Simplified weak Galerkin and new finite difference schemes for the Stokes equation

Liu, Yujie; Wang, Junping

doi:10.1016/j.cam.2019.04.024

Cited by 37 publications

(27 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Table 8. 23 shows that the convergence rate for ∇ d e b 0 is of order O(h 1.9 ) which outperforms the result O(h 1.5 ) in Theorem 7.4. Table 8.23 Test Case 8 : Convergence of the lowest order WG-FEM on the unit cubic domain with exact solution u = sin(x) sin(y) sin(z), non-uniform cubic partitions, stabilization parameter ρ = 1, h = max(|ex|, |ey|, |ez|), and L 2 projection of the Dirichlet boundary data g. The coefficient matrix is a 11 = 1 + x 2 , a 12 = xy/4, a 13 = xz/4, a 22 = 1 + y 2 , a 23 = yz/4, and a 33 = 1 + z 2 .…”

Section: Numerical Experiments For Piecewise Constant Diffusion Tensomentioning

confidence: 67%

Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

Nie

Wang

2019

Computers & Mathematics with Applications

View full text Add to dashboard Cite

A superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak Galerkin finite element scheme on the uniform and non-uniform cubic partitions. Due to the loss of the symmetric property from two dimensions to three dimensions, this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions [22] from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the L 2 -norm arrives at a superconvergence order of O(h r )(1.5 ≤ r ≤ 2) when the lowest order weak Galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed. A series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions.

show abstract

Section: Numerical Experiments For Piecewise Constant Diffusion Tensomentioning

confidence: 67%

Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

Nie

Wang

2019

Computers & Mathematics with Applications

View full text Add to dashboard Cite

show abstract

“…According to (24), (52), the random initial condition (20), and the regularity condition of f, we complete the proof.…”

Section: Semidiscrete Error Analysis For Simplicity We Introduce Twmentioning

confidence: 73%

“…en, the WG method is successfully applied to elliptic interface problems [11,12] and linear parabolic problems [13][14][15][16] and further developed for other applications, such as biharmonic problems [17][18][19][20][21][22], Cahn-Hilliard problems [23], and Stokes problems [24][25][26][27][28][29]. To ensure the method is highly flexible in element construction and mesh generation, the idea of a stabilization term is introduced in [30], which allows for arbitrary piecewise polynomial shape functions in deformed and honeycomb meshes.…”

Section: Introductionmentioning

confidence: 99%

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

Zhou

Zou

Chai

et al. 2020

Mathematical Problems in Engineering

View full text Add to dashboard Cite

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

show abstract

“…In fact,

s (v_{b})

can be viewed as an extension of v b from ∂T to T through a least‐squares fitting, and its computation is local and well‐defined. Readers are referred to for a detailed derivation.…”

Section: Swg On Polymeshmentioning

confidence: 99%

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

Liu

Wang

2019

Numerical Methods Partial

Self Cite

View full text Add to dashboard Cite

This article establishes a discrete maximum principle (DMP) for the approximate solution of convection–diffusion–reaction problems obtained from the weak Galerkin (WG) finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the WG involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin (SWG) method has a reduced computational complexity over the usual WG, and indeed provides a discretization scheme different from the WG when the reaction terms are present. An application of the SWG on uniform rectangular partitions yields some 5‐ and 7‐point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the DMP and the accuracy of the scheme, particularly the finite difference scheme.

show abstract

Simplified weak Galerkin and new finite difference schemes for the Stokes equation

Cited by 37 publications

References 37 publications

Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

Contact Info

Product

Resources

About