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A variational formulation based on velocity and stress is developed for linear fluid–structure interaction problems. The well-posedness and energy stability of this formulation are established. A hybridizable discontinuous Galerkin method is employed to discretize the problem. An hp-convergence analysis is performed for the resulting semi-discrete scheme. The Crank–Nicolson method is used for temporal discretization, and the convergence properties of the fully discrete scheme are examined. Numerical experiments are presented to validate the theoretical results and demonstrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

We propose and analyze a hybridizable discontinuous Galerkin (HDG) method for solving a mixed magnetic advection–diffusion problem within a more general Friedrichs system framework. With carefully constructed numerical traces, we introduce two distinct stabilization parameters: τ t for the tangential trace and τ n for the normal trace. These parameters are tailored to satisfy different requirements, ensuring the stability and convergence of the method. Furthermore, we incorporate a weight function to facilitate the establishment of stability conditions. We also investigate an elementwise postprocessing technique that proves to be effective for both two-dimensional and three-dimensional problems in terms of broken H (curl) semi-norm accuracy improvement. Extensive numerical examples are presented to showcase the performance and effectiveness of the HDG method and the postprocessing techniques. [ABSTRACT FROM AUTHOR]

In this paper, a hybridizable discontinuous Galerkin (HDG) model order reduction technique is proposed to solve the variable coefficient advection equation. In order to obtain a high precision original full order model (FOM), the HDG and diagonally implicit Runge–Kutta (DIRK) methods are used for space and time discretization, respectively. The obtained FOM can achieve higher order accuracy in both space and time. Then, we introduce POD method and Galerkin projection to construct the reduced order model (ROM). Compared with the FOM, the proposed ROM can maintain the same higher order accuracy and greatly reduce the computational cost. Finally, some numerical results are illustrated to confirm the validity and higher order accuracy of the proposed reduced order HDG method. [ABSTRACT FROM AUTHOR]

We present the first a priori error analysis of a class of space-discretizations by Hybridizable Discontinuous Galerkin (HDG) methods for the time-dependent Maxwell's equations introduced in Sánchez et al. [Comput. Methods Appl. Mech. Eng. 396 (2022) 114969]. The distinctive feature of these discretizations is that they display a discrete version of the Hamiltonian structure of the original Maxwell's equations. This is why they are called ''Hamiltonian" HDG methods. Because of this, when combined with symplectic time-marching methods, the resulting methods display an energy that does not drift in time. We provide a single analysis for several of these methods by exploiting the fact that they only differ by the choice of the approximation spaces and the stabilization functions. We also introduce a new way of discretizing the static Maxwell's equations in order to define the initial condition in a manner consistent with our technique of analysis. Finally, we present numerical tests to validate our theoretical orders of convergence and to explore the convergence properties of the method in situations not covered by our analysis. [ABSTRACT FROM AUTHOR]

In this paper, we address the study of elliptic boundary value problems in presence of a boundary condition of integral type (IBC) where the potential is an unknown constant and the flux (the integral of the flux density) over a portion of the boundary is given by a value or a coupling condition. We first motivate our work with realistic examples from nano-electronics, high field magnets and ophthalmology. We then define a general framework stemming from the Hybridizable Discontinuous Galerkin method that accounts naturally for the IBC and we provide a complete analysis at continuous and discrete levels. The implementation in the Feel++framework is then detailed and the convergence and scalability properties are verified. Finally, numerical experiments performed on the real-life motivating applications are used to illustrate our methodology. [ABSTRACT FROM AUTHOR]

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 hybridizable discontinuous Galerkin (HDG) scheme with harmonic averaging technique. • Combine the robustness of finite volume Scharfetter-Gummel method with the hp-flexibility of HDG scheme. • Test with the real-world semiconductor devices. [ABSTRACT FROM AUTHOR]

