Discontinuous Galerkin Approximation Research Articles

Note on quasi-optimal error estimates for the pressure for shear-thickening fluids

In this paper, we derive quasi-optimal a priori error estimates for the kinematic pressure for a Local Discontinuous Galerkin (LDG) approximation of steady systems of $p$-Navier--Stokes type in the case of shear-thickening, \textit{i.e.}, $p>2$, imposing a new Muckenhoupt regularity condition on the viscosity of the fluid, which is mild in the two-dimensional case but potentially restrictive in the three-dimensional case.

An entropy conserving/stable discontinuous Galerkin solver in entropy variables based on the direct enforcement of entropy balance

Numerical schemes that guarantee the preservation of flow properties as prescribed by physical laws, ensure high-fidelity results and improved computational robustness. This work presents a highly accurate entropy conserving/stable solver for the solution of the compressible Euler equations. The method uses a modal Discontinuous Galerkin approximation and takes advantage of both entropy variables and the Direct Enforcement of Entropy Balance (DEEB), proposed by Abgrall [1] to fulfill the second law of thermodynamics at the discrete level. Thanks to the use of an orthonormal modal basis, the cost of adding the DEEB correction to the spatially discretized equations is negligible. Entropy variables are used directly, as unknowns, or to assemble the discrete operators while solving for the conservative variables. In the latter case, the conservative variables are “converted” into the entropy ones through an L2 “entropy projection”. If this strategy is coupled with DEEB, a very efficient alternative to the direct use of the entropy variables is obtained, with a dramatic improvement in robustness over the direct use of the conservative variables. For the time integration both an explicit Strong Stability Preserving Runge-Kutta scheme and the implicit Generalized Crank-Nicolson method are considered. The latter allows for the construction of a discrete entropy conservative/stable scheme both in space and time to be used as a reference. Convergence and conservation properties are demonstrated together with robustness and computational efficiency on a suite of two-dimensional test cases, showing that by coupling DEEB with entropy projection the solver robustness is improved while not affecting the stability limit.

Local discontinuous Galerkin method coupled with the implicit‐explicit Runge–Kutta method for the time‐dependent micropolar fluid equations

AbstractIn this article, the spatial local discontinuous Galerkin (LDG) approximation coupled with the temporal implicit‐explicit Runge–Kutta (RK) evolution for the micropolar fluid equations are adopted to construct the discretization method. To avoid the incompressibility constraint, the artificial compressibility strategy method is used to convert the micropolar fluid equations into the Cauchy–Kovalevskaja type equations. Then the LDG method based on the modal expansion and the implicit‐explicit RK method are properly combined to construct the expected third‐order method. Theoretically, the unconditionally stable of the fully discrete method are derived in multidimensions for triangular meshs. And the numerical experiments are given to verify the theoretical and effectiveness of the presented methods.

The Discontinuous Galerkin Approximation of the Grad-Div and Curl-Curl Operators in First-Order Form Is Involution-Preserving and Spectrally Correct

The Discontinuous Galerkin Approximation of the Grad-Div and Curl-Curl Operators in First-Order Form Is Involution-Preserving and Spectrally Correct

Higher-order methods for the Stokes equations based on the coupling of discontinuous Galerkin method and spectral deferred correction method

In this paper, the spatial discontinuous Galerkin (DG) approximation coupled with the temporal spectral deferred correction (SDC) evolution for the Stokes equations is adopted to construct the higher-order discretization method. First, the artificial compressibility strategy method is used to convert the Stokes equations into the Cauchy–Kovalevskaja type equations. Second, the corresponding equations can be rewritten as a first-order system by introducing the new variable equal to the gradient of the velocity. Then, the DG and the SDC methods are properly combined to construct the expected higher-order method. Theoretically, the stability analysis of the second-order fully discrete method is proved. The numerical experiments are given to verify the effectiveness of the presented methods.

