Hybridizable Discontinuous Galerkin Research Articles

A Coupled Overset Mesh and Hybridizable Discontinuous Galerkin Algorithm for Pseudo-Compressible Flow

Abstract A previously presented overset mesh enabled hybridizable discontinuous Galerkin (HDG) finite element method is extended in this work to an isentropic compressible (pseudo-compressible) fluid. This formulation is a first-principles approach and is complementary to the augmented Lagrangian approach that was utilized in the previous HDG incompressible Navier–Stokes formulations which eliminate the global pressure field. This is the first original presentation combining overset meshes, HDG, and fluid flow, specifically isentropic flow for low Mach number applications. Verification of the code implementation of the proposed overset-HDG formulation is performed via the method of manufactured solutions (MMS) on a successively refined overset mesh configuration containing five meshes, and for order k=1,…,4, Lagrange polynomial elements in both two and three dimensions. Optimal order convergence, k + 1, can be observed in all fields for both the two- and three-dimensional simulations, for each mesh. A two-dimensional benchmark problem is also presented to enable code-to-code comparison as a preliminary validation exercise.

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.

Hybrid coupling of CG and HDG discretizations based on Nitsche’s method

A strategy to couple continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) discretizations based only on the HDG hybrid variable is presented for linear thermal and elastic problems. The hybrid CG-HDG coupling exploits the definition of the numerical flux and the trace of the solution on the mesh faces to impose the transmission conditions between the CG and HDG subdomains. The continuity of the solution is imposed in the CG problem via Nitsche's method, whereas the equilibrium of the flux at the interface is naturally enforced as a Neumann condition in the HDG global problem. The proposed strategy does not affect the core structure of CG and HDG discretizations. In fact, the resulting formulation leads to a minimally-intrusive coupling, suitable to be integrated in existing CG and HDG libraries. Numerical experiments in two and three dimensions show optimal global convergence of the stress and superconvergence of the displacement field, locking-free approximation, as well as the potential to treat structural problems of engineering interest featuring multiple materials with compressible and nearly incompressible behaviors.

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.

A projective hybridizable discontinuous Galerkin mixed method for second-order diffusion problems

Abstract In this paper, we present a hybridizable discontinuous Galerkin (HDG) mixed method for second-order diffusion problems using a projective stabilization function and broken Raviart–Thomas functions to approximate the dual variable. The proposed HDG mixed method is inspired by the primal HDG scheme with reduced stabilization suggested by Lehrenfeld and Schoberl in 2010, and the standard hybridized version of the Raviart–Thomas (H-RT) method. Indeed, we use the broken Raviart–Thomas space of degree k ≥ 0 for the flux, a piecewise polynomial of degree k + 1 for the potential, and a piecewise polynomial of degree k for its numerical trace. This unconventional polynomial combination is made possible by the projective Lehrenfeld–Schoberl (LS) stabilization function. Its introduction and the use of Raviart-Thomas spaces will have beneficial effects: no postprocessing is required to improve the accuracy of the potential uh, and a straightforward flux reconstruction is sufficient to obtain a H(div)–conforming flux variable. The convergence and accuracy of our method are investigated through numerical experiments in two-dimensional space by using h and p refinement strategies. An optimal convergence order ( k + 1 ) for the H(div)-conforming flux and superconvergence ( k + 2 ) for the potential is observed. Comparative tests with the classical H-RT and the well-known hybridizable local discontinuous Galerkin (H-LDG) mixed methods are also performed and exposed in terms of CPU time.

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.

