### Electronic Research Archive

2021, Issue 6: 3609-3627. doi: 10.3934/era.2021053
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# A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh

• Received: 01 November 2020 Revised: 01 June 2021 Published: 22 July 2021
• Primary: 65N15, 65N30, 76D07; Secondary: 35B45, 35J50

• A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than the optimal-order for velocity of the WG approximation is proved in both an energy norm and the $L^2$ norm. Optimal order error estimate for pressure in the $L^2$ norm is also established. The numerical examples cover low and high order approximations, and 2D and 3D cases.

Citation: Xiu Ye, Shangyou Zhang. A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh[J]. Electronic Research Archive, 2021, 29(6): 3609-3627. doi: 10.3934/era.2021053

### Related Papers:

• A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than the optimal-order for velocity of the WG approximation is proved in both an energy norm and the $L^2$ norm. Optimal order error estimate for pressure in the $L^2$ norm is also established. The numerical examples cover low and high order approximations, and 2D and 3D cases.

 [1] A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method. Appl. Numer. Math. (2020) 150: 444-451. [2] The lowest-order stabilizer free weak Galerkin finite element method. Appl. Numer. Math. (2020) 157: 434-445. [3] Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions. Comput. Math. Appl. (2019) 78: 905-928. [4] Superconvergence of the gradient approximation for weak Galerkin finite element methods on rectangular partitions. Appl. Numer. Math. (2020) 150: 396-417. [5] New error estimates of nonconforming mixed finite element methods for the Stokes problem. Math. Methods Appl. Sci. (2014) 37: 937-951. [6] Lowest-order weak Galerkin finite element method for Darcy flow on convex polygonal meshes. SIAM J. Sci. Comput. (2018) 40: 1229-1252. [7] L. Mu, Pressure robust weak Galerkin finite element methods for Stokes problems, SIAM J. Sci. Comput., 42 (2020), B608-B629. doi: 10.1137/19M1266320 [8] Stability and approximability of the P1-P0 element for Stokes equations. Internat. J. Numer. Methods Fluids (2007) 54: 497-515. [9] Stability of the finite elements $9/(4c+1)$ and $9/5c$ for stationary Stokes equations. Comput. $&$ Structures (2005) 84: 70-77. [10] A Weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comp. (2014) 83: 2101-2126. [11] X. Ye and S. Zhang, A stabilizer-free weak Galerkin finite element method on polytopal meshes, J. Comput. Appl. Math, 371 (2020), 112699. arXiv: 1906.06634. doi: 10.1016/j.cam.2019.112699 [12] X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method: Part Ⅱ, Int. J. Numer. Anal. Model., 17 (2020), 110-117. arXiv: 1904.03331. [13] X. Ye and S. Zhang, A stabilizer free weak Galerkin finite element method on polytopal mesh: Part Ⅱ, J. Comput. Appl. Math., 394 (2021), 113525, 11 pp. arXiv: 2008.13631. doi: 10.1016/j.cam.2021.113525 [14] X. Ye and S. Zhang, A stabilizer free weak Galerkin finite element method on polytopal mesh: Part Ⅲ, J. Comput. Appl. Math., 394 (2021), 113538, 9 pp. arXiv: 2009.08536. doi: 10.1016/j.cam.2021.113538 [15] X. Ye and S. Zhang, A stabilizer-free pressure-robust finite element method for the Stokes equations, Adv. Comput. Math., 47 (2021), Paper No. 28, 17 pp. doi: 10.1007/s10444-021-09856-9 [16] A 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations. Int. J. Numer. Anal. Model. (2017) 14: 730-743. [17] A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comp. (2005) 74: 543-554. [18] On the P1 Powell-Sabin divergence-free finite element for the Stokes equations. J. Comput. Math. (2008) 26: 456-470. [19] Divergence-free finite elements on tetrahedral grids for $k\geq 6$. Math. Comp. (2011) 80: 669-695. [20] Quadratic divergence-free finite elements on Powell-Sabin tetrahedral grids. Calcolo (2011) 48: 211-244. [21] $C_0P_2$-$P_0$ Stokes finite element pair on sub-hexahedron tetrahedral grids. Calcolo (2017) 54: 1403-1417.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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