Conjugate heat transfer (CHT) problem of flow around a fixed cylinder is examined by using a high-order method which is based on the hybridizable discontinuous Galerkin (HDG) method. The present numerical method based on HDG discretization produces a system of equations in which the energy equation of fluid is coupled with that of solid while the continuity of heat-flux at the fluid-solid interface is automatically satisfied. We Investigate the effect of the conductivity ratio on the temperature distribution inside the cylinder and more importantly, the constraint of heat-flux continuity at the fluid-solid interface. The present high-order solutions are compared with low-order solutions by finite volume method of ANSYS, especially in terms of the constraint of heat-flux continuity at the interface. We show that the present high-order method provides accurate solutions and satisfies the constraint of heat-flux continuity better than ANSYS even with the use of a coarse grid. Furthermore, we have derived a numerical correlation between the Nusselt and the Reynolds number by using the fact that the surface temperature of the cylinder is nearly constant when conductivity ratio is larger than order of hundred. The proposed numerical correlation was found to be close to that from the exiting experiment. • The HDG formulation for high-order simulation of conjugate heat transfer is employed. • Th present coupled formulation of energy equation can trivially satisfy the heat-flux continuity. • The proposed high-order method outperforms a low-order method based on Finite Volume Method. • The present method can accurately resolve a very steep thermal boundary layer of fluid region. • A correlation for Nu-Re is derived for a flow around a cylinder by using energy balance. [ABSTRACT FROM AUTHOR]

We present the hybridization of flux reconstruction methods for advection-diffusion problems. Hybridization introduces a new variable into the problem so that it can be reduced via static condensation. This allows the solution of implicit discretizations to be done more efficiently. We derive an energy statement from a stability analysis considering a range of correction functions on hybridized and embedded flux reconstruction schemes. Then, we establish connections to standard formulations. We devise a post-processing scheme that leverages existing flux reconstruction operators to enhance accuracy for diffusion-dominated problems. Results show that the implicit convergence of these methods for advection-diffusion problems can result in performance benefits of over an order of magnitude. In addition, we observe that the superconvergence property of hybridized methods can be extended to the family of FR schemes for a range of correction functions. • We compare hybridized and standard flux reconstruction schemes for advection-diffusion. • Super-accuracy can be achieved for a range of HFR schemes in the diffusion-dominated regime. • Results show that hybridization results in superior accuracy in diffusion-dominated problems. • A series of numerical examples showed speedups of one to two orders of magnitude. [ABSTRACT FROM AUTHOR]

This paper presents p -adaptive hybridized flux reconstruction schemes to reduce the computational cost of implicit discretizations. We first introduce spatial and temporal discretization and discuss the adaptation algorithm via a nondimensional vorticity indicator for hybridized methods with globally continuous and globally discontinuous numerical traces. At each adaptation level, projection operations are applied to determine the new space based on the element-wise projected solution and transmission conditions. We validate our implementation and analyze performance via numerical examples. Specifically, we show via an isentropic vortex that p -adaptive hybridization of both HFR and EFR methods results in comparable numerical error to standard p -adaptive and p -uniform FR discretizations with a fraction of degrees of freedom. Results for a cylinder at Re = 150 showcase speedup factors in excess of 6 for hybridized methods in comparison with p -adaptive standard FR schemes and up to 40 against p -uniform FR discretizations. Similarly, results for a NACA 0012 airfoil at Re = 10 , 000 demonstrate speedup factors close to 6 against p -adaptive FR discretizations and up to 33 against p -uniform conventional FR. Hence, combining hybridization with adaptation yields a significant reduction in computational cost compared with standard implicit discretizations. • We introduce polynomial adaptation in the context of hybridized flux reconstruction schemes. • The proposed formulation generalizes to discontinuous and continuous traces. • The cost of our adaptation procedure is akin to a single time step. • We observed speedup factors over 5 against p-adaptive standard FR schemes. • Speedup factors over 30 against p-uniform standard FR schemes were also observed. [ABSTRACT FROM AUTHOR]

