuous Galerkin approximation is therefore significantly reduced. If the problem is time-dependent, we discretize the time derivative by means of backward difference formulae. This results in efficient schemes capable of producing high-order accurate solutions in space and time. Indeed, when the time-marching method is ( p + 1 ) th order accurate and when polynomials of degree p ⩾ 0 are used to represent the scalar variable, the flux and the approximate trace, we observe that the approximations for the scalar variable, the flux and the trace of the scalar variable converge with the optimal order of p + 1 in the L 2 -norm. Finally, we introduce a simple element-by-element postprocessing scheme to obtain new approximations of the flux and the scalar variable. The new approximate flux, which has a continuous inter-element normal component, is shown to converge with order p + 1 in the L 2 -norm. The new approximate scalar variable is shown to converge with order p + 2 in the L 2 -norm. For the time-dependent case, the postprocessing does not need to be applied at each time-step but only at the times for which an enhanced solution is required. Moreover, the postprocessing procedure is less expensive than the solution procedure, since it is performed at the element level. Extensive numerical results are presented to demonstrate the convergence properties of the method.