An indicator-based hybrid limiter in discontinuous Galerkin methods for hyperbolic conservation laws

This paper proposes a hybrid limiter in discontinuous Galerkin (DG) methods for hyperbolic conservation laws. This hybrid limiter combines the high-order DG approximation with the low-order limited solution to achieve better resolution for multiscale structures. Meanwhile, the essentially non-oscillatory property can also be retained. The indicator, based on the jump information at the cell interface, provides the non-linear weight for the hybrid limiter, which also acts as a shock detection technique to identify troubled cells in the DG approximation. The presented limiter also maintains the conservation of mass cell by cell. In addition, the hybrid limiter maintains the local data structure and can, therefore, preserve the parallel efficiency of the DG method. The hybrid limiter is very simple in design and can be easily implemented with arbitrary high-order accuracy on both structured and unstructured meshes. Benchmark examples, mainly on Euler equations, are presented to demonstrate the performance of these DG methods with associated hybrid limiters.

Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems

We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined [Formula: see text] discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non-homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined [Formula: see text] space, the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.

ADER discontinuous Galerkin Material Point Method

AbstractThe first‐order accurate discontinuous Galerkin Material Point Method (DGMPM), initially introduced by Renaud et al. (J Comput Phys. 2018;369:80–102.), considers a solid body discretized by a collection of material points carrying the history of the matter, embedded in an arbitrary grid on which a nodal discontinuous Galerkin approximation is defined, and that serves to solve balance equations. This method has been shown to be promising, especially for solving hyperbolic problems in finite deforming solids (Renaud et al. Int J Numer Methods Eng. 2020;121(4):664–689. Renaud et al. Comput Methods Appl Mech Eng. 2020;365:112987.). The main goal of this research is to extend the first‐order DGMPM to arbitrary high‐order accurate approximations. This is performed by adapting the ADER (Arbitrary high order DErivative Riemann problem) approach (Busto et al. Front Phys. 2020;8:32.) to the particular spatial discretization of the DGMPM. First, the predictor step permits to design a particle‐to‐grid projection of arbitrary high order of accuracy, consistent with that of the nodal discontinuous Galerkin approximation defined on the arbitrary grid. This is performed using a moving least square approximation for the ADER predictor field. Second, since the degrees of freedom of the predictor field are now defined at material points, the computation of the constitutive response of the material is ensured to be always performed at these material points. This is of crucial importance for history‐dependent constitutive models because it avoids any diffusive transfer of internal variables on a new computational grid. Finally, a total Lagrangian formulation of equations is kept, which allows to precompute once and for all both the nodal discontinuous Galerkin approximation and that of the ADER predictor field, until the arbitrary grid is discarded if required. The method is illustrated on a few two‐dimensional numerical examples, on which comparisons are shown with the ADER‐DGFEM and Runge–Kutta‐DGFEM.

The Well-Posedness and Discontinuous Galerkin Approximation for the Non-Newtonian Stokes–Darcy–Forchheimer Coupling System

The Well-Posedness and Discontinuous Galerkin Approximation for the Non-Newtonian Stokes–Darcy–Forchheimer Coupling System

The regularised inertial Dean–Kawasaki equation: discontinuous Galerkin approximation and modelling for low-density regime

The Regularised Inertial Dean–Kawasaki model (RIDK) – introduced by the authors and J. Zimmer in earlier works – is a nonlinear stochastic PDE capturing fluctuations around the meanfield limit for large-scale particle systems in both particle density and momentum density. We focus on the following two aspects. Firstly, we set up a Discontinuous Galerkin (DG) discretisation scheme for the RIDK model: we provide suitable definitions of numerical fluxes at the interface of the mesh elements which are consistent with the wave-type nature of the RIDK model and grant stability of the simulations, and we quantify the rate of convergence in mean square to the continuous RIDK model. Secondly, we introduce modifications of the RIDK model in order to preserve positivity of the density (such a feature only holds in a “high-probability sense” for the original RIDK model). By means of numerical simulations, we show that the modifications lead to physically realistic and positive density profiles. In one case, subject to additional regularity constraints, we also prove positivity. Finally, we present an application of our methodology to a system of diffusing and reacting particles. Our Python code is available in open-source format.