A high-order, degree-adaptive hybridizable discontinuous Galerkin (HDG) method is presented for two-fluid incompressible Stokes flows, with boundaries and interfaces described using NURBS. The NURBS curves are embedded in a fixed Cartesian grid, yielding an unfitted HDG scheme capable of treating the exact geometry of the boundaries/interfaces, circumventing the need for fitted, high-order, curved meshes. The framework of the NURBS-enhanced finite element method (NEFEM) is employed for accurate quadrature along immersed NURBS and in elements cut by NURBS curves. A Nitsche's formulation is used to enforce Dirichlet conditions on embedded surfaces, yielding unknowns only on the mesh skeleton as in standard HDG, without introducing any additional degree of freedom on non-matching boundaries/interfaces. The resulting unfitted HDG-NEFEM method combines non-conforming meshes, exact NURBS geometry and high-order approximations to provide high-fidelity results on coarse meshes, independent of the geometric features of the domain. Numerical examples illustrate the optimal accuracy and robustness of the method, even in the presence of badly cut cells or faces, and its suitability to simulate microfluidic systems from CAD geometries. • High-order HDG method with immersed NURBS surfaces. • No additional unknowns introduced on non-matching boundaries/interfaces. • Suitability of coarse meshes independent of the geometric features of the domain. • Optimal convergence and robustness even in the presence of badly cut cells or faces. • Application to incompressible one-fluid and two-fluid Stokes flows. [ABSTRACT FROM AUTHOR]

In this work, we propose an ensemble Monte Carlo hybridizable discontinuous Galerkin (EMC-HDG) algorithm to simulate the convection-diffusion equation with random diffusion coefficients. The EMC-HDG algorithm reduces the computation cost compared with Monte Carlo HDG (MC-HDG) method. This algorithm is a semi-implicit scheme which shares a common coefficient matrix with multiple right-hand-vectors by introducing an ensemble average of coefficient functions. Moreover, for the numerical approximation, optimal convergence rate O(hk+ 1) in space and O(△t) in time are obtained. To confirm our theoretical results, several numerical experiments are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we derive improved a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a variable penalty for second-order elliptic problems. The strategy is to use a penalization function of the form O (1 / h 1 + δ) , where h denotes the mesh size and δ is a user-dependent parameter. We then quantify its direct impact on the convergence analysis, namely, the (strong) consistency, discrete coercivity and boundedness (with h δ -dependency), and we derive updated error estimates for both discrete energy- and L 2 -norms. The originality of the error analysis relies specifically on the use of conforming interpolants of the exact solution. All theoretical results are supported by numerical evidence. • Families of Hybridizable Interior Penalty methods using variable penalty for diffusion problems. • Unified convergence analysis & improved a priori error estimates. • The h -dependency of the coercivity condition and boundedness. • The stabilization term strongly influences estimated convergence rates and robustness. • Superconvergence of the symmetric scheme is achieved on κ -orthogonal grids by selecting an appropriate penalty parameter. [ABSTRACT FROM AUTHOR]

We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter. [ABSTRACT FROM AUTHOR]

This paper investigates the efficiency, robustness, and scalability of approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in the all-at-once solution of a space-time hybridizable discontinuous Galerkin discretization of advection-dominated flows. The motivation for this study is that the time-dependent advection-diffusion equation can be seen as a "steady" advection-diffusion problem in (d+1)-dimensions and AIR has been shown to be a robust solver for steady advection-dominated problems. Numerical examples demonstrate the effectiveness of AIR as a preconditioner for advection-diffusion problems on fixed and time-dependent domains, using both slab-by-slab and all-at-once space-time discretizations, and in the context of uniform and space-time adaptive mesh refinement. A closer look at the geometric coarsening structure that arises in AIR also explains why AIR can provide robust, scalable, space-time convergence on advective and hyperbolic problems, while most multilevel parallel-in-time schemes struggle with such problems. [ABSTRACT FROM AUTHOR]

We 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. [ABSTRACT FROM AUTHOR]

