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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

A Continuous-Discontinuous Galerkin Method for Electromagnetic Simulations Based on an All-Frequency Stable Formulation

In this paper, a potential-based partial-differential formulation, called the all-frequency stable formulation, is presented for the accurate and robust simulation of electromagnetic problems at all frequencies. Due to its stability from (near) dc to microwave frequencies, this formulation can be applied to simulate wide-band and multiscale problems without encountering the infamous low-frequency breakdown issue or applying basis function decompositions such as the tree-cotree splitting technique. To provide both efficient and flexible numerical solutions to the electromagnetic formulation, a mixed continuous-discontinuous Galerkin (CDG) method is proposed and implemented. In regions with homogeneous media, the continuous Galerkin method is employed to avoid the introduction of duplicated degrees of freedom (DoFs) on the elemental interfaces, while on the interfaces of two different media, the discontinuous Galerkin method is applied to permit the jump of the normal components of the electromagnetic fields. Numerical examples are provided to validate and demonstrate the proposed numerical solver for problems in a wide electromagnetic spectrum.

Yan, Su↗
DOE OSTI2020

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1999

Efficient Implementations of the Quadrature-Free Discontinuous Galerkin Method

The efficiency of the quadrature-free form of the dis- continuous Galerkin method in two dimensions, and briefly in three dimensions, is examined. Most of the work for constant-coefficient, linear problems involves the volume and edge integrations, and the transformation of information from the volume to the edges. These operations can be viewed as matrix-vector multiplications. Many of the matrices are sparse as a result of symmetry, and blocking and specialized multiplication routines are used to account for the sparsity. By optimizing these operations, a 35% reduction in total CPU time is achieved. For nonlinear problems, the calculation of the flux becomes dominant because of the cost associated with polynomial products and inversion. This component of the work can be reduced by up to 75% when the products are approximated by truncating terms. Because the cost is high for nonlinear problems on general elements, it is suggested that simplified physics and the most efficient element types be used over most of the domain.

Lockard, David P.↗
NASA NTRS2002

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.

Yan, Jue↗
DOE OSTI2023

A Refinement-by-Superposition -Method for (curl)- and (div)-Conforming Discretizations

Here, we present refinement-by-superposition (RBS) hp-refinement infrastructure for computational electromagnetics (CEMs), which permits exponential rates of convergence. In contrast to dominant approaches to hp-refinement for continuous Galerkin methods, which rely on explicit constraint equations, the multilevel strategy presented drastically reduces the implementation complexity. Through the RBS methodology, enforcement of continuity occurs by construction, enabling arbitrary levels of refinement with ease, and without the practical (but not theoretical) limitations of constrained-node refinement. We outline the construction of the RBS hp-method for refinement with H (curl)- and H (div)-conforming finite cells. Numerical simulations for the 2-D finite element method (FEM) solution of the Maxwell eigenvalue problem demonstrate the effectiveness of RBS hp-refinement. As an additional goal of this work, we aim to promote the use of mixed-order (low- and high-order) elements in practical CEM applications.

42 ENGINEERING↗
DOE OSTI2020

Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods

In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on finite element method--spectral element method (FEM-SEM) equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order (p = 1) finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in the polynomial degree of the high-order operator, requiring specialized solvers to treat the anisotropy. We make use of an element-structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in h and p. The method is extended to interior penalty and Bassi and Rebay (BR2) discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Finally, numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A Non-perturbative Approach to Computing Seismic Normal Modes in Rotating Planets

In this work, a continuous Galerkin method based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filtering eigensolver. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method. Our discretization utilizes fully unstructured tetrahedral meshes for both solid and fluid regions. The relevant eigenvalue problem is solved by a combination of several highly parallel and computationally efficient methods. We validate our three-dimensional results in the non-rotating case using analytical results for constant elastic balls, as well as numerical results for an isotropic Earth model from standard “radial” algorithms. We also validate the computations in the rotating case, but only in the slowly-rotating regime where perturbation theory applies, because no other independent algorithms are available in the general case. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Earth and Mars models studying the effects of heterogeneity on a large range of scales.

58 GEOSCIENCES↗
DOE OSTI2020

Fourier Analyses of High-Order Continuous and Discontinuous Galerkin Methods

In this paper, we present a Fourier analysis of wave propagation problems subject to a class of continuous and discontinuous discretizations using high-degree Lagrange polynomials. This allows us to obtain explicit analytical formulas for the dispersion relation and group velocity and, for the first time to our knowledge, characterize analytically the emergence of gaps in the dispersion relation at specific wavenumbers, when they exist, and compute their specific locations. Wave packets with energy at these wavenumbers will fail to propagate correctly, leading to significant numerical dispersion. We also show that the Fourier analysis generates mathematical artifacts, and we explain how to remove them through a branch selection procedure conducted by analysis of eigenvectors and associated reconstructed solutions. The higher frequency eigenmodes, named erratic in this study, are also investigated analytically and numerically.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations

A stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented herein. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Continuously bounds-preserving discontinuous Galerkin methods for hyperbolic conservation laws

For finite element approximations of transport phenomena, it is often necessary to apply a form of limiting to ensure that the discrete solution remains well-behaved and satisfies physical constraints. However, these limiting procedures are typically performed at discrete nodal locations, which is not sufficient to ensure the robustness of the scheme when the solution must be evaluated at arbitrary locations (e.g., for adaptive mesh refinement, remapping in arbitrary Lagrangian–Eulerian solvers, overset meshes, etc.). In this work, a novel limiting approach for discontinuous Galerkin methods is presented which ensures that the solution is continuously bounds-preserving (i.e., across the entire solution polynomial) for any arbitrary choice of basis, approximation order, and mesh element type. Through a modified formulation for the constraint functionals, the proposed approach requires only the solution of a single spatial scalar minimization problem per element for which a highly efficient numerical optimization procedure is presented. Here, the efficacy of this approach is shown in numerical experiments by enforcing continuous constraints in high-order unstructured discontinuous Galerkin discretizations of hyperbolic conservation laws, ranging from scalar transport with maximum principle preserving constraints to compressible gas dynamics with positivity-preserving constraints.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A high accuracy/resolution spectral element/Fourier–Galerkin method for the simulation of shoaling non-linear internal waves and turbulence in long domains with variable bathymetry

A high-order hybrid continuous-Galerkin numerical method, designed for the simulation of non-linear, non -hydrostatic internal waves and turbulence in long computational domains with complex bathymetry, is presented. The spatial discretization in the non-periodic wave-propagating directions, utilizes the nodal spectral element method. Such a high-order element-based discretization allows the highly accurate representation of complex domain geometry along with the flexibility of concentrating resolution in areas of interest. Under the assumption of the normal-to-isobath propagation of non-linear internal waves, a third periodic direction is incorporated via a Fourier-Galerkin discretization. The distinct non-hydrostatic nature of non-linear internal waves and, any instabilities and turbulence therein, necessitates the numerically challenging solution of the pressure Poisson problem. A defining feature of this work is the application of a domain decomposition approach, combined with block-Jacobi/deflation-based preconditioning to the pressure Poisson problem. Such a combined approach is particularly suitable for the long high aspect-ratio complex domains of interest and enables the efficient high-accuracy reproduction of the non-hydrostatic dynamics of non-linear internal waves. Implementation details are also described in the context of the stability of the solver and its parallelization strategy. A series of benchmarks of increasing complexity demonstrate the robustness of the flow solver. The benchmarks culminate with the three-dimensional simulation of a convectively breaking mode-one non-linear internal wave over a realistic South-China-Sea bathymetric transect and background current/stratification profiles.

Deflation↗
DOE OSTI2024

A face-upwinded spectral element method

Here we present a new high-order accurate discretisation on unstructured meshes of quadrilateral elements. Our Face Upwinded Spectral Element (FUSE) method uses the same node distribution as a high-order continuous Galerkin (CG) method, but with a particular choice of node locations within each element and an upwinded stencil on the face nodes. This results in a number of benefits, including fewer degrees of freedom and straight-forward integration with CG. We present the derivation of the scheme and the analysis of its properties, in particular showing stability using von Neumann analysis. We show numerical evidence for its accuracy and efficiency on multiple classes of problems including convection-dominated flows, Poisson's equation, and the incompressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A note on higher-order and nonlinear limiting approaches for continuously bounds-preserving discontinuous Galerkin methods

In Dzanic (2024), a limiting approach for high-order discontinuous Galerkin schemes was introduced which allowed for imposing constraints on the solution continuously (i.e., everywhere within the element). While exact for linear constraint functionals, this approach only imposed a sufficient (but not the minimum necessary) amount of limiting for nonlinear constraint functionals. This short note shows how this limiting approach can be extended to allow exactness for general nonlinear quasiconcave constraint functionals through a nonlinear limiting procedure, reducing unnecessary numerical dissipation. Finally, some examples are shown for nonlinear pressure and entropy constraints in the compressible gas dynamics equations, where both analytic and iterative approaches are used.

97 MATHEMATICS AND COMPUTING↗
NASA NTRSDate not supplied

Discontinuous Galerkin and Related Methods for ODE

Starting from the standard integral formulation, the DG method is derived here in differential form. The key ingredient is a polynomial called the correction function, which helps ‘correct’ the discontinuous solution by approximating the jump and yields a continuous one. Under the right Radau quadrature, this continuous solution is identical to the solutions by the right Radau collocation and the continuous Galerkin (CG) methods. Next, the correction function facilitates the construction of the associated implicit Runge-Kutta schemes (IRK-DG). Different quadratures for DG result in different IRK-DG methods: left Radau quadrature in Radau IA, right Radau quadrature in Radau IIA or right Radau collocation, and Gauss quadrature in a method called DG-Gauss. The construction of IRK-DG clarifies the meaning and facilitates the proofs of various 𝐵(𝑝), 𝐶(𝜂), and 𝐷(𝜁) conditions for accuracy. The two consequences of these conditions are that all 𝑠-stage IRK-DG methods are accurate to order 2𝑠− 1, and the IRK-DG methods of Radau type are unique. Numerical examples showing the behavior of the DG solutions are provided. In all, the correction function plays a key role and helps establish the relations among the DG, IRK DG, collocation, and CG methods.

Numerical Methods for Ordinary Differential Equati↗
NASA NTRSDate not supplied

Discontinuous Galerkin and Related Methods for ODE

A defining feature of the discontinuous Galerkin (DG) method for ODE is that the piecewise polynomial solution can have a jump discontinuity at the beginning of each step. Starting from the standard integral formulation, the DG method is derived here in differential form. The key ingredient is a polynomial called the correction function, which helps ‘correct’ the discontinuous solution by approximating the jump and yields a continuous one. Under the right Radau quadrature, this continuous solution is identical to the solutions by the right Radau collocation and the continuous Galerkin (CG) methods. Next, the correction function facilitates the construction of the associated implicit Runge-Kutta schemes (IRK-DG). Different quadratures for DG result in different IRK-DG methods: left Radau quadrature in Radau IA, right Radau quadrature in Radau IIA or right Radau collocation, and Gauss quadrature in a method called DG-Gauss. The construction of IRK-DG clarifies the meaning and facilitates the proofs of various 𝐵(𝑝), 𝐶(𝜂), and 𝐷(𝜁) conditions for accuracy. The two consequences of these conditions are that all 𝑠-stage IRK-DG methods are accurate to order 2𝑠 − 1, and the IRK-DG methods of Radau type are unique. Numerical examples showing the behavior of the DG solutions are provided. In all, the correction function plays a key role and helps establish the relations among the DG, IRK-DG, collocation, and CG schemes.

numerical methods↗
DOE OSTI2026

High-Fidelity, Low-Dissipation/Symmetry-Preserving Numerical Scheme for Solving the Euler Equations with Unstructured, Metric-Based Mesh Adaptation

This work presents an overview of a high-fidelity compressible Euler solver that utilizes the continuous Galerkin (CG) method with added artificial numerical diffusion for stabilization to solve a variety of unsteady and steady benchmark inviscid flow problems. This work shows that discretizing the Euler equations with this CG approach and first order basis functions produces a cost-effective stencil as well as simple well-posed boundary conditions. We show through convergence testing with manufactured solutions that the reduced stencil of CG, combined with the low amount of artificial diffusion required when using the stabilization method outlined in this work, leads to stable and highly accurate results for a variety of unsteady and steady applications. When combined with the adaptive mesh refinement approach used for many of the cases in this work, our results show that the flow solver achieves even more accurate results. A variety of inviscid flow cases are presented in this work, including transient 2D cases with complex shock structures and several steady 3D airfoils sections with a constant span.

Doetsch, Kevin [ORNL] (ORCID:0000000267051705)↗
NASA NTRS1997

Continued Development of the Discontinuous Galerkin Method for Computational Aeroacoustic Applications

The formulation and the implementation of boundary conditions within the context of the quadrature-free form of the discontinuous Galerkin method are presented for several types of boundary conditions for the Euler equations. An important feature of the discontinuous Galerkin method is that the interior point algorithm is well behaved in the neighborhood of the boundary and requires no modifications. This feature leads to a simple and accurate treatment for wall boundary conditions and simple inflow and outflow boundary conditions. Curved walls are accurately treated with only minor changes to the implementation described in earlier work. The 'perfectly matched layer' approach to nonreflecting boundary conditions is easily applied to the discontinuous Galerkin. The compactness of the discontinuous Galerkin method makes it better suited for buffer-zone-type methods than high-order finite-difference methods. Results are presented for wall, characteristic inflow and outflow, and nonreflecting boundary conditions.

Atkins, H. L.↗
NASA NTRS2013

ONERA-NASA Cooperative Effort on Liner Impedance Eduction

As part of a cooperation between ONERA and NASA, the liner impedance eduction methods developed by the two research centers are compared. The NASA technique relies on an objective function built on acoustic pressure measurements located on the wall opposite the test liner, and the propagation code solves the convected Helmholtz equation in uniform ow using a finite element method that implements a continuous Galerkin discretization. The ONERA method uses an objective function based either on wall acoustic pressure or on acoustic velocity acquired above the liner by Laser Doppler Anemometry, and the propagation code solves the linearized Euler equations by a discontinuous Galerkin discretization. Two acoustic liners are tested in both ONERA and NASA ow ducts and the measured data are treated with the corresponding impedance eduction method. The first liner is a wire mesh facesheet mounted onto a honeycomb core, designed to be linear with respect to incident sound pressure level and to grazing ow velocity. The second one is a conventional, nonlinear, perforate-over-honeycomb single layer liner. Configurations without and with ow are considered. For the nonlinear liner, the comparison of liner impedance educed by NASA and ONERA shows a sensitivity to the experimental conditions, namely to the nature of the source and to the sample width.

Primus, Julien↗