Mixed Discontinuous Galerkin Research Articles

A locking-free mixed enriched Galerkin method of arbitrary order for linear elasticity using the stress–displacement formulation

In this paper, we develop an arbitrary-order locking-free enriched Galerkin method for the linear elasticity problem using the stress–displacement formulation in both two and three dimensions. The method is based on the mixed discontinuous Galerkin method in Wang et al. (2020), but with a different stress approximation space that enriches the arbitrary order continuous Galerkin space with some piecewise symmetric-matrix valued polynomials. The enriched Galerkin method maintains locking-free property of the mixed discontinuous Galerkin method in Wang et al. (2020), but has fewer degrees of freedom for commonly used and appropriately high orders. We present some numerical examples in two and three dimensions to demonstrate the effectiveness of the proposed method.

Hybrid mixed discontinuous Galerkin finite element method for incompressible miscible displacement problem

A new hybrid mixed discontinuous Galerkin finite element (HMDGFE) method is constructed for incompressible miscible displacement problem. In this method, the hybrid mixed finite element (HMFE) procedure is considered to solve the pressure and velocity equations, and a new hybrid mixed discontinuous Galerkin finite element procedure is constructed to solve the concentration equation with upwind technique. Compared with other traditional discontinuous Galerkin methods, the new method can reach a global system with less unknowns and sparser stencils. The consistency and conservation of the method are analyzed, the stability and optimal error estimates are also derived by the new technique. Meanwhile, numerical examples are provided to confirm the corresponding theoretical analysis.

Optimal order error estimates of a mixed discontinuous Galerkin method with BDF-2 scheme for transient Darcy flow

In this paper, we study the numerical solution of Darcy flow via a mixed discontinuous Galerkin method. The two-step Euler backward formula is used for the time variable discretization and a mixed discontinuous Galerkin method is applied to spatial variable discretization. The stability results are established for our mixed discontinuous Galerkin numerical scheme in a fully discrete framework. Optimal error estimates for our unconditional scheme are derived for both the velocity and pressure variables. Finally, a numerical experiment is provided to approve the accuracy and efficiency of the proposed numerical method.

Shock capturing for a high-order ALE discontinuous Galerkin method with applications to fluid flows in time-dependent domains

In recent decades, the arbitrary Lagrangian–Eulerian (ALE) approach has been one of the most popular choices to deal with the fluid flows having moving boundaries. The ALE finite volume (FV) method is well-known for its capability of shock capturing. Several ALE discontinuous Galerkin (DG) methods are developed to fully explore the advantages of high-order methods, especially in complex flows. The DG scheme is highly efficient and has high resolution in smooth parts of the flow while the FV scheme has advantage on the robustness against shocks. In this paper, we present a novel algorithm to combine both the ALE FV and ALE DG methods into one stable and efficient hybrid approach. The main challenge for a mixed ALE FV method and ALE DG method is to reduce the inconsistency between both discretizations. Of particular concern are the same mesh velocity distribution between DG and FV elements and a continuous mesh velocity at the coupling interfaces. Since the computational efficiency of the proposed algorithm is of major concern, performance aspects will be considered during the construction of the scheme. To investigate the accuracy of the new scheme, several benchmark test cases with prescribed movements are included. Afterwards, the algorithm is applied to more general and complex scenarios to show its ability to cope with complex setups. In our paper, the scheme is implemented into a loosely-coupled fluid–structure interaction (FSI) framework. With a shock-driven cylinder movement in a channel flow and a transonic flow-induced airfoil vibration, we demonstrate our method for FSI applications.

Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem

Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem

A hybrid mixed finite element method for convection-diffusion-reaction equation with local exponential fitting technique

A new hybrid mixed finite element method is proposed for solving the convection-diffusion-reaction equation with local exponential fitting technique. In this method, the exponential fitting technique in [7] is used to discretize the convection-diffusion-reaction equation by introducing a new variable at element level; then the hybrid mixed discontinuous Galerkin finite element method is used to approximate the discretized problem. The convergence of the proposed method is analyzed, and the corresponding a priori error estimate is derived. The numerical results are presented to confirm our theoretical analysis.