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this work, we provide an algorithm to compute efficiently and accurately the full outgoing modal Green's kernel for the scalar wave equation in local helioseismology under spherical symmetry. Due to the high computational cost of a full Green's function, current helioseismic studies rely on single-source computations. However, a more realistic modelization of the helioseismic products (cross-covariance and power spectrum) requires the full Green's kernel. In the classical approach, the Dirac source is discretized and one simulation gives the Green's function on a line. Here, we propose a two-step algorithm which, with two simulations, provides the full kernel on the domain. Moreover, our method is more accurate, as the singularity of the solution due to the Dirac source is described exactly. In addition, it is coupled with the exact Dirichlet-to-Neumann boundary condition, providing optimal accuracy in approximating the outgoing Green's kernel, which we demonstrate in our experiments. In addition, we show that high-frequency approximations of the nonlocal radiation boundary conditions can represent accurately the helioseismic products. [ABSTRACT FROM AUTHOR]

Summary: We propose a method to solve the acoustic wave equation on an immersed domain using the hybridizable discontinuous Galerkin method for spatial discretization and the arbitrary derivative method with local time stepping (LTS) for time integration. The method is based on a cut finite element approach of high order and uses level set functions to describe curved immersed interfaces. We study under which conditions and to what extent small time step sizes balance cut instabilities, which are present especially for high‐order spatial discretizations. This is done by analyzing eigenvalues and critical time steps for representative cuts. If small time steps cannot prevent cut instabilities, stabilization by means of cell agglomeration is applied and its effects are analyzed in combination with local time step sizes. Based on two examples with general cuts, performance gains of the LTS over the global time stepping are evaluated. We find that LTS combined with cell agglomeration is most robust and efficient. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we propose an adaptive refinement strategy for phase‐field models of brittle fracture, which is based on a novel hybridizable discontinuous Galerkin (HDG) formulation of the problem. The adaptive procedure considers standard elements and only one type of h‐refined elements, dynamically located along the propagating cracks. Thanks to the weak imposition of interelement continuity in HDG methods, and in contrast with other existing adaptive approaches, hanging nodes or special transition elements are not needed, which simplifies the implementation. Various numerical experiments, including one branching test, show the accuracy, robustness, and applicability of the presented approach to quasistatic phase‐field simulations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• Projective HDG mixed method is presented for second-order diffusion problems. • The HDG method uses Raviart-Thomas spaces for approximating the dual variable and the projective (LS) stabilization function. • No post-processing of the scalar variable is required since it converges naturally at the rate k + 2. • A straightforward flux reconstruction is proposed to obtain a H(div)-conforming variable converging at the rate k + 1. • Comparative tests with the standard H-RT and H-LDG mixed methods are performed in terms of h and p refinements and CPU time. 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 Schöberl 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–Schöberl (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 u h , 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. [ABSTRACT FROM AUTHOR]

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. • We present a hybridizable discontinuous Galerkin method for thin and thick structures. • A technique of elimination of unknowns significantly reduces the computational cost. • We present a penalization that both alleviates locking and stabilizes the HDG method. • Displacements converge optimally, and cheap postprocessing improves the accuracy. • The method gives accurate results for various classical nonlinear shell problems. [ABSTRACT FROM AUTHOR]

Abstract In this research, we give projection-based error analysis on a multiscale hybridizable discontinuous Galerkin method to numerically solve parabolic problem with a heterogeneous coefficient. We modified the spectral multiscale HDG method introduced in [22] to fit to the time-dependent PDE. The method uses multiscale spaces generated by eigenfunctions of local spectral problems. By considering two different grids, one relatively coarser than the other, we give bounds for the error between the actual solution and the approximate one derived from multiscale HDG method. One of the main focuses of the paper is to derive error analysis that depends on the size of fine and coarse grids and eigenvalues of local spectral problems. To solve a given coarse problem, the more eigenfunction we choose, the more accurate the approximation becomes: we shall see that the numerical result indicates that when we fix fine and coarse grids, the error between the reference solution and the derived one decreases whenever we have more multiscale basis functions. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the a priori error analysis for a convection-dominated diffusion 2D problem, when applying the HDG method on a family of anisotropic triangulations. It is known that in this case, boundary or interior layers may appear. Therefore, it is important to resolve these layers in order to recover, if possible, the expected order of approximation. In this work, we extend the use of HDG method on anisotropic meshes. In this context, when the discrete local spaces are polynomials of degree k ≥ 0 , this approach is able to recover an order of convergence k + 1 2 in L 2 for all the variables, under certain assumptions on the stabilization parameter and family of triangulations. Numerical examples confirm our theoretical results. Highlights • We develop an a priori error analysis for a HDG scheme defined on anisotropic meshes that are made of triangles. • We require that the family of triangulations satisfy the maximum angle condition, which is usual in this case. • We include numerical examples that validate our theoretical results. [ABSTRACT FROM AUTHOR]

