Weak Galerkin Method Research Articles

Optimal convergence analysis of the energy-preserving immersed weak Galerkin method for second-order hyperbolic interface problems in inhomogeneous media

This article reports our explorations for solving second-order hyperbolic interface problems by immersed weak Galerkin (IWG) method on interface independent meshes. The method presented here uses IWG functions for the discretization in spatial variable. The study includes the Newmark algorithm which has been used extensively in applications. The stability analysis based on the energy method is presented for semi-discrete and fully-discrete schemes under some conditions on parameters of the Newmark algorithm. In this work, we carried out a convergence analysis and obtained optimal a priori error estimates in both energy and L2 norms for the semi-discrete and fully-discrete schemes under piecewise H2 regularity assumption in space and some conditions on parameters of the Newmark algorithm. We demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(h2+τr(γ)), where r(γ)=1 if γ≠1/2, r(1/2)=2, h and τ are the mesh size and the time step, respectively. Numerical examples are provided to confirm theoretical findings and illustrate the efficiency of the method for standing and traveling waves.

Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems

We propose and analyze a modified weak Galerkin finite element method (MWG-FEM) for solving singularly perturbed problems of convection-dominated type. The proposed method is constructed over piecewise polynomials of degree $k\geq1$ on interior of each element and piecewise constant on the boundary of each element. The present method is parameter-free and has less degrees of freedom compared to the classical weak Galerkin finite element method. The method is shown uniformly convergent for small perturbation parameters. An uniform convergence rate of $\mathcal {O}((N^{-1}\ln N)^k)$ in the energy-like norm is established on the piecewise uniform Shishkin mesh, where $N$ is the number of elements. Various numerical examples are presented to confirm the theoretical results. Moreover, we numerically confirm that the proposed method has the optimal order error estimates of $\mathcal {O}(N^{-(k+1)})$ in a discrete $L^2$-norm and converges at superconvergence order of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L^\infty$-norm.

Superconvergence of a modified weak Galerkin method for singularly perturbed two-point elliptic boundary-value problems

Superconvergence of a modified weak Galerkin method for singularly perturbed two-point elliptic boundary-value problems

Higher order weak Galerkin methods for the Navier–Stokes equations with large Reynolds number

AbstractIn this article, we provide a class of numerical schemes to solve the steady state and non‐steady state Navier–Stokes equations with large Reynolds number. The high order weak Galerkin methods are numerically proposed and tested. We numerically find that the stabilizing term has a great impact on the robustness of the Jacobian matrices associated with the Newton's method in the procedure of solving the discrete nonlinear systems generated by weak Galerkin discretizations, especially for the case with large Reynolds number. The numerical investigations of several benchmarks are then carried out to demonstrate the efficiency and robustness of our proposed numerical methods. The errors for the Taylor–Green vortex flow are computed and compared with different weak Galerkin techniques. The lid driven flow and the flow around a cylinder are further numerically investigated and compared with existing references. The Navier–Stokes equations with large Reynolds number can be efficiently solved by the high order weak Galerkin methods with proper stabilizers.

New discontinuous Galerkin algorithms and analysis for linear elasticity with symmetric stress tensor

This paper presents a new and unified approach to the derivation and analysis of many existing, as well as new discontinuous Galerkin methods for linear elasticity problems. The analysis is based on a unified discrete formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor \(\varvec{\sigma }_h\) and displacement \(u_h\) inside each element, and the modifications of these two variables \(\check{\varvec{\sigma }}_h\) and \({\check{u}}_h\) on elementary boundaries of elements. Motivated by many relevant methods in the literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problems and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to zero, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problems. Two families of inf-sup conditions, one known as \(H^1\)-based and the other known as \(H(\mathrm{div})\)-based, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed methods and also offer a new and unified analysis for many existing methods in the literature as a by-product. Some numerical examples are provided to verify the theoretical analysis including the optimal convergence of the new proposed methods.

A [formula omitted] weak Galerkin method for linear Cahn–Hilliard–Cook equation with random initial condition

This paper introduces a C0 weak Galerkin finite element method for a linear Cahn–Hilliard–Cook equation. The highlights of the proposed method are that the complexity of constructing the C1 finite element space for fourth order problem is avoided and the number of degree of freedom is apparently reduced compared to the fully discontinuous weak Galerkin finite element method. With the redefined discrete weak Laplace operator and the classical C0 Lagrange elements, the L2 optimal error estimates in spatial variable are obtained. In time, the classical Euler scheme is then used to do the numerical simulation. Finally, numerical experiments are presented to demonstrate the efficiency of the proposed numerical method.

