Hybridizable Discontinuous Galerkin Research Articles

Accelerated iterative DG finite element solvers for large-scale time-harmonic acoustic problems

Finite element methods are widely used to solve time-harmonic wave propagation problems, but solving large cases can be extremely difficult even with the computational power of parallel computers. In this work, the linear system resulting from the finite element discretization is solved with iterative solution methods, which are efficient in parallel but can require a large number of iterations. In standard discontinuous Galerkin (DG) methods, the numerical solution is discontinuous at the interfaces between the elements. In hybridizable DG methods, additional unknowns are introduced at the interfaces between the finite elements, and the physical unknowns are eliminated from the global system, resulting in a hybridized system. We have recently proposed a new strategy, called CHDG, where the additional unknowns correspond to transmission variables, whereas in the standard approach they are numerical fluxes. This strategy improves the properties of the hybridized system for faster iterative solution procedures. In this talk, we present and study a 3D CHDG implementation with nodal finite element basis functions. The resulting scheme has properties amenable to efficient parallel computing. Numerical results are presented to validate the method, and preliminary 3D computational results are proposed.

Detective generalized multiscale hybridizable discontinuous Galerkin(GMsHDG) method for porous media

The Detective Generalized Multiscale Hybridizable Discontinuous Galerkin (Detective GMsHDG) method aims to further reduce the computational cost of the GMsHDG method. The GMsHDG method itself reduces the computational cost of the HDG method by employing an upscaled structure on a two-grid mesh. Given a PDE within a specified domain, we subdivide the domain into polygonal subdomains and transforms a HDG problem into globular and local problems. Globular problem concerns whether the solutions on smaller domains glue well to form a globular solution. The process involves generation of multiscale spaces, which is a vector space of functions defined on edges of the polygonal regions. A naive approximation by polynomials fails, especially in porous media, necessitating the generation of problem-specific spaces. The Detective GMsHDG method improves this process by replacing the generation of the multiscale space with the detective algorithm. The Detective GMsHDG method has two stages. First is called an offline stage. During the offline stage, we construct a detective function which, given a permeability distribution, it gives a multiscale space. Later stage is called the offline stage where, given the multiscale space, we use GMsHDG method to solve a given PDE numerically. We show numerical results to argue the liability of the solution using the detective GMsHDG method.

An adaptive hybridizable discontinuous Galerkin method for Darcy–Forchheimer flow in fractured porous media

In this paper, we consider an adaptive hybridizable discontinuous Galerkin (HDG) method based on the discrete fracture model for approximation of a single-phase flow in fractured porous media. We are interested in the case that the flow rate in fracture is large enough to justify the use of Forchheimer’s law for modeling flow within the fracture, while Darcy’s law is applied to the surrounding matrix. The HDG method could be designed to simulate the flow in porous media with reduced fractures which consist of many straight lines or planes. More specifically, we use piecewise polynomials of degree [Formula: see text] to approximate the velocity and pressure in fracture and surrounding porous media. The existence and uniqueness of discrete solutions are proved by the Brouwer fixed point theorem, and an efficient and reliable a posteriori error estimator is obtained with respect to an energy norm. Moreover, the HDG scheme, the existence and uniqueness of discrete solutions, and the a posteriori error estimates are also extended to the problem with non-planar, embedded, and intersecting fractures. Finally, several numerical examples are provided to validate the performance of the obtained a posteriori error estimator.

A robust hybridizable discontinuous Galerkin scheme with harmonic averaging technique for steady state of real-world semiconductor devices

Solving real-world nonlinear semiconductor device problems modeled by the drift-diffusion equations coupled with the Poisson equation (also known as the Poisson-Nernst-Planck equations) necessitates an accurate and efficient numerical scheme which can avoid non-physical oscillations even for problems with extremely sharp doping profiles. In this paper, we propose a flexible and high-order hybridizable discontinuous Galerkin (HDG) scheme with harmonic averaging (HA) technique to tackle these challenges. The proposed HDG-HA scheme combines the robustness of finite volume Scharfetter-Gummel (FVSG) method with the high-order accuracy and hp-flexibility offered by the locally conservative HDG scheme. The coupled Poisson equation and two drift-diffusion equations are simultaneously solved by the Newton method. Indicators based on the gradient of net doping and solution variables are proposed to switch between cells with HA technique and high-order conventional HDG cells, utilizing the flexibility of HDG scheme. Numerical results suggest that the proposed scheme does not exhibit oscillations or convergence issues, even when applied to heavily doped and sharp PN-junctions. Devices with circular junctions and realistic doping profiles are simulated in two dimensions, qualifying this scheme for practical simulation of real-world semiconductor devices.