We considered an hybridizable discontinuous Galerkin (HDG) method for discrete elliptic PDEs with random coefficients. By an approach of projection, we obtained the error analysis under the assumption that a(ω,x) is uniformly bounded. Together with the HDG method, we applied a multilevel Monte Carlo (MLMC) method (MLMC-HDG method) to simulate the random elliptic PDEs. We derived the overall convergence rate and total computation cost estimate. Finally, some numerical experiments are presented to confirm the theoretical results.

This work introduces a hybridizable discontinuous Galerkin formulation for the simulation of ideal plasmas governed by the Euler–Maxwell equations. The approach is based on a monolithic source-based coupling of the fluid and electromagnetic subproblems, along with a fully implicit time integration scheme, and a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. The numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation and validate it against problems of increasing complexity, ranging from single-physics cases to weakly and fully coupled electromagnetic plasma simulations. • Novel Hybridizable Discontinuous Galerkin formulation for Euler-Maxwell plasma model. • Monolithic source-based coupling and fully implicit time integration scheme. • Projection-based divergence correction method to enforce Gauss' laws. • Simulation of high-frequency plasma phenomena. • Accuracy, efficiency and robustness demonstrated on problems of increasing complexity. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

A coupling strategy between hybridizable discontinuous Galerkin (HDG) and continuous Galerkin (CG) methods is proposed in the framework of second-order elliptic operators. The coupled formulation is implemented and its convergence properties are established numerically by using manufactured solutions. Afterwards, the solution of the coupled Navier-Stokes/convection-diffusion problem, using Boussinesq approximation, is formulated within the HDG framework and analysed using numerical experiments. Results of Rayleigh-Bénard convection flow are also presented and validated with literature data. Finally, the proposed coupled formulation between HDG and CG for heat equation is combined with the coupled Navier-Stokes/convection diffusion equations to formulate a new CG-HDG model for solving conjugate heat transfer problems. Benchmark examples are solved using the proposed model and validated with literature values. The proposed CG-HDG model is also compared with a CG-CG model, where all the equations are discretized using the CG method, and it is concluded that CG-HDG model can have a superior computational efficiency when compared to CG-CG model. [ABSTRACT FROM AUTHOR]

This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element postprocess procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types. [ABSTRACT FROM AUTHOR]

This work presents a dispersion analysis of the Hybrid Discontinuous Galerkin (HDG) method. Considering the Helmholtz system, we quantify the discrepancies between the exact and discrete wavenumbers. In particular, we obtain an analytic expansion for the wavenumber error for the lowest order Single Face HDG (SFH) method. The expansion shows that the SFH method exhibits convergence rates of the wavenumber errors comparable to that of the mixed hybrid Raviart-Thomas method. In addition, we observe the same behavior for the higher order cases in numerical experiments. [ABSTRACT FROM AUTHOR]

This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin (HDG) method. Modern implementations of high-order methods with state-of-the-art multigrid solvers for the Poisson equation are considered, including fast matrix-free implementations with sum factorization on quadrilateral and hexahedral elements. For the HDG method, a multigrid approach that combines a grid transfer from the trace space to the space of linear finite elements with algebraic multigrid on further levels is developed. It is found that high-order continuous finite elements give best time to solution for smooth solutions, closely followed by the matrix-free solvers for the other two discretizations. Their performance is up to an order of magnitude higher than that of the best matrix-based methods, even after including the superconvergence effects in the matrix-based HDG method. This difference is because of the vastly better performance of matrix-free operator evaluation as compared to sparse matrix-vector products. A rooine performance model confirms the superiority of the matrix-free implementation. [ABSTRACT FROM AUTHOR]