Discrete maximum principle for the weak Galerkin method on triangular and rectangular meshes

In this paper, the discrete maximum principle (DMP) of the weak Galerkin (WG) method for the second order elliptic equations is considered. The WG schemes are proposed on triangular and rectangular meshes, respectively. For both schemes, the discrete maximum principle for non-negative source and general source are proved. Numerical experiments are also presented to verify the convergence and the DMP-preserving property.

A study of mixed problem for second order elliptic problems using modified weak Galerkin finite element method

In this article, a modified weak Galerkin finite element method is developed for a second order elliptic problem with mixed boundary conditions. The basic idea of modified weak Galerkin method is to reduce the degrees of freedom in comparison to the standard weak Galerkin method. The optimal convergence orders in a discrete L2 norm and H1 norm are established. Numerical examples are presented to verify the theoretical estimates.

Robust a-posteriori error estimates for weak Galerkin method for the convection-diffusion problem

We present a robust a posteriori error estimator for a weak Galerkin finite element method applied to stationary convection-diffusion equations in the convection-dominated regime. The estimator provides global upper and lower bounds of the error and is robust in the sense that upper and lower bounds are uniformly bounded with respect to the diffusion coefficient. The weak Galerkin method we use was developed by Lin, Ye, Zhang, and Zhu (2018) for the convection-diffusion problem without assuming any additional conditions on the convection coefficient and, has a simple formulation. The motivation for our work comes from the fact that while this method performs very well in the strongly convection-dominated regime, it continues to exhibit poor behavior in the intermediate regime. In this proposed work we show that by relying on adaptively refined meshes based on a posteriori residual-type estimator, we can retrieve the optimal order of convergence for all the regimes not just for the strongly convection-dominated regime. Results of the numerical experiments are presented to illustrate the performance of the error estimator.

Numerical analysis of fully discrete energy stable weak Galerkin finite element Scheme for a coupled Cahn-Hilliard-Navier-Stokes phase-field model

The Cahn-Hilliard phase-field model of two-phase incompressible flows, namely the Cahn-Hilliard-Navier-Stokes (CH-NS) problem represents the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Since finding solution (numerically and theoretically) of the CH-NS system is non-trivial (owing to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation), in this paper, we propose and analyze a numerical scheme with the following properties: (1) first-order in time, (2) nonlinear, (3) fully coupled and (4) energy stable; for solving CH-NS system, in the framework of weak Galerkin (WG) method. More precisely, we employ the WG method, which uses discontinuous functions to construct the approximation space, and the first-order backward Euler (implicit) method for space and time discretizations, respectively. We first recall the corresponding variational formulation, and then summarize the main WG method ingredients that are required for our discrete analysis. In particular, in order to define the weak discrete bilinear (and trilinear) form, whose continuous version involves classical differential operators, we propose two well-known alternatives for gradient and divergence operators onto a suitable polynomial subspace. Next, we show that the weak global discrete bilinear form satisfies the hypotheses required by the Lax-Milgram lemma. In this way, we derive the associated a priori error estimates for phase field variable, chemical potential, velocity and further the pressure in L2 norm. Finally, several numerical results confirming the theoretical rates of convergence and illustrating the good performance of the method for simulation influence of surface tension, small density variations, and the driven cavity are presented.

Uniform Stability and Error Analysis for Some Discontinuous Galerkin Methods

In this paper, we provide a number of new estimates on the stability and convergence of both hybrid discontinuous Galerkin (HDG) and weak Galerkin (WG) methods. By using the standard Brezzi theory on mixed methods, we carefully define appropriate norms for the various discretization variables and then establish that the stability and error estimates hold uniformly with respect to stabilization and discretization parameters. As a result, by taking appropriate limit of the stabilization parameters, we show that the HDG method converges to a primal conforming method and the WG method converge to a mixed conforming method.

A weak Galerkin-mixed finite element method for the Stokes-Darcy problem

In this paper, we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation. A discrete inf-sup condition is proved and the optimal error estimates are also derived. Numerical experiments validate the theoretical analysis.