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>

Analysis of a mixed discontinuous Galerkin method for the time-harmonic Maxwell equations with minimal smoothness requirements

AbstractAn error analysis of a mixed discontinuous Galerkin (DG) method with lifting operators as numerical fluxes for the time-harmonic Maxwell equations with minimal smoothness requirements is presented. The key difficulty in the error analysis for the DG method is that due to the low regularity the tangential trace of the exact solution is not well defined on the faces of the computational mesh. This difficulty is addressed by adopting the face-to-cell lifting introduced by Ern & Guermond (2021, Quasi-optimal nonconforming approximation of elliptic PDEs with contrasted coefficients and $H^{1+r}$, $r>0$, regularity. Found. Comput. Math., 1–36). To obtain optimal local interpolation estimates, we introduce Scott–Zhang-type interpolations that are well defined for $H(\textrm {curl})$ and $H(\textrm {div})$ functions with minimal regularity requirements. As a by-product of penalizing the lifting of the tangential jumps, an explicit and easily computable stabilization parameter is given.

A three-dimensional modal discontinuous Galerkin method for the second-order Boltzmann-Curtiss-based constitutive model of rarefied and microscale gas flows

A three-dimensional mixed modal discontinuous Galerkin (DG) method based on tetrahedral meshes was developed for simulating all flow regimes from subsonic to hypersonic rarefied and microscale gas flows within a single framework. The mixed modal DG scheme was used for solving conservation laws in conjunction with the second-order Boltzmann-Curtiss-based constitutive model of diatomic and polyatomic gases in strong thermal nonequilibrium. A decomposition algorithm based on the compression-expansion and velocity shear sub-problems was presented for solving the multi-dimensional second-order constitutive model. The Langmuir and Maxwell-Smoluchowski velocity-slip and temperature-jump boundary conditions were also implemented into the DG framework. To assess the ability of the new computational model to capture correct physical phenomena, we applied the new model to various gas flows in a wide range of continuum-rarefied and microscale regimes. The computational results in the rarefied and microscale flow regimes showed that the second-order constitutive model yielded solutions that were in better agreement with the direct simulation Monte Carlo and experimental data than the first-order constitutive model.

Superconvergence of Discontinuous Galerkin Methods for Elliptic Boundary Value Problems

In this paper, we present a unified analysis of the superconvergence property for a large class of mixed discontinuous Galerkin methods. This analysis applies to both the Poisson equation and linear elasticity problems with symmetric stress formulations. Based on this result, some locally postprocess schemes are employed to improve the accuracy of displacement by order $$\min (k+1, 2)$$ if polynomials of degree k are employed for displacement. Some numerical experiments are carried out to validate the theoretical results.

Unconditionally energy stable discontinuous Galerkin schemes for the Cahn–Hilliard equation

In this paper, we introduce novel discontinuous Galerkin (DG) schemes for the Cahn–Hilliard equation, which arises in many applications. The method is designed by integrating the mixed DG method for the spatial discretization with the Invariant Energy Quadratization (IEQ) approach for the time discretization. Coupled with a spatial projection, the resulting IEQ–DG schemes are shown to be unconditionally energy dissipative, and can be efficiently solved without resorting to any iteration method. Both one and two dimensional numerical examples are provided to verify the theoretical results, and demonstrate the good performance of IEQ–DG in terms of efficiency, accuracy, and preservation of the desired solution properties.

A mixed discontinuous Galerkin method for the wave equation

We study a mixed discontinuous Galerkin (DG) method for solving the second-order wave equation. The stress variable p and the displacement variable u are discretized by the mixed DG element pair Pk+1–Pk (k≥0). Under appropriate regularity assumptions on the solution pair, we derive optimal error estimates for the spatially semi-discrete scheme. Specifically, we prove the optimal convergence orders for u in L2 norm and p in L2 norm. Some test problems are presented, and the numerical results show that the orders of convergence coincide with the predicted error estimates.

A Discontinuous Galerkin Method for the Coupled Stokes and Darcy Problem

Combining the mixed discontinuous Galerkin method for the Darcy flow and the interior penalty discontinuous Galerkin methods for the Stokes problem, a locally conservative discrete scheme is proposed for numerically solving the coupled Stokes and Darcy problem. We prove the well-posedness of the solution of the proposed numerical scheme by boundedness, K-ellipticity and a discrete inf-sup condition. A priori error estimates, in proper norms are derived, and to verify the theoretical analysis, some numerical experiments are given.