We show that two widely used Galerkin formulations for second-order elliptic problems provide approximations which are actually superclose, that is, their difference converges faster than the corresponding errors. In the framework of linear elasticity, the two formulations correspond to using either the stiffness tensor or its inverse the compliance tensor. We find sufficient conditions, for a wide class of methods (including mixed and discontinuous Galerkin methods), which guarantee a supercloseness result. For example, for the HDGk method using polynomial approximations of degree k>0, we find that the difference of approximate fluxes superconverges with order k+2 and that the difference of the scalar approximations superconverges with order k+3. We provide numerical results verifying our theoretical results. [ABSTRACT FROM AUTHOR]

This article proves that a particular space-time, hybridizable discontinuous Galerkin method is entropy stable for the compressible Navier-Stokes equations. In order to facilitate the proof, 'entropy variables' are utilized to rewrite the compressible Navier-Stokes equations in a symmetric form. The resulting form of the equations is discretized with a hybridizable discontinuous finite element approach in space, and a classical discontinuous finite element approach in time. Thereafter, the initial solution is shown to continually bound the solutions at later times. [ABSTRACT FROM AUTHOR]

We propose and analyze a new coupling procedure for the Hybridizable Discontinuous Galerkin Method with Galerkin Boundary Element Methods based on a double layer potential representation of the exterior component of the solution of a transmission problem. We show a discrete uniform coercivity estimate for the non-symmetric bilinear form and prove optimal convergence estimates for all the variables, as well as superconvergence for some of the discrete fields. Some numerical experiments support the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we develop a priori estimates for the convergence of outputs, output error estimates, and localizations of output error estimates for Galerkin finite element methods. Output error estimates for order p finite element solutions are constructed using the Dual-Weighted Residual (DWR) method with a higher-order p ′ > p dual solution. Specifically, we analyze these DWR estimates for Continuous Galerkin (CG), Discontinuous Galerkin (DG), and Hybridized DG (HDG) methods applied to the Poisson problem. For all discretizations, as h → 0 , we prove that the output and output error estimate converge at order 2 p and 2 p ′ (assuming sufficient smoothness), while localizations of the output and output error estimate converge at 2 p + d and p + p ′ + d . For DG, the results use a new post processing for the error associated with the lifting operator. For HDG, these rates improve an additional order when the stabilization is based upon an O ( 1 ) length scale. [ABSTRACT FROM AUTHOR]

This work presents a novel application of the hybridizable discontinuous Galerkin (HDG) finite element method to the multi-physics simulation of coupled fluid–structure interaction (FSI) problems. Recent applications of the HDG method have primarily been for single-physics problems including both solids and fluids, which are necessary building blocks for FSI modeling. Utilizing these established models, HDG formulations for linear elastostatics, a nonlinear elastodynamic model, and arbitrary Lagrangian–Eulerian Navier–Stokes are derived. The elasticity formulations are written in a Lagrangian reference frame, with the nonlinear formulation restricted to hyperelastic materials. With these individual solid and fluid formulations, the remaining challenge in FSI modeling is coupling together their disparate mathematics on the fluid–solid interface. This coupling is presented, along with the resultant HDG FSI formulation. Verification of the component models, through the method of manufactured solutions, is performed and each model is shown to converge at the expected rate. The individual components, along with the complete FSI model, are then compared to the benchmark problems proposed by Turek and Hron [1] . The solutions from the HDG formulation presented in this work trend towards the benchmark as the spatial polynomial order and the temporal order of integration are increased. [ABSTRACT FROM AUTHOR]

We build a bridge between the hybrid high-order (HHO) and the hybridizable discontinuous Galerkin (HDG) methods in the setting of a model diffusion problem. First, we briefly recall the construction of HHO methods and derive some new variants. Then, by casting the HHO method in mixed form, we identify the numerical flux so that the HHO method can be compared to HDG methods. In turn, the incorporation of the HHO method into the HDG framework brings up new, efficient choices of the local spaces and a new, subtle construction of the numerical flux ensuring optimal orders of convergence on meshes made of general shape-regular polyhedral elements. Numerical experiments comparing two of these methods are shown. [ABSTRACT FROM AUTHOR]