Weak Galerkin Finite Element Method Research Articles

A globally divergence-free weak Galerkin finite element method with IMEX-SAV scheme for the Kelvin–Voigt viscoelastic fluid flow model with high Reynolds number

In this manuscript, we present a globally divergence-free weak Galerkin finite element method with the IMEX-SAV scheme for solving the Kelvin–Voigt viscoelastic fluid flow model. Firstly, the weak Galerkin finite element method is used to analyze the spatial discretization, this method can yield globally divergence-free velocity solutions. Secondly, the IMEX-SAV difference scheme is adopted in the temporal discretization for the fully discrete scheme, and we can obtain the unconditional stability result for the velocity. In addition, combined with the stabilization idea, the method can not only reduce the spurious numerical oscillation at the high Reynolds number but also yield the uniform results that the constants in the error estimates do not explicitly depend on the inverse power of the physical parameters of the Kelvin–Voigt viscoelastic fluid flow model. Finally, several numerical experiments are performed to verify the theoretical results.

The energy-diminishing weak Galerkin finite element method for the computation of ground state and excited states in Bose-Einstein condensates

In this paper, we employ the weak Galerkin (WG) finite element method and the imaginary time method to compute both the ground state and the excited states in Bose-Einstein condensate (BEC) which is governed by the Gross-Pitaevskii equation (GPE). First, we use the imaginary time method for GPE to get the nonlinear parabolic partial differential equation. Subsequently, we apply the WG method to spatially discretize the parabolic equation. This yields a semi-discrete scheme, in which an energy function is explicitly defined. For the case β⩾0, we demonstrate that the energy is diminishing with respect to time t at each time step. Applying the backward Euler scheme for temporal discretization yields a fully discrete scheme. For the case β=0, we provide a mathematical justification, establishing the convergence analysis for the numerical solution of the ground state. Moreover, based on the theory of solving eigenvalue problems using the WG method, we present the error estimates between the ground state and its numerical solution under the H1 and L2 norms. Numerical experiments are provided to illustrate the effectiveness of the proposed schemes. Moreover, the results indicate that our method also can compute the first excited state, achieving optimal convergence orders.

Anisotropic error analysis of weak Galerkin finite element method for singularly perturbed biharmonic problems

We consider the weak Galerkin finite element approximation of the singularly perturbed biharmonic elliptic problem on a unit square domain with clamped boundary conditions. Shishkin mesh is used for domain discretization as the solution exhibits boundary layers near the domain boundary. Error estimates in the equivalent H2− norm have been established and the uniform convergence of the proposed method has been proved. Numerical examples are presented corroborating our theoretical findings.

The weak Galerkin finite element method for Stokes interface problems with curved interface

In this paper, we develop a weak Galerkin (WG) finite element scheme for the Stokes interface problems with curved interface. The conventional numerical schemes rely on the use of straight segments to approximate the curved interface and the accuracy is limited by geometric errors. Hence in our method, we directly construct the weak Galerkin finite element space on the curved cells to avoid geometric errors. For the integral calculation on curved cells, we employ non-affine transformations to map curved cells onto the reference element. The optimal error estimates are obtained in both the energy norm and the L2 norm. A series of numerical experiments are provided to validate the efficiency of the proposed WG method.

Streamline Diffusion Weak Galerkin Finite Element Methods for Linear Unsteady State Convection Diffusion Equations and Error Analysis

In this paper, the streamline diffusion weak Galerkin finite element method is proposed and analyzed for solving unsteady time convection diffusion problem in two dimension. The v-elliptic property and the stability of this scheme are proved in terms of some conditions. We derive an error estimate in L2(μ) and H1(μ) norm. Numerical experiments have demonstrated the effectiveness of the method in solving convection propagation problems, and the theoretical analysis has been validated.

A weak Galerkin finite element method for fourth-order parabolic singularly perturbed problems on layer adapted Shishkin mesh

In this paper, we propose a weak Galerkin finite element approximation for a class of fourth-order singularly perturbed parabolic problems. The problem exhibits boundary layers and so we have considered layer adapted triangulations, in particular Shishkin triangular mesh in the spatial domain. For temporal discretization, we utilize the Crank-Nicolson scheme on a uniform mesh. Stability and error estimates along with the uniform convergence of the method has been proved. Numerical examples are included which verifies our analysis.

Interpolated coefficients stabilizer-free weak Galerkin method for semilinear parabolic convection–diffusion problem

We continue our effort in Li et al. (2024) to explore an interpolated coefficients stabilizer-free weak Galerkin finite element method (IC SFWG-FEM) to solve a one-dimensional semilinear parabolic convection–diffusion equation. Due to the introduction of interpolated coefficients and the design without stabilizers, this method not only possesses the capability of approximating functions and sparsity in the stiffness matrix, but also reduces the complexity of analysis and programming. Theoretical analysis of stability for the semi-discrete IC SFWG finite element scheme is provided. Moreover, numerical experiments are carried out to demonstrate the effectivity and stability.

Generalized weak Galerkin finite element method for linear elasticity interface problems

Generalized weak Galerkin finite element method for linear elasticity interface problems

Error Analysis of the Classical Artificial Diffusion Weak Galerkin Finite Element Method for the steady state- convection diffusion-reaction Equation in 2-D

In this study, we modified the error in the weak Galerkin method when solving problems in which diffusion is the dominant convection( h) in two dimensions. This is done by adding the artificial diffusion term (-δΔw, where δ=h-ϵ). The finite element method for discrete functions using a weakly defined gradient operator is presented in this study. The concept of the weak discrete gradient is introduced, which plays an important role when using numerical methods to solve partial differential equations. The goal of this study is to enhance the accuracy and stability of the solutions by studying the ellipticity and stability properties of the method, which works to ensure that the numerical method retains the properties of the original equation while reducing the fluctuations occurring with the weak galerkin finite element method. Specific theories have been used to estimate the error in parameters -norm, and . Practical examples demonstrate how this method improves the handling of partial equations characterized by convection-dominated diffusion, enhancing its potential for advancing numerical simulations in engineering and physics.

A defect correction weak Galerkin finite element method for the Kelvin–Voigt viscoelastic fluid flow model

In this paper, we present a defect correction weak Galerkin finite element method for solving the Kelvin–Voigt viscoelastic fluid flow model, which can guarantee that the discrete velocity is globally divergence-free. The second-order Crank–Nicolson difference scheme is considered for temporal discretization. In particular, combined with the defect correction method, the numerical oscillation effect is reduced at high Reynolds number. Our algorithm not only gives the stability and convergence of the numerical solutions of velocity and pressure but also is uniformly valid at the retardation time κ→0. Several numerical experiments are performed to verify the method has good behaviors.

An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order O(N−(k−1/2)) on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order O((N−1ln⁡N)(k−1/2)) on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

Optimal [formula omitted] error estimates of stabilizer-free weak Galerkin finite element method for the drift-diffusion problem

We continue our effort in Li et al. (2024) to develop a stabilizer-free weak Galerkin finite element method (SFWG-FEM) for the two-dimensional unsteady-state drift-diffusion (DD) model. The transient state equation includes both the first derivative convection and the second derivative diffusion terms, and it couples with a Poisson potential equation. The piecewise Pk(k≥1) polynomials elements in the interior and on the boundaries are utilized for both electron concentration and electric potential functions. The vector function of [Pj(K)]2(j≥j0) for the discrete weak gradient space is used on each element K. The main difficulty is the treatment of the non-linearity and coupling of the system. Based on an introduced Ritz projection and two L2 projection operators, optimal error estimates of O(Δt+hk+1) in the L2 norm and O(Δt+hk) in the energy-like norm are derived. Numerical experiments are presented to illustrate the theoretical analysis.

Weak Galerkin finite element methods for semilinear Klein–Gordon equation on polygonal meshes

Weak Galerkin finite element methods for semilinear Klein–Gordon equation on polygonal meshes

Weak Galerkin finite element methods for optimal control problems governed by second order elliptic equations

This paper is concerned with the development of weak Galerkin (WG) finite element methods for optimal control problems governed by second order elliptic partial differential equations. WG methods are advantageous over other existing methods in that the control variable along the domain boundary is a natural variable by design in the numerical scheme. A convergence theory was established for the numerical solutions in various Sobolev norms. Numerical results are presented to demonstrate the efficiency and accuracy of the new scheme.

A modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes

We introduce a new numerical method for solving time-harmonic Maxwell’s equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite elements are replaced by the average of the two discontinuous polynomial functions on the two sides of the polygon, in the modified weak Galerkin (MWG) finite element method. With the dependent inter-element functions, the weak curl and the weak gradient are defined directly on totally discontinuous polynomials. Optimal-order convergence of the method is proved. Numerical examples confirm the theory and show effectiveness of the modified weak Galerkin method over the existing methods.

Analysis and Computation of a Weak Galerkin Scheme for Solving the 2D/3D Stationary Stokes Interface Problems with High-Order Elements

Abstract In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ d (d = 2, 3). This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ≥ 1 for the velocity and polynomials of degree k – 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L 2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.

A posteriori error estimate of the weak Galerkin finite element method solving the Stokes problems on polytopal meshes

AbstractIn this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG‐FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG‐FEM. Error analysis is proved to be valid under the mesh assumptions of the WG‐FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples.

A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖uN−INu‖WG is presented, where INu denotes the vertex-edge-cell interpolation of the solution u, and ‖⋅‖WG denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method.

The lowest-order weak Galerkin finite element method for linear elasticity problems on convex polygonal grids

This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of CW0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H1 and L2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property.