Discontinuous Galerkin Finite Element Method Research Articles

A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems

<abstract><p>For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using $ \mathbb{P}_{k}-\mathbb{P}_{k-1} $ element $ (k\geq 1) $ is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order $ O(dof^{\frac{-2k}{d}}) $.</p></abstract>

Discontinuous Galerkin method for blow-up solutions of nonlinear wave equations

e develop and study an explicit time-space discrete discontinuous Galerkin finite element method to approximate the solution of one-dimensional nonlinear wave equations. We show that the numerical scheme is stable if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that under weak convergence assumptions, the numerical blow-up time tends toward the theoretical one. The validity of our results is confirmed throughout several examples and benchmarks.

The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis

<abstract><p>In this paper, we investigate the local discontinuous Galerkin (LDG) finite element method for the fractional Allen-Cahn equation with Caputo-Hadamard derivative in the time domain. First, the regularity of the solution is analyzed, and the results indicate that the solution of this equation generally possesses initial weak regularity in the time dimension. Due to this property, a logarithmic nonuniform L1 scheme is adopted to approximate the Caputo-Hadamard derivative, while the LDG method is used for spatial discretization. The stability and convergence of this fully discrete scheme are proven using a discrete fractional Gronwall inequality. Numerical examples demonstrate the effectiveness of this method.</p></abstract>

High Order Central Discontinuous Galerkin-Finite Element Methods for the abcd Boussinesq System

High Order Central Discontinuous Galerkin-Finite Element Methods for the abcd Boussinesq System

Unified discontinuous Galerkin finite element methods for second order Dirichlet boundary control problem

In this article, we study the Dirichlet boundary control problem governed by Poisson equation, therein the control is penalized in H1(Ω) space and various symmetric discontinuous Galerkin finite element methods are designed and analyzed for its numerical approximation. Symmetric property of the bilinear forms is exploited to obtain the discrete optimality system. By a careful use of various intermediate problems, the optimal order convergence rates are obtained for the control in the energy and L2-norms. Moreover, using an auxiliary system of equations, a posteriori error estimator is derived which is shown to be reliable and efficient. Numerical experiment results are included to confirm the theoretical findings.

The discontinuous Galerkin and the nonconforming ECR element approximations for an MHD Stokes eigenvalue problem

In this paper, we adopt the discontinuous Galerkin finite element method and the enriched Crouzeix‐Raviart finite element method to study the magnetohydrodynamic (MHD) Stokes eigenvalue problem describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field. We give the convergence and error analysis for the approximations, and the theoretical analysis and numerical experiments show that the methods are effective and can be applied to general domains. We also explore the influence of the Hartmann number on the eigenpairs and the consequential variation of the eigenstructure with the magnetic field by numerical experiments.

A reconstructed discontinuous approximation on unfitted meshes to [formula omitted] and [formula omitted] interface problems

We propose a high-order discontinuous Galerkin finite element method for solving the H(div)- and H(curl)-elliptic interface problems on unfitted meshes. The vector-valued approximation space is constructed by the patch reconstruction with at most d degrees of freedom per element. The C2-smooth interface is allowed to intersect elements in a quite general setup. The patch reconstruction provides the stability near the interface naturally without any additional stabilization strategy. The method is based on the symmetric interior penalty method and the optimal and suboptimal convergence rates under the energy norm and L2 norm are derived. Numerical examples in both two and three dimensions are presented to demonstrate the accuracy of the method.

A hybridizable discontinuous Galerkin method for the coupled Navier–Stokes and Darcy problem

We present and analyze a strongly conservative hybridizable discontinuous Galerkin finite element method for the coupled incompressible Navier–Stokes and Darcy problem with Beavers–Joseph–Saffman interface condition. An a priori error analysis shows that the velocity error does not depend on the pressure, and that velocity and pressure converge with optimal rates. These results are confirmed by numerical examples.

