×

A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes. (English) Zbl 1452.65362

Summary: A new stabilizer free weak Galerkin (WG) method is introduced and analyzed for the biharmonic equation. Stabilizing/penalty terms are often necessary in the finite element formulations with discontinuous approximations to ensure the stability of the methods. Removal of stabilizers will simplify finite element formulations and will reduce programming complexity. This stabilizer free WG method has an ultra simple formulation and can work on general partitions with polygons/polyhedra. Optimal order error estimates in a discrete \(H^2\) norm for \(k\ge 2\) and in an \(L^2\) norm for \(k>2\) are established for the corresponding weak Galerkin finite element solutions, where \(k\) is the degree of the polynomial in the approximation. Numerical results are provided to confirm the theories.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J35 Variational methods for higher-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
35B45 A priori estimates in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci., 2 (1980), pp. 556-581. · Zbl 0445.35023
[2] L. Morley, The triangular equilibrium element in the solution of plate bending problems, Aero. Quart., 19 (1968), pp. 149-169.
[3] L. Mu, J. Wang, and X. Ye, A weak Galerkin finite element method for biharmonic equations on polytopal meshes, Numer. Methods Partial Differential Equations, 30 (2014), pp. 1003-1029. · Zbl 1314.65151
[4] L. Mu, J. Wang, X. Ye, and S. Zhang, \(C^0\) Weak Galerkin finite element methods for the biharmonic equation, J. Sci. Comput., 59 (2014), pp. 437-495. · Zbl 1305.65233
[5] L. Mu, X. Ye, and S. Zhang, Development of a \(P_2\) element with optimal \(L^2\) convergence for biharmonic equation, Numer. Methods Partial Differential Equations, 21 (2019), pp. 1497-1508. · Zbl 1418.65181
[6] C. Wang and J. Wang, An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes, Comput. Math. Appl., 68, (2014), pp. 2314-2330. · Zbl 1361.35058
[7] X. Ye, S. Zhang, and Z. Zhang, A new \(P_1\) weak Galerkin method for the biharmonic equation, J. Comput. Appl. Math., 364 (2020), 112337, https://doi.org/10.1016/j.cam.2019.07.002. · Zbl 1427.65342
[8] R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order, J. Sci. Comput., 64 (2015), pp. 559-585. · Zbl 1331.65163
[9] C. Wang and H. Zhou, A weak Galerkin finite element method for a type of fourth order problem arising from fluorescence tomography, J. Sci. Comput., 71 (2017), pp. 897-918. · Zbl 1375.65156
[10] J. Wang and X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., 83 (2014), pp. 2101-2126. · Zbl 1308.65202
[11] J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comp. Appl. Math., 241 (2013), pp. 103-115. · Zbl 1261.65121
[12] X. Ye and S. Zhang, A stabilizer-free weak Galerkin finite element method on polytopal meshes, J. Comput. Appl. Math., 372 (2020), 112699. · Zbl 1434.65285
[13] X. Ye, S. Zhang, and Y. Zhu, Stabilizer-free weak Galerkin methods for monotone quasilinear elliptic PDEs, Results Appl. Math., 8 (2020), 100097, https://doi.org/10.1016/j.rinam.2020.100097. · Zbl 1451.65207
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.