A mixed discontinuous Galerkin method for an unsteady incompressible Darcy equation

We study a mixed discontinuous Galerkin (MDG) method for solving a time-dependent Darcy problem, which simulates an incompressible fluid such as water flowing in a rigid porous medium. The discretization of the unsteady Darcy problem relies on a backward Euler scheme for temporal variable and MDG method for spatial variables. Spatially semi-discrete and fully discrete schemes are analyzed. Existence and uniqueness of the numerical solutions are proved, and optimal order error estimates are derived for both the velocity and pressure variables. Finally, some test problems are provided to display the performance of the MDG method, and numerical results are reported to support the theoretical predictions.

A mixed discontinuous Galerkin method with symmetric stress for Brinkman problem based on the velocity–pseudostress formulation

The Brinkman equations can be regarded as a combination of the Stokes and Darcy equations which model transitions between the fast flow in channels (governed by Stokes equations) and the slow flow in porous media (governed by Darcy’s law). The numerical challenge for this model is the designing of a numerical scheme which is stable for both the Stokes-dominated (high permeability) and the Darcy-dominated (low permeability) equations. In this paper, we solve the Brinkman model in n dimensions (n=2,3) by using the mixed discontinuous Galerkin (MDG) method, which meets this challenge. This MDG method is based on the pseudostress–velocity formulation and uses a discontinuous piecewise polynomial pair P̲k+1S-Pk(k≥0), where the stress field is symmetric. The main unknowns are the pseudostress and the velocity, whereas the pressure is easily recovered through a simple postprocessing. A key step in the analysis is to establish the parameter-robust inf–sup stability through specific parameter-dependent norms at both continuous and discrete levels. Therefore, the stability results presented here are uniform with respect to the permeability. Thanks to the parameter-robust stability analysis, we obtain optimal error estimates for the stress in broken H̲(div)-norm and velocity in L2-norm. Furthermore, the L̲2 error estimate for pseudostress is derived under certain conditions. Finally, numerical experiments are provided to support the theoretical results and to show the robustness, accuracy, and flexibility of the MDG method.

A Mixed Discontinuous Galerkin Method for the Helmholtz Equation

In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. The mixed discontinuous Galerkin method is designed by using a discontinuous Pp+1−1−Pp−1 finite element pair for the flux variable and the scattered field with p≥0. We can get optimal order convergence for the flux variable in both Hdiv-like norm and L2 norm and the scattered field in L2 norm numerically. Moreover, we conduct the numerical experiments on the Helmholtz equation with perturbation and the rectangular waveguide, which also demonstrate the good performance of the mixed discontinuous Galerkin method.

A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry

In this paper, we study a mixed discontinuous Galerkin (MDG) method to solve linear elasticity problem with arbitrary order discontinuous finite element spaces in $d$-dimension ($d=2,3$). This method uses polynomials of degree $k+1$ for the stress and of degree $k$ for the displacement ($k\geq 0$). The mixed DG scheme is proved to be well-posed under proper norms. Specifically, we prove that, for any $k \geq 0$, the $H({\rm div})$-like error estimate for the stress and $L^2$ error estimate for the displacement are optimal. We further establish the optimal $L^2$ error estimate for the stress provided that the $\mathcal{P}_{k+2}-\mathcal{P}_{k+1}^{-1}$ Stokes pair is stable and $k \geq d$. We also provide numerical results of MDG showing that the orders of convergence are actually sharp.

Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lame system of linear elasticity in polyhedral domains in $${\mathbb {R}}^3$$. In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we illustrate numerically that the proposed mixed DG schemes converge exponentially in a natural DG norm.

An a-priori error analysis for discontinuous Lagrangian finite elements applied to nonconforming dual-mixed formulations: Poisson and Stokes problems

In this paper, we discuss the well-posedness of a mixed discontinuous Galerkin (DG) scheme for the Poisson and Stokes problems in 2D, considering only piecewise Lagrangian finite elements. The complication here lies in the fact that the classical Babuška-Brezzi theory is difficult to verify for low-order finite elements, so we proceed in a non-standard way. First, we prove uniqueness, and then we apply a discrete version of Fredholm's alternative theorem to ensure existence. The a-priori error analysis is done by introducing suitable projections of the exact solution. As a result, we prove that the method is convergent, and, under standard additional regularity assumptions on the exact solution, the optimal rate of convergence of the method is guaranteed.

A new mixed discontinuous Galerkin method for the electrostatic field

We introduce and analyze a new mixed discontinuous Galerkin method for approximation of an electric field. We carry out its error analysis and prove an error estimate that is optimal in the mesh size. Some numerical results are given to confirm the theoretical convergence.