Imaging features of different defects in metals using laser ultrasonic techniques

Different types of defects such as pores, porosity, cracks in different directions, have various impact on the load bearing capacity and fatigue strength of metal components used in the field of aerospace, rail transportation, electrical equipment, etc. Therefore, feature recognition and classification of defects contribute greatly to the performance evaluation of the metal components. In this paper, the imaging features of six typical defects in metals were obtained by using laser ultrasonic (LU) scan inspection techniques. First, the numerical models of LU inspection of different defects were established based on the discontinuous Galerkin finite element method (dG-FEM) in COMSOL/Multiphysics. Then, multiple groups of FE simulations were carried out. The generation and receiving lasers were moved along the scanning direction according to a fixed step during the simulation. All the simulated A-scan signals were stacked up and plotted in the B-scan and D-scan maps. At last, six samples with different artificial defects were manufactured for the LU inspection experiment. The defect imaging features in the experimental B-scan maps are consistent with the numerical results. Besides, the mathematical calculation methods of defect position, dimension and orientation were put forward.

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An over-penalized weak Galerkin method for parabolic interface problems with time-dependent coefficients

Based on the idea of classical discontinuous Galerkin and weak Galerkin finite element methods, we introduce an over-penalized term in the new scheme as a part of stabilization for solving parabolic interface problems. From the double-valued functions defined on interior edges of elements, it is natural to generate jumps of the over-penalized term. An over-penalized weak Galerkin (OPWG) finite element method can be applied very well to interface problems with general imperfect interface. Importantly, the diffusion coefficients of the interface problems depend on both temporal and spacial variables, not only limited in space as usual. With the use of (Pk(K),Pk−1(e),[Pk−1(K)]d) elements, semi-discrete and fully discrete schemes with backward Euler approximation in time are presented, and then the semi-discrete one is analyzed to be unconditionally stable. To analyze error estimates of semi-discrete and fully discrete schemes directly by error equation, we can just have optimal convergence order in energy norm. By virtue of the introduction of an elliptic projection operator, optimal error estimates of those schemes in L2 norm can be proved. Numerical examples are given to validate the efficiency and optimal convergence orders of the new schemes.

Numerical Analysis of a Discontinuous Galerkin Method for the Borrvall--Petersson Topology Optimization Problem

Divergence-free discontinuous Galerkin (DG) finite element methods offer a suitable discretization for the pointwise divergence-free numerical solution of Borrvall and Petersson's model for the topology optimization of fluids in Stokes flow [Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77--107]. The convergence results currently found in literature only consider $H^1$-conforming discretizations for the velocity. In this work, we extend the numerical analysis of Papadopoulos and Suli to divergence-free DG methods with an interior penalty [I. P. A. Papadopoulos and E. Suli, Numerical analysis of a topology optimization problem for Stokes flow, arXiv preprint arXiv:2102.10408, (2021)]. We show that, given an isolated minimizer to the analytical problem, there exists a sequence of DG finite element solutions, satisfying necessary first-order optimality conditions, that strongly converges to the minimizer.

A Discontinuous Galerkin Finite Element Method based axial SN for the 2D/1D transport method

The 2D/1D transport method is a prominent solver for the 3D Boltzmann equation due to its strong geometric capability. To pursue high accuracy, the discrete ordinate (SN) method for transport equation is a straightforward choice as the axial 1D solver instead of the traditional nodal expansion method (NEM) for diffusion equation. In practice, the axial SN solver is usually combined with the transverse leakage splitting method to deal with the negative source issue, which strongly affects the convergence of the 2D/1D method. However, transverse leakage splitting breaks the consistency of scalar fluxes between the 2D MOC and 1D SN equations. To solve this problem, a discontinuous Galerkin finite element method (DGFEM) for axial SN together with an improved 2D/1D transport approach is proposed in the present study to ensure that the scalar flux from the 1D axial solver is the same as that from the 2D solver. The accuracy, consistency and computational efficiency of the new scheme are evaluated using various well-known numerical problems, including heterogeneous assembly problem, KUCA benchmark, and C5G5-3D benchmark.

Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes

This article considers the extension of two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when agglomerated polygonal/polyhedral meshes are employed for the coarse mesh approximation. We recall that within the two-grid setting, while it is necessary to solve a nonlinear problem on the coarse approximation space, only a linear problem must be computed on the original fine finite element space. In this article, the coarse space will be constructed by agglomerating elements from the original fine mesh. Here, we extend the existing a priori and a posteriori error analysis for the two-grid hp-version discontinuous Galerkin finite element method from Congreve et al. [1] for coarse meshes consisting of standard element shapes to include arbitrarily agglomerated coarse grids. Moreover, we develop an hp-adaptive two-grid algorithm to adaptively design the fine and coarse finite element spaces; we stress that this is undertaken in a fully automatic manner, and hence can be viewed as blackbox solver. Numerical experiments are presented for two- and three-dimensional problems to demonstrate the computational performance of the proposed hp-adaptive two-grid method.