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part II

A stabilizer free weak Galerkin (WG) finite element method on polytopal mesh has been introduced in Part I of this paper (Ye and Zhang (2020)). Removing stabilizers from discontinuous finite element methods simplifies formulations and reduces programming complexity. The purpose of this paper is to introduce a new WG method without stabilizers on polytopal mesh that has convergence rates one order higher than optimal convergence rates. This method is the first WG method that achieves superconvergence on polytopal mesh. Numerical examples in 2D and 3D are presented verifying the theorem.

Weak Galerkin finite element methods with or without stabilizers

The purpose of this paper is to investigate the connections between the weak Galerkin (WG) methods with and without stabilizers. The choices of stabilizers directly affect the convergence rates of the corresponding WG methods in general. However, we observed that the convergence rates are independent of the choices of stabilizers for these WG elements with stabilizers being optional. In this paper, we will verify such phenomena theoretically as well as numerically.

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part III

A weak Galerkin (WG) finite element method without stabilizers was introduced in Ye and Zhang (2020) on polytopal mesh. Then it was improved in Ye and Zhang (2021) with order one superconvergence. The goal of this paper is to develop a new stabilizer free WG method on polytopal mesh. This method has convergence rates two orders higher than the optimal convergence rates for the corresponding WG solution in both an energy norm and the L2 norm. The numerical examples are tested for low and high order elements in two and three dimensional spaces.

SUPERCLOSENESS ANALYSIS OF STABILIZER FREE WEAK GALERKIN FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION EQUATIONS

<abstract> Recently, a stabilizer free weak Galerkin (SFWG) method is proposed in [<xref ref-type="bibr" rid="b14">14</xref>], which is easier to implement and more efficient. In this paper, we developed an SFWG scheme for solving the general second-order elliptic problem on triangular meshes in 2D. This new SFWG method will dramatically reduce the error between the $ L^{2} $-projection of the exact solution and the numerical solution. </abstract>

A posteriori error estimates for weak Galerkin methods for second order elliptic problems on polygonal meshes

In this paper, a posteriori error estimates for the Weak Galerkin finite element methods (WG-FEMs) for second order elliptic problems are derived in terms of an H1−equivalent energy norm. Corresponding estimators based on the helmholtz decomposition yield globally upper and locally lower bounds for the approximation errors of the WG-FEMs. Especially, the error analysis of our methods is proved to be valid for polygonal meshes (e.g., hybrid, polytopal non-convex meshes and those with hanging nodes) under general assumptions. In addition, the work can make adaptive WG-FEMs solving partial differential equations such as stokes equations and biharmonic equations on polygonal meshes possible. Finally, we verify the theoretical findings by a few numerical examples.

A family of interior-penalized weak Galerkin methods for second-order elliptic equations

Interior-penalized weak Galerkin (IPWG) finite element methods are proposed and analyzed for solving second order elliptic equations. The new methods employ the element $(\mathbb{P}_{k}, \mathbb{P}_{k}, \mathcal{RT}_{k})$, with dimensions of space $d = 2, 3$, and the optimal a priori error estimates in discrete $H^1$-norm and $L^2$-norm are established. Moreover, provided enough smoothness of the exact solution, superconvergence in $H^1$ and $L^2$ norms can be derived. Some numerical experiments are presented to demonstrate flexibility, effectiveness and reliability of the IPWG methods. In the experiments, the convergence rates of the IPWG methods are optimal in $L^2$-norm, while they are suboptimal for NIPG and IIPG if the polynomial degree is even.

A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction

The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous approximations, which maintains simple finite element formulation. Stabilizer free weak Galerkin methods further simplify the WG methods and reduce computational complexity. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the stabilizer free WG schemes without compromising the accuracy of the numerical approximation. A new stabilizer free weak Galerkin finite element method is proposed and analyzed with polynomial degree reduction. To achieve such a goal, a new definition of weak gradient is introduced. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete \begin{document}$ H^1 $\end{document} norm and the standard \begin{document}$ L^2 $\end{document} norm. The numerical examples are tested on various meshes and confirm the theory.

A Least Square Based Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form

This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator. Naturally, the resulting linear system is symmetric and positive definite, and thus the algorithm is easy to implement and analyze. Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh. A superconvergence result is proved when the coefficient matrix is constant or piecewise constant. Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.