Weak Galerkin Finite Element Method Research Articles

An analysis of weak Galerkin finite element method for a steady state Boussinesq problem

In this article, we present and analyze a weak Galerkin finite element method (WG-FEM) for the coupled Navier–Stokes/temperature (or Boussinesq) problems. In this WG-FEM, discontinuous functions are applied to approximate the velocity, temperature, and the normal derivative of temperature on the boundary while piecewise constants are used to approximate the pressure. The stability, existence and uniqueness of solution of the associated WG-FEM are proved in detail. An optimal a priori error estimate is then derived for velocity in the discrete H1 and L2 norms, pressure in the L2 norm, temperature in the discrete H1 and L2 norms, and the normal derivative of temperature in H−1/2 norm. Finally, to complete this study some numerical tests are presented which illustrate that the numerical errors are consistent with theoretical results.

A reduced-order Weak Galerkin finite element algorithm based on POD technique for parabolic problem on polytopal mesh

Weak Galerkin (WG method) is a numerical method based on the weak functions and weak partial derivatives, which has been widely developed in recent decades. In the implementation, different degrees of freedom (DOFs) are distributed in the element interior and boundary respectively, which leads to a great impact on the computational complexity. In our work, we propose a new reduced-order weak galerkin finite element (RO-WG) method with seldom DOFs, where the proper orthogonal decomposition (POD) technique is adopted to save CPU time. The parabolic equation is considered for a test problem. Some numerical examples are given to prove the superiority of this method, where the CPU time can be reduced a lot.

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.

A recovery-based a posteriori error estimator of the weak Galerkin finite element method for elliptic problems

In this paper, we propose a recovery-type a posteriori error estimator of the weak Galerkin finite element method for the second order elliptic equation. The reliability and efficiency of the estimator are analyzed by a discrete H1-norm of the exact error. The estimator is further used in the adaptive weak Galerkin algorithm on the triangular, quadrilateral and other polygonal meshes. Numerical results are provided to demonstrate the effectiveness of the adaptive mesh refinement guided by this estimator.

A modified weak Galerkin finite element method for nonmonotone quasilinear elliptic problems

A modified weak Galerkin finite element method is studied for nonmonotone quasilinear elliptic problems. Using the contraction mapping theorem, the uniqueness of the solution to the discrete problem is proved. Moreover, optimal order a priori error estimates are established in both a discrete H1 norm and the L2 norm. Numerical experiments are conducted to confirm the theoretical results.

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.

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.

A priori and a posteriori error estimates of the weak Galerkin finite element method for parabolic problems

We derive the priori and a posteriori error estimates of the weak Galerkin finite element method with the Crank-Nicolson time discretization for the parabolic equation in this paper. The priori error estimates are deduced based on existing priori error results of the corresponding elliptic projection problem. For the a posteriori error estimates, the elliptic reconstruction technique is introduced to decompose the true error into elliptic error and parabolic error. Then the elliptic part is bounded by the a posteriori error estimates of the auxiliary elliptic reconstruction problem. The a posteriori error estimator is further used to develop the temporal and spatial adaptive algorithm. Numerical results in the uniform and adaptive meshes are provided to validate the proposed estimators.

A locking-free P0 finite element method for linear elasticity equations on polytopal partitions

Abstract This article presents a $P_0$ finite element method for boundary value problems for linear elasticity equations. The new method makes use of piecewise constant approximating functions on the boundary of each polytopal element and is devised by simplifying and modifying the weak Galerkin finite element method based on $P_1/P_0$ approximations for the displacement. This new scheme includes a tangential stability term on top of the simplified weak Galerkin to ensure the necessary stability due to the rigid motion. The new method involves a small number of unknowns on each element, it is user friendly in computer implementation and the element stiffness matrix can be easily computed for general polytopal elements. The numerical method is of second-order accurate, locking-free in the nearly incompressible limit, and ease polytopal partitions in practical computation. Error estimates in $H^1$, $L^2$ and some negative norms are established for the corresponding numerical displacement. Numerical results are reported for several two-dimensional and three-dimensional test problems, including the classical benchmark Cook’s membrane problem in two dimensions as well as some three-dimensional problems involving shear-loaded phenomena. The numerical results show clearly the simplicity, stability, accuracy and efficiency of the new method.