Space-Time Petrov-Discontinuous Galerkin Finite Element Method for Solving Linear Convection-Diffusion Problems

The paper presents the theory of the space-time Petrov-discontinuous Galerkin finite element (PDGFE) method for the discretization of the nonstationary linear convection-diffusion problems. The PDGFE method is modified for the discontinuous Galerkin finite element (DGFE) method in the case of the symmetric interior penalty Galerkin (SIPG) scheme. PDGFE method is applied separately in space using different space gride on different time levels. We prove the properties of the bilinear form aPD, m (u, ν) (V − elliptic and continuity) stability and prove the approximate solution converges with the error of order o (h 2 + τ 3). A numerical experiment is carried out to confirm the theoretical conclusions.

Discontinuous Galerkin FEM with Hot Element Addition for the Thermal Simulation of Additive Manufacturing

Despite its promising advantages, the application of directed energy deposition (DED) to produce large metal parts is hindered by challenges inherent to the process. Undesired residual stresses, distortions and heterogeneous material properties mainly originate from a part’s thermal history. Fast part-scale thermal models therefore facilitate improved applicability of DED by enabling the prediction and mitigation of these unwanted effects. In this work, the efficiency of a discontinuous Galerkin-based thermal model with heat input by hot element addition, is evaluated and improved to allow such fast simulations. It is found that the model permits the use of a coarse discretization around the heat source, which significantly reduces simulation time while maintaining accurate solutions. It is also shown that the model naturally facilitates the use of local time stepping, which can considerably improve the efficiency of thermal additive manufacturing simulations.

On Mathematical and Numerical Modelling of Multiphysics Wave Propagation with Polytopal Discontinuous Galerkin Methods: a Review

In this work we review discontinuous Galerkin finite element methods on polytopal grids (PolydG) for the numerical simulation of multiphysics wave propagation phenomena in heterogeneous media. In particular, we address wave phenomena in elastic, poro-elastic, and poro-elasto-acoustic materials. Wave propagation is modeled by using either the elastodynamics equation in the elastic domain, the acoustics equations in the acoustic domain and the low-frequency Biot’s equations in the poro-elastic one. The coupling between different models is realized by means of (physically consistent) transmission conditions, weakly imposed at the interface between the subdomains. For all models configuration, we introduce and analyse the PolydG semi-discrete formulation, which is then coupled with suitable time marching schemes. For the semi-discrete problem, we present the stability analysis and derive a-priori error estimates in a suitable energy norm. A wide set of two-dimensional verification tests with manufactured solutions are presented in order to validate the error analysis. Examples of physical interest are also shown to demonstrate the capability of the proposed methods.

High-Order CFD Solvers on Three-Dimensional Unstructured Meshes: Parallel Implementation of RKDG Method with WENO Limiter and Momentum Sources

In aerospace engineering, high-order computational fluid dynamics (CFD) solvers suitable for three-dimensional unstructured meshes are less developed than expected. The Runge–Kutta discontinuous Galerkin (RKDG) finite element method with compact weighted essentially non-oscillatory (WENO) limiters is one of the candidates, which might give high-order solutions on unstructured meshes. In this article, we provide an efficient parallel implementation of this method for simulating inviscid compressible flows. The implemented solvers are tested on lower-dimensional model problems and real three-dimensional engineering problems. Results of lower-dimensional problems validate the correctness and accuracy of these solvers. The capability of capturing complex flow structures even on coarse meshes is shown in the results of three-dimensional applications. For solving problems containing rotary wings, an unsteady momentum source model is incorporated into the solvers. Such a finite element/momentum source hybrid method eliminates the reliance on advanced mesh techniques, which might provide an efficient tool for studying rotorcraft aerodynamics.

Optimal error estimates to smooth solutions of the central discontinuous Galerkin methods for nonlinear scalar conservation laws

In this paper, we study the error estimates to sufficiently smooth solutions of the nonlinear scalar conservation laws for the semi-discrete central discontinuous Galerkin (DG) finite element methods on uniform Cartesian meshes. A general approach with an explicitly checkable condition is established for the proof of optimal L2 error estimates of the semi-discrete CDG schemes, and this condition is checked to be valid in one and two dimensions for polynomials of degree up to k = 8. Numerical experiments are given to verify the theoretical results.

Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems

Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems