Discontinuous Galerkin Discretization Research Articles

Polytopal discontinuous Galerkin discretization of brain multiphysics flow dynamics

A comprehensive mathematical model of the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain can be expressed as the coupling of a poromechanics system and Stokes' equations: the first describes fluids filtration through the cerebral tissue and the tissue's elastic response, while the latter models the flow of the CSF in the brain ventricles. This model describes the functioning of the brain's waste clearance mechanism, which has been recently discovered to play an essential role in the progress of neurodegenerative diseases. To model the interactions between different scales in the porous medium, we propose a physically consistent coupling between Multi-compartment Poroelasticity (MPE) equations and Stokes' equations. In this work, we introduce a numerical scheme for the discretization of such coupled MPE-Stokes system, employing a high-order discontinuous Galerkin method on polytopal grids to efficiently account for the geometric complexity of the domain. We analyze the stability and convergence of the space semidiscretized formulation, we prove a-priori error estimates, and we present a temporal discretization based on a combination of Newmark's β-method for the elastic wave equation and the θ-method for the other equations of the model. Numerical simulations carried out on test cases with manufactured solutions validate the theoretical error estimates. We also present numerical results on a two-dimensional slice of a patient-specific brain geometry reconstructed from diagnostic images, to test in practice the advantages of the proposed approach.

Port-Hamiltonian discontinuous Galerkin finite element methods

Abstract A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a nonzero energy flow through the boundaries. In this paper, we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving semidiscrete finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes–Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.

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. 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.

Trefftz discontinuous Galerkin discretization for the Stokes problem

AbstractWe introduce a new discretization based on a polynomial Trefftz-DG method for solving the Stokes equations. Discrete solutions of this method fulfill the Stokes equations pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows us to easily incorporate inhomogeneous right-hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to other (hybrid) discontinuous Galerkin Stokes discretizations and present numerical examples.

On the stability and convergence of discontinuous Galerkin schemes for incompressible flows

Abstract The property that the velocity $\textbf{u}$ belongs to $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretization methods are uniformly stable in the $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$-norm. In this work, we establish that this is indeed the case for discontinuous Galerkin (DG) discretizations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretization is equivalent to the RadauIIA Implicit Runge–Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo–Nirenberg-type inequalities on DG spaces, which might be of independent interest.

Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.

Local discontinuous Galerkin for the functional renormalisation group

We apply a Local Discontinuous Galerkin discretisation to flow equations of the O(N)-model in the Local Potential Approximation. The improved stability is directly observed by solving the flow equation for various N and space-time dimensions d. A particular focus of this work is the numerical discretisation and its implementation. It is realised as a module within the high performance PDE framework DUNE. A preliminary version of the module is available on GitHub, but it is not submitted for publication as it is too early to be archived.

A leap-frog nodal discontinuous Galerkin method for Maxwell polynomial chaos Debye model

The Maxwell polynomial chaos Debye model uses the parameter distribution in the Debye model to represent the Cole-Cole dispersive mechanism. In this paper, a leap-frog nodal discontinuous Galerkin (DG) method is studied for solving this model. Under the Courant-Friedrichs-Lewy (CFL) condition, the stability of the fully discrete DG scheme is demonstrated and its error estimate is derived. Numerical examples are provided for supporting the correctness of the theoretical analysis and the effectiveness of our proposed method.

Duality analysis of interior penalty discontinuous Galerkin methods under minimal regularity and application to the a priori and a posteriori error analysis of Helmholtz problems

We consider interior penalty discontinuous Galerkin discretizations of time-harmonic wave propagation problems modeled by the Helmholtz equation, and derive novel a priori and a posteriori estimates. Our analysis classically relies on duality arguments of Aubin–Nitsche type, and its originality is that it applies under minimal regularity assumptions. The estimates we obtain directly generalize known results for conforming discretizations, namely that the discrete solution is optimal in a suitable energy norm and that the error can be explicitly controlled by a posteriori estimators, provided the mesh is sufficiently fine.

Fully higher-order coupling of finite element and level set methods for two-phase flow with a new explicit projection method

This work presents the development of a higher-order finite element method for modeling unsteady, incompressible and inviscid two-phase flows using the level set method. A new explicit projection method is introduced to solve the incompressible Euler equation, where the pressure and the velocity fields are solved separately. To implement high order temporal schemes, the Hodge decomposition in this projection method is not applied to the velocity but to its time derivative. A discontinuous Galerkin discretization is used to solve both the level set and the nonlinear advection terms present in the Euler equations, while a Galerkin discretization is used to solve the pressure Poisson equation. The present method is a fully high order numerical approach, both in space and temporal dimensions. Numerical results provided by several 2D benchmark test cases in naval hydrodynamics (i.e hydrostatic equilibrium, sloshing and dam break) validate the computational code, illustrating the power of the proposed method. Compared to the CFD codes based on low order VOF/finite-volume methods, the present approach demonstrates greater accuracy on coarse meshes, leading to reduced CPU time.

A space–time DG method for the Schrödinger equation with variable potential

We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal h-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.

Preconditioned discontinuous Galerkin method and convection‐diffusion‐reaction problems with guaranteed bounds to resulting spectra

AbstractThis paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection‐diffusion‐reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of nonsymmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection‐)diffusion‐reaction problems.

