A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions. (English) Zbl 1453.65408
Summary: In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete \(H^1\) norm and the standard \(L^2\) norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35J50 | Variational methods for elliptic systems |
Keywords:
Galerkin finite element methods; discrete gradient; second-order elliptic problems; mixed boundary conditionsReferences:
| [1] | D. N. Arnold,An interior penalty finite element method with discontinuous elements, SIAM Journal on Numerical Analysis, 1982, 19(4), 742-760. · Zbl 0482.65060 |
| [2] | D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini,Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 2002, 39(5), 1749-1779. · Zbl 1008.65080 |
| [3] | C. Athanasiadis and I. G. Stratis,On some elliptic transmission problems, in Annales Polonici Mathematici, 1996, 63, 137-154. · Zbl 0848.35135 |
| [4] | G. A. Baker,Finite element methods for elliptic equations using nonconforming elements, Mathematics of Computation, 1977, 31(137), 45-59. · Zbl 0364.65085 |
| [5] | C. E. Baumann and J. T. Oden,A discontinuous hp finite element method for convectiondiffusion problems, Computer Methods in Applied Mechanics and Engineering, 1999, 175(3-4), 311-341. · Zbl 0924.76051 |
| [6] | S. Brenner and R. Scott,The Mathematical Theory Of Finite Element Methods, Springer Science & Business Media, 2007. |
| [7] | P. G. Ciarlet,The finite element method for elliptic problems, Classics in Applied Mathematics, 2002, 40, 1-511. · Zbl 0999.65129 |
| [8] | L. Mu, J. Wang, Y. Wang and X. Ye,A computational study of the weak Galerkin method for second-order elliptic equations, Numerical Algorithms, 2013, 63(4), 753-777. · Zbl 1271.65140 |
| [9] | L. Mu, J. Wang, Y. Wang and X. Ye,A Weak Galerkin Mixed Finite Element Method For Biharmonic Equations, Springer, 2013. · Zbl 1283.65109 |
| [10] | L. Mu, J. Wang and X. Ye,Weak Galerkin finite element methods on polytopal meshes, International Journal of Numerical Analysis & Modeling, 2012, 12(1). · Zbl 1332.65172 |
| [11] | L. Mu, J. Wang and X. Ye,Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, Numerical Methods for Partial Differential Equations, 2014, 30(3), 1003-1029. · Zbl 1314.65151 |
| [12] | L. Mu, J. Wang and X. Ye,A weak Galerkin finite element method with polynomial reduction, Journal of Computational and Applied Mathematics, 2015, 285, 45-58. · Zbl 1315.65099 |
| [13] | L. Mu, J. Wang, X. Ye and S. Zhang,A Cˆ0-weak Galerkin finite element method for the biharmonic equation, Journal of Scientific Computing, 2014, 59(2), 473-495. · Zbl 1305.65233 |
| [14] | L. Mu, J. Wang, X. Ye and S. Zhang,A weak Galerkin finite element method for the Maxwell equations, Journal of Scientific Computing, 2015, 65(1), 363-386. · Zbl 1327.65220 |
| [15] | L. Mu, J. Wang, X. Ye and S. Zhao,A numerical study on the weak Galerkin method for the helmholtz equation with large wave numbers, arXiv preprint arXiv:1111.0671, 2011. |
| [16] | B. Rivi‘ere, M. F. Wheeler and V. Girault,Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. part i, Computational Geosciences, 1999, 3(3-4), 337-360. · Zbl 0951.65108 |
| [17] | B. Rivi‘ere, M. F. Wheeler and V. Girault,A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM Journal on Numerical Analysis, 2001, 39(3), 902-931. · Zbl 1010.65045 |
| [18] | J. Wang and X. Ye,A weak Galerkin finite element method for second-order elliptic problems, Journal of Computational and Applied Mathematics, 2013, 241(1), 103-115. · Zbl 1261.65121 |
| [19] | J. Wang and X. Ye,A weak Galerkin finite element method for the stokes equations, Advances in Computational Mathematics, 2016, 42(1), 155-174. · Zbl 1382.76178 |
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.