A PROJECTION-BASED ERROR ANALYSIS OF HDG METHODS FOR A NONLINEAR SYSTEM: COUPLED IN BOTH DIRECTIONS

A technique for the error analysis of hybrid discontinuous Galerkin methods (HDG) is applied to a coupled problem. The technique relies on the use of a projection whose design is inspired by the numerical traces of the methods. When this technique appeared in the literature (Cockburn et al., 2010), the authors performed an analysis for an elliptical problem. In this work, we will analyze two coupled elliptical subproblems. The main idea is to apply this projection technique, defined based on the shape of the numerical traces, in the numerical analysis of the HDG method for the coupled problem to verify whether this numerical analysis technique still maintains its main characteristics, such as simplicity and consistency, even with this coupling. The analysis of the coupled problem in two directions was carried out following the strategy of first analyzing the partially coupled problem (De Oliveira, 2024), and then the coupled problem in both directions. The differences between both problems were the analytical development of the coupling and the computation of the approximate solution, which in the problem treated here requires a fixed point algorithm. We expect that the orders of convergence for the approximate solutions correspond to theoretical expectations with the approximation spaces used, which are made up of natural polynomials, that is, order k for vector variables and k+1 for scalar variables. All theoretical and and experimental objectives were achieved, i.e., the convergence in the HDG simulations corresponded to what was theoretically expected.

A Combined Mixed Hybrid and Hybridizable Discontinuous Galerkin Method for Darcy Flow and Transport

A Combined Mixed Hybrid and Hybridizable Discontinuous Galerkin Method for Darcy Flow and Transport

Gradient-Robust Hybrid DG Discretizations for the Compressible Stokes Equations

This paper studies two hybrid discontinuous Galerkin (HDG) discretizations for the velocity-density formulation of the compressible Stokes equations with respect to several desired structural properties, namely provable convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The later property dramatically enhances the accuracy in well-balanced situations, such as the hydrostatic balance where the pressure gradient balances the gravity force. One of the studied schemes employs an H(div)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H(\ extrm{div})$$\\end{document}-conforming velocity ansatz space which ensures all mentioned properties, while a fully discontinuous method is shown to satisfy all properties but the gradient-robustness. Also higher-order schemes for both variants are presented and compared in three numerical benchmark problems. The final example shows the importance also for non-hydrostatic well-balanced states for the compressible Navier–Stokes equations.

An exactly divergence-free hybridized discontinuous Galerkin method for the generalized Boussinesq equations with singular heat source

The purpose of this work is to propose and analyze a hybridized discontinuous Galerkin (HDG) method for the generalized Boussinesq equations with singular heat source. We use polynomials of order k, k − 1 and k to approximate the velocity, the pressure and the temperature. By introducing Lagrange multipliers for the pressure, the approximate velocity field obtained by the HDG method is shown to be exactly divergence-free and H(div)-conforming. Under a smallness assumption on the problem data and solutions, we prove by Brouwer’s fixed point theorem that the discrete system has a solution in two dimensions. If the viscosity and thermal conductivity are further assumed to be positive constants, a priori error estimates with convergence rate O(h) and efficient and reliable a posteriori error estimates are derived. Finally numerical examples illustrate the theoretical analysis and show the performance of the obtained a posteriori error estimator.

A hybridizable discontinuous Galerkin method for the coupled Navier–Stokes/Biot problem

In this paper we present a hybridizable discontinuous Galerkin method for the time-dependent Navier–Stokes equations coupled to the quasi-static poroelasticity equations via interface conditions. We determine a bound on the data that guarantees stability and well-posedness of the fully discrete problem and prove a priori error estimates. A numerical example confirms our analysis.

Hybridizable Discontinuous Galerkin Methods for the Two-Dimensional Monge–Ampère Equation

Hybridizable Discontinuous Galerkin Methods for the Two-Dimensional Monge–Ampère Equation

A flux‐based HDG method

AbstractIn this article, we present a flux‐based formulation of the hybridizable discontinuous Galerkin (HDG) method for steady‐state diffusion problems and propose a new method derived by letting a stabilization parameter tend to infinity. Assuming an inf‐sup condition, we prove its well‐posedness and error estimates of optimal order. We show that the inf‐sup condition is satisfied by some triangular elements. Numerical results are also provided to support our theoretical results.

A hybridizable discontinuous Galerkin method for the dual-porosity-Stokes problem

We introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the dual-porosity-Stokes problem. This coupled problem describes the interaction between free flow in macrofractures/conduits, governed by the Stokes equations, and flow in microfractures/matrix, governed by a dual-porosity model. We prove that the HDG method is strongly conservative, well-posed, and give an a priori error analysis showing dependence on the problem parameters. Our theoretical findings are corroborated by numerical examples.