A Local Discontinuous Galerkin Approximation for the \(\boldsymbol{p}\) -Navier–Stokes System, Part III: Convergence Rates for the Pressure

A Local Discontinuous Galerkin Approximation for the \(\boldsymbol{p}\) -Navier–Stokes System, Part III: Convergence Rates for the Pressure

A Local Discontinuous Galerkin Approximation for the \(\boldsymbol{{p}}\) -Navier–Stokes System, Part II: Convergence Rates for the Velocity

A Local Discontinuous Galerkin Approximation for the \(\boldsymbol{{p}}\) -Navier–Stokes System, Part II: Convergence Rates for the Velocity

A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems

A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems

A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids

A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids

Discontinuous Galerkin Methods with Generalized Numerical Fluxes for the Vlasov-Viscous Burgers’ System

In this paper, semi-discrete numerical scheme for the approximation of the periodic Vlasov-viscous Burgers’ system is developed and analyzed. The scheme is based on the coupling of discontinuous Galerkin approximations for the Vlasov equation and local discontinuous Galerkin approximations for the viscous Burgers’ equation. Both these methods use generalized numerical fluxes. The proposed scheme is both mass and momentum conservative. Based on generalized Gauss–Radau projections, optimal rates of convergence in the case of smooth compactly supported initial data are derived. Finally, computational results confirm our theoretical findings.

Convergence analysis of a Local Discontinuous Galerkin approximation for nonlinear systems with balanced Orlicz-structure

In this paper, we investigate a Local Discontinuous Galerkin (LDG) approximation for systems with balanced Orlicz-structure. We propose a new numerical flux, which yields optimal convergence rates for linear ansatz functions. In particular, our approach yields a unified treatment for problems with (p, δ)-structure for arbitrary p ∈ (1, ∞) and δ ≥ 0.

Discontinuous Galerkin Approximation of the Fully Coupled Thermo-poroelastic Problem

We present and analyze a discontinuous Galerkin method for the numerical modeling of the nonlinear fully coupled thermo-poroelastic problem. For the spatial discretization, we design a high-order discontinuous Galerkin method on polygonal and polyhedral grids based on a novel four-field formulation of the problem. To handle the nonlinear convective transport term in the energy conservation equation we adopt a fixed-point linearization strategy. We perform a robust stability analysis for the linearized semidiscrete problem under mild requirements on the problem’s physical parameters. A priori hp-version error estimates in suitable energy norms are also derived. A complete set of numerical simulations is presented in order to validate the theoretical analysis, to inspect numerically the robustness properties, and to test the capability of the proposed method in a practical scenario inspired by a geothermal problem.

An adaptive C0 Interior Penalty Discontinuous Galerkin method and an equilibrated a posteriori error estimator for the von Kármán equations

We consider an adaptive C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of the fourth order von Kármán equations with homogeneous Dirichlet boundary conditions and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a six-field formulation of the finite element discretized von Kármán equations. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W02,2 norm in terms of the associated primal and dual energy functionals. It requires the construction of equilibrated fluxes and equilibrated moment tensors which can be computed on local patches around interior nodal points of the triangulations. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.

Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source

The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.

An h-Adaptive Poly-Sinc-Based Local Discontinuous Galerkin Method for Elliptic Partial Differential Equations

For the purpose of solving elliptic partial differential equations, we suggest a new approach using an h-adaptive local discontinuous Galerkin approximation based on Sinc points. The adaptive approach, which uses Poly-Sinc interpolation to achieve a predetermined level of approximation accuracy, is a local discontinuous Galerkin method. We developed an a priori error estimate and demonstrated the exponential convergence of the Poly-Sinc-based discontinuous Galerkin technique, as well as the adaptive piecewise Poly-Sinc method, for function approximation and ordinary differential equations. In this paper, we demonstrate the exponential convergence in the number of iterations of the a priori error estimate derived for the local discontinuous Galerkin technique under the condition that a reliable estimate of the precise solution of the partial differential equation at the Sinc points exists. For the purpose of refining the computational domain, we employ a statistical strategy. The numerical results for elliptic PDEs with Dirichlet and mixed Neumann-Dirichlet boundary conditions are demonstrated to validate the adaptive greedy Poly-Sinc approach.