A collision-based hybrid method for the BGK equation

We apply the collision-based hybrid method introduced by Hauck and McClarren [1] to the Boltzmann equation with the BGK operator and a hyperbolic scaling. An implicit treatment of the source term is used to handle stiffness associated with the BGK operator. Although it helps the numerical scheme become stable with a large time step size, it is still not obvious to achieve the desired order of accuracy due to the relationship between the size of the spatial cell and the mean free path. Without asymptotic preserving property, a very restricted grid size is required to resolve the mean free path, which is not practical. Our approaches are based on the noncollision-collision decomposition of the BGK equation. We introduce the arbitrary order of nodal discontinuous Galerkin (DG) discretization in space with a semi-implicit time-stepping method; we employ the backward Euler time integration for the uncollided equation and the 2nd order predictor-corrector scheme for the collided equation, i.e., both source terms in uncollided and collided equations are treated implicitly and only streaming term in the collided equation is solved explicitly. This improves the computational efficiency without the complexity of the numerical implementation. Numerical results are presented for various Knudsen numbers to present the effectiveness and accuracy of our hybrid method. Also, we compare the solutions of the hybrid and non-hybrid schemes.

Reduced-Order Modeling for Heston Stochastic Volatility Model

In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.

Strong Stability Preserving Second Derivative General Linear Methods Based on Taylor Series Conditions for Discontinuous Galerkin Discretizations

Strong Stability Preserving Second Derivative General Linear Methods Based on Taylor Series Conditions for Discontinuous Galerkin Discretizations

Adjoint-based aerodynamic shape optimization with hybridized discontinuous Galerkin methods

We present a discrete adjoint approach to aerodynamic shape optimization (ASO) based on a hybridized discontinuous Galerkin (HDG) discretization. Our implementation is designed to tie in as seamlessly as possible into a solver architecture written for general balance laws, thus adding design capability to a tool with a wide range of applicability. Design variables are introduced on designated surfaces using the knots of a 2D spline-based geometry representation, while gradients are computed from the adjoint solution using a difference approximation of residual perturbations. A suitable optimization algorithm, such as an in-house steepest descent or the Preconditioned Sequential Quadratic Programming (PSQP) approach from the pyOpt framework, is then employed to find an improved geometry. We present verification of the implementation, including drag or heat flux minimization in compressible flows, as well as inverse design.

Entropy dissipative higher order accurate positivity preserving time-implicit discretizations for nonlinear degenerate parabolic equations

We develop entropy dissipative higher order accurate local discontinuous Galerkin (LDG) discretizations coupled with Diagonally Implicit Runge–Kutta (DIRK) methods for nonlinear degenerate parabolic equations with a gradient flow structure. Using the simple alternating numerical flux, we construct DIRK-LDG discretizations that combine the advantages of higher order accuracy, entropy dissipation and proper long-time behavior. We theoretically prove the entropy dissipation of the implicit Euler-LDG discretization without any time-step restrictions when no positivity constraint is imposed. Next, in order to ensure the positivity of the numerical solution, we use the Karush–Kuhn–Tucker (KKT) limiter, which achieves a positive solution by coupling the positivity preserving KKT conditions with higher order accurate DIRK-LDG discretizations using Lagrange multipliers. In addition, mass conservation of the positivity-limited solution is ensured by imposing a mass conservation equality constraint to the KKT equations. Under a time step restriction, the unique solvability and entropy dissipation for implicit first order accurate in time, but higher order accurate in space, positivity-preserving LDG discretizations with periodic boundary conditions are proved, which provide a first theoretical analysis of the KKT limiter. Finally, numerical results demonstrate the higher order accuracy and entropy dissipation of the positivity-preserving DIRK-LDG discretizations for problems requiring a positivity limiter. In addition, we can observe from the numerical results that the implicit time-discrete methods alleviate the time-step restrictions needed for the stability of the numerical discretizations, which improves computational efficiency.

A space-time discontinuous Galerkin discretization for the linear transport equation

We consider a full-upwind DG approximation in space and time for the linear transport equation. Based on our results for linear symmetric Friedrichs systems we establish inf-sup stability and convergence in a mesh-dependent DG norm, and we construct an error indicator with respect to this norm. Numerical results of test problems with known solution demonstrate the efficiency of the a priori and a posteriori results as well for smooth and for non-smooth solutions. Then, we show that by introducing suitable degrees of freedom on the space-time element boundaries the corresponding hybrid formulation yields a reduction to a considerably smaller linear system.

Domain-specific implementation of high-order Discontinuous Galerkin methods in spherical geometry

In recent years, domain-specific languages (DSLs) have achieved significant success in large-scale efforts to reimplement existing meteorological models in a performance portable manner. The dynamical cores of these models are based on finite difference and finite volume schemes, and existing DSLs are generally limited to supporting only these numerical methods. In the meantime, there have been numerous attempts to use high-order Discontinuous Galerkin (DG) methods for atmospheric dynamics, which are currently largely unsupported in main-stream DSLs. In order to link these developments, we present two domain-specific languages which extend the existing GridTools (GT) ecosystem to high-order DG discretization. The first is a C++-based DSL called G4GT, which, despite being no longer supported, gave us the impetus to implement extensions to the subsequent Python-based production DSL called GT4Py to support the operations needed for DG solvers. As a proof of concept, the shallow water equations in spherical geometry are implemented in both DSLs, thus providing a blueprint for the application of domain-specific languages to the development of global atmospheric models. We believe this is the first GPU-capable DSL implementation of DG in spherical geometry. The results demonstrate that a DSL designed for finite difference/volume methods can be successfully extended to implement a DG solver, while preserving the performance-portability of the DSL.

Homogeneous multigrid for HDG applied to the Stokes equation

Abstract We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart–Thomas, or a hybrid Brezzi–Douglas–Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Beyond this, we establish relations, which resemble those in Cockburn & Gopalakrishnan (2008, Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput., 74, 1653–1677) for elliptic problems, between the approximates that are obtained by the single-face hybridizable, hybrid Raviart–Thomas and hybrid Brezzi–Douglas–Marini methods. Numerical experiments underline our analytical findings.