HDG Method for Nonlinear Parabolic Integro-Differential Equations

Abstract The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ ( h k + 1 ) O(h^{k+1}) when polynomials of degree k ≥ 0 k\geq 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

Conservative solution transfer between anisotropic meshes for time‐accurate hybridized discontinuous Galerkin methods

AbstractWe present a hybridized discontinuous Galerkin (HDG) solver for general time‐dependent balance laws. In particular, we focus on a coupling of the solution process for unsteady problems with our anisotropic mesh refinement framework. The goal is to properly resolve all relevant unsteady features with the smallest possible number of mesh elements, and hence to reduce the computational cost of numerical simulations while maintaining its accuracy. A crucial step is then to transfer the numerical solution between two meshes, as the anisotropic mesh adaptation is producing highly skewed, non‐nested sequences of triangular grids. For this purpose, we adopt the Galerkin projection for the HDG solution transfer as it preserves the conservation of physically relevant quantities and does not compromise the accuracy of high‐order method. We present numerical experiments verifying these properties of the anisotropically adaptive HDG method.

A strongly conservative hybridizable discontinuous Galerkin method for the coupled time-dependent Navier–Stokes and Darcy problem

We present a strongly conservative and pressure-robust hybridizable discontinuous Galerkin method for the coupled time-dependent Navier–Stokes and Darcy problem. We show existence and uniqueness of a solution and present an optimal a priori error analysis for the fully discrete problem when using Backward Euler time stepping. The theoretical results are verified by numerical examples.

A Coupled Hybridizable Discontinuous Galerkin and Boundary Integral Method for Analyzing Electromagnetic Scattering

A Coupled Hybridizable Discontinuous Galerkin and Boundary Integral Method for Analyzing Electromagnetic Scattering

A unified hp-HDG framework for Friedrichs' PDE systems

This work proposes a unified hp-adaptivity framework for hybridized discontinuous Galerkin (HDG) method for a large class of partial differential equations (PDEs) of Friedrichs' type. In particular, we present unified hp-HDG formulations for abstract one-field and two-field structures and prove their well-posedness. In order to handle non-conforming interfaces we simply take advantage of HDG built-in mortar structures. With split-type mortars and the approximation space of trace, a numerical flux can be derived via Godunov approach and be naturally employed without any additional treatment. As a consequence, the proposed formulations are parameter-free. We perform several numerical experiments for time-independent and linear PDEs including elliptic, hyperbolic, and mixed-type to verify the proposed unified hp-formulations and demonstrate the effectiveness of hp-adaptation. Two adaptivity criteria are considered: one is based on a simple and fast error indicator, while the other is rigorous but more expensive using an adjoint-based error estimate. The numerical results show that these two approaches are comparable in terms of convergence rate even for problems with strong gradients, discontinuities, and singularities.

Toward discretization-consistent closure schemes for large eddy simulation using reinforcement learning

This study proposes a novel method for developing discretization-consistent closure schemes for implicitly filtered large eddy simulation (LES). Here, the induced filter kernel and, thus, the closure terms are determined by the properties of the grid and the discretization operator, leading to additional computational subgrid terms that are generally unknown in a priori analysis. In this work, the task of adapting the coefficients of LES closure models is thus framed as a Markov decision process and solved in an a posteriori manner with reinforcement learning (RL). This optimization framework is applied to both explicit and implicit closure models. The explicit model is based on an element-local eddy viscosity model. The optimized model is found to adapt its induced viscosity within discontinuous Galerkin (DG) methods to homogenize the dissipation within an element by adding more viscosity near its center. For the implicit modeling, RL is applied to identify an optimal blending strategy for a hybrid DG and finite volume (FV) scheme. The resulting optimized discretization yields more accurate results in LES than either the pure DG or FV method and renders itself as a viable modeling ansatz that could initiate a novel class of high-order schemes for compressible turbulence by combining turbulence modeling with shock capturing in a single framework. All newly derived models achieve accurate results that either match or outperform traditional models for different discretizations and resolutions. Overall, the results demonstrate that the proposed RL optimization can provide discretization-consistent closures that could reduce the uncertainty in implicitly filtered LES.

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.

EXtended Hybridizable Discontinuous Galerkin (X-HDG) method for linear convection-diffusion equations on unfitted domains

In this work, we propose a novel strategy for the numerical solution of linear convection diffusion equation (CDE) over unfitted domains. In the proposed numerical scheme, strategies from high order Hybridized Discontinuous Galerkin method and eXtended Finite Element method are combined with the level set definition of the boundaries. The proposed scheme and hence, is named as eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted domains; i.e., the computational mesh does not need to fit to the domain boundary; instead, the boundary is defined by a level set function and cuts through the background mesh arbitrarily. The original unknown structure of HDG and its hybrid nature ensuring the local conservation of fluxes is kept, while developing a modified bilinear form for the elements cut by the boundary. At every cut element, an auxiliary nodal trace variable on the boundary is introduced, which is eliminated afterwards while imposing the boundary conditions. Both stationary and time dependent CDEs are studied over a range of flow regimes from diffusion to convection dominated; using high order (p≤4) XHDG through benchmark numerical examples over arbitrary unfitted domains. Results proved that XHDG inherits optimal (p+1) and super (p+2) convergence properties of HDG while removing the fitting mesh restriction.