A WEAK GALERKIN FINITE ELEMENT METHOD FOR THE SECOND ORDER ELLIPTIC PROBLEMS WITH MIXED BOUNDARY CONDITIONS

Saqib Hussain; Nolisa Malluwawadu; Peng Zhu

doi:10.11948/2018.1452

2018 Volume 8 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(5): 1452-1463

Next Article Previous Article

Saqib Hussain, Nolisa Malluwawadu, Peng Zhu. A WEAK GALERKIN FINITE ELEMENT METHOD FOR THE SECOND ORDER ELLIPTIC PROBLEMS WITH MIXED BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1452-1463. doi: 10.11948/2018.1452

Citation:

Saqib Hussain, Nolisa Malluwawadu, Peng Zhu. A WEAK GALERKIN FINITE ELEMENT METHOD FOR THE SECOND ORDER ELLIPTIC PROBLEMS WITH MIXED BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(5): 1452-1463. doi: 10.11948/2018.1452

A WEAK GALERKIN FINITE ELEMENT METHOD FOR THE SECOND ORDER ELLIPTIC PROBLEMS WITH MIXED BOUNDARY CONDITIONS

1 Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA;
2 College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, China

Fund Project:

Abstract

Abstract

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 H¹ norm and the standard L² norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.
- Galerkin finite element methods /
- discrete gradient /
- second-order elliptic problems /
- mixed boundary conditions
MSC: 65N15;65N30;35J50

FullText(HTML)

References

[1]	D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM Journal on Numerical Analysis, 1982, 19(4), 742-760. Google Scholar
[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. Google Scholar
[3]	C. Athanasiadis and I. G. Stratis, On some elliptic transmission problems, in Annales Polonici Mathematici, 1996, 63, 137-154. Google Scholar
[4]	G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Mathematics of Computation, 1977, 31(137), 45-59. Google Scholar
[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. Google Scholar
[6]	S. Brenner and R. Scott, The Mathematical Theory Of Finite Element Methods, Springer Science & Business Media, 2007. Google Scholar
[7]	P. G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, 2002, 40, 1-511. Google Scholar
[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. Google Scholar
[9]	L. Mu, J. Wang, Y. Wang and X. Ye, A Weak Galerkin Mixed Finite Element Method For Biharmonic Equations, Springer, 2013. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	L. Mu, J. Wang, X. Ye and S. Zhang, A C^0-weak Galerkin finite element method for the biharmonic equation, Journal of Scientific Computing, 2014, 59(2), 473-495. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	B. Rivière, 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. Google Scholar
[17]	B. Rivière, 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	R. Zhang and Q. Zhai, A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order, Journal of Scientific Computing, 2015, 64(2), 559-585. Google Scholar