Hybridizable Discontinuous Galerkin Method Contrast Source Inversion of 2-D and 3-D Dielectric and Magnetic Targets

We present a microwave imaging algorithm capable of simultaneously reconstructing electric and magnetic targets from both electric and magnetic field measurement data, with support for arbitrary imaging system boundaries and inhomogeneous background media. A high-order time-harmonic discontinuous Galerkin method (DGM) forward solver is adopted within a contrast source inversion (CSI) algorithm formulated for electric and magnetic targets. Due to its high-order capabilities, the DGM forward solver effectively decouples the contrast and field discretizations without introducing a dual mesh. The drawback of standard DGM forward solvers, namely, their high computational cost, is addressed by a hybridizable DGM (HDGM) formulation. Synthetic and experimental results for both DGM-CSI and HDGM-CSI are presented for a variety of 2-D and 3-D measurement configurations and both dielectric and magnetic targets. HDGM-CSI is shown to have imaging performance comparable to FEM-CSI for dielectric targets, and significant additional flexibility that will enable future electromagnetic imaging applications and system design.

A hybridizable discontinuous Galerkin method for both thin and 3D nonlinear elastic structures

We present a 3D hybridizable discontinuous Galerkin (HDG) method for nonlinear elasticity which can be efficiently used for thin structures with large deformation. The HDG method is developed for a three-field formulation of nonlinear elasticity and is endowed with a number of attractive features that make it ideally suited for thin structures. Regarding robustness, the method avoids a variety of locking phenomena such as membrane locking, shear locking, and volumetric locking. Regarding accuracy, the method yields optimal convergence for the displacements, which can be further improved by an inexpensive postprocessing. And finally, regarding efficiency, the only globally coupled unknowns are the degrees of freedom of the numerical trace on the interior faces, resulting in substantial savings in computational time and memory storage. This last feature is particularly advantageous for thin structures because the number of interior faces is typically small. In addition, we discuss the implementation of the HDG method with arc-length algorithms for phenomena such as snap-through, where the standard load incrementation algorithm becomes unstable. Numerical results are presented to verify the convergence and demonstrate the performance of the HDG method through simple analytical and popular benchmark problems in the literature.

Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin (HDG) method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the "hybridized" versions of several of the most commonly-used finite element methods, including mixed, nonconforming, and discontinuous Galerkin methods. (Interestingly, for the continuous Galerkin method in dimension greater than one, we show that multisymplecticity only holds in a weaker sense.) Consequently, these general-purpose finite element methods may be used for structure-preserving discretization (or semidiscretization) of canonical Hamiltonian systems of ODEs or PDEs. This establishes multisymplecticity for a large class of arbitrarily-high-order methods on unstructured meshes.

An HDG formulation for incompressible and immiscible two-phase porous media flow problems

ABSTRACTWe develop a high-order hybridisable discontinuous Galerkin (HDG) formulation to solve the immiscible and incompressible two-phase flow problem in a heterogeneous porous media. The HDG method is locally conservative, has fewer degrees of freedom than other discontinuous Galerkin methods due to the hybridisation procedure, provides built-in stabilisation for arbitrary polynomial degrees and, if the error of the temporal discretisation is low enough, the pressure, the saturation and their fluxes converge with order P+1 in -norm, being P the polynomial degree. In addition, an element-wise post-process can be applied to obtain a convergence rate of P+2 in -norm for the scalar variables. All of these advantages make the HDG method suitable for solving multiphase flow trough porous media. We show numerical evidence of the convergences rates. Finally, to assess the capabilities of the proposed formulation, we apply it to several cases of water-flooding technique for oil recovery.

An HDG Method for the Time-dependent Drift–Diffusion Model of Semiconductor Devices

We propose a hybridizable discontinuous Galerkin (HDG) finite element method to approximate the solution of the time dependent drift-diffusion problem. This system involves a nonlinear convection diffusion equation for the electron concentration $u$ coupled to a linear Poisson problem for the the electric potential $\phi$. The non-linearity in this system is the product of the $\nabla \phi$ with $u$. An improper choice of a numerical scheme can reduce the convergence rate. To obtain optimal HDG error estimates for $\phi$, $u$ and their gradients, we utilize two different HDG schemes to discretize the nonlinear convection diffusion equation and the Poisson equation. We prove optimal order error estimates for the semidiscrete problem. We also present numerical experiments to support our theoretical results.