Application of weak Galerkin finite element method for nonlinear chemotaxis and haptotaxis models

In this paper, we introduce a weak Galerkin finite element method for 2D Keller-Segel chemotaxis models and closely related haptotaxis models to simulate tumor invasion into surrounding healthy tissue. The chemotaxis and haptotaxis models consist of a system of time-dependent nonlinear reaction-diffusion-taxis partial differential equations. It is well known that solutions of chemotaxis systems may blow-up in finite time in the cell density. This blow-up is a mathematical description of a cell concentration phenomenon, which mathematically results in rapidly growing solutions in small neighborhoods of concentration points or curves. Capturing such singular solutions numerically is a challenging problem. We applied the weak Galerkin finite element method (WGFEM) to approximate the spatial variables. Weak Galerkin finite element method can be considered as an extension of the standard finite element method where classical derivatives are replaced in the variational equation by the weakly defined derivatives on discontinuous weak functions. The weak derivatives of weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the WGFEM is impacted by the selection of such polynomials. We show that the proposed method produced numerical results that are nonnegative, oscillation-free and suitable for solving models that have a blow-up of the cell density. Error bounds for cell density and chemoattractant equation are obtained. The proposed method is applied to several Keller-Segel chemotaxis models and extended to the haptotaxis model for simulating the behavior of cancer cell invasion of surrounding healthy tissue. Numerical results demonstrate the accuracy and efficiency of the proposed scheme.

$(1+s)$-Order Convergence Analysis of Weak Galerkin Finite Element Methods for Second Order Elliptic Equations

$(1+s)$-Order Convergence Analysis of Weak Galerkin Finite Element Methods for Second Order Elliptic Equations

Low Regularity Error Analysis for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

Low Regularity Error Analysis for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

A weak Galerkin finite element method based on [formula omitted] virtual element for Darcy flow on polytopal meshes

This paper presents a weak Galerkin finite element method based on H(div) virtual element for Darcy flow on polytopal meshes. Specifically, the pressure is approximated by constant functions on both the polytopal elements and the element sides, respectively. Their discrete weak gradients are calculated in the local H(div) virtual element spaces whose basis functions never need to be explicitly computed. The method works on very general polytopal meshes. After obtaining the numerical pressure, the numerical velocity that is locally mass-conservative and has continuous normal fluxes is easily obtained by a simple L2 projection. The error estimates for the numerical pressure and numerical velocity are proved. The assembly of element stiffness matrix is introduced. Numerical experiments in two and three dimensions are reported to illustrate the good performance of the method.

A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients

In this paper, a weak Galerkin finite element method for solving the time fractional reaction-convection diffusion problem is proposed. We use the well known L1 discretization in time and a weak Galerkin finite element method on uniform mesh in space. Both continuous and discrete time weak Galerkin finite element method are considered and analyzed. The stability of the discrete time scheme is proved. The error estimates for both schemes are given. Finally, we give some numerical experiments to show the efficiency of the proposed method.

De Rham complexes for weak Galerkin finite element spaces

Two de Rham complex sequences of the finite element spaces are introduced for weak finite element functions and weak derivatives developed in the weak Galerkin (WG) finite element methods on general polyhedral elements. One of the sequences uses polynomials of equal order for all the finite element spaces involved in the sequence and the other one uses polynomials of naturally descending orders. It is shown that the diagrams in both de Rham complexes commute for general polyhedral elements. The exactness of one of the complexes is established for the lowest order element.

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.

Low regularity primal–dual weak Galerkin finite element methods for convection–diffusion equations

We consider finite element discretizations for convection–diffusion problems under low regularity assumptions. The derivation and analysis use the primal–dual weak Galerkin (PDWG) finite element framework. The Euler–Lagrange formulation resulting from the PDWG scheme yields a system of equations involving not only the equation for the primal variable but also its adjoint for the dual variable. We show that the proposed PDWG method is stable and convergent. We also derive a priori error estimates for the primal variable in the Hϵ-norm for ϵ∈[0,12). A series of numerical tests that validate the theory is presented as well.