A conforming discontinuous Galerkin finite element method on rectangular partitions

Yue Feng; Yujie Liu; Ruishu Wang; Shangyou Zhang; Yue Feng; Yujie Liu; Ruishu Wang; Shangyou Zhang

doi:10.3934/era.2020120

Electronic Research Archive

2021, Volume 29, Issue 3: 2375-2389. doi: 10.3934/era.2020120

Previous Article Next Article

Special Issues

A conforming discontinuous Galerkin finite element method on rectangular partitions

Yue Feng ^{1
,},
Yujie Liu ^{2
,},
Ruishu Wang ^{1
,
,},
Shangyou Zhang ^{3
,}

1.
Department of Mathematics, Jilin University, Changchun, China
2.
Artificial Intelligence Research Center, Peng Cheng Laboratory, Shenzhen 518005, China
3.
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

Received: 01 December 2019 Revised: 01 September 2020 Published: 26 November 2020
Primary:65N15, 65N30;Secondary:35B45, 35J50

This article presents a conforming discontinuous Galerkin (conforming DG) scheme for second order elliptic equations on rectangular partitions. The new method is based on DG finite element space and uses a weak gradient arising from local Raviart Thomas space for gradient approximations. By using the weak gradient and enforcing inter-element continuity strongly, the scheme maintains the simple formulation of conforming finite element method while have the flexibility of using discontinuous approximations. Hence, the programming complexity of this new conforming DG scheme is significantly reduced compared to other existing DG methods. Error estimates of optimal order are established for the corresponding conforming DG approximations in various discrete Sobolev norms. Numerical results are presented to confirm the developed convergence theory.
- Conforming discontinuous Galerkin finite element method,
- second order elliptic problem,
- rectangular partitions,
- error estimates
Citation: Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions[J]. Electronic Research Archive, 2021, 29(3): 2375-2389. doi: 10.3934/era.2020120

Related Papers:

Abstract

This article presents a conforming discontinuous Galerkin (conforming DG) scheme for second order elliptic equations on rectangular partitions. The new method is based on DG finite element space and uses a weak gradient arising from local Raviart Thomas space for gradient approximations. By using the weak gradient and enforcing inter-element continuity strongly, the scheme maintains the simple formulation of conforming finite element method while have the flexibility of using discontinuous approximations. Hence, the programming complexity of this new conforming DG scheme is significantly reduced compared to other existing DG methods. Error estimates of optimal order are established for the corresponding conforming DG approximations in various discrete Sobolev norms. Numerical results are presented to confirm the developed convergence theory.

References

[1]	Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. (2001/02) 39: 1749-1779.
[2]	The finite element method with penalty. Math. Comp. (1973) 27: 221-228.
[3]	Finite element methods for elliptic equations using nonconforming elements. Math. Comp. (1977) 31: 45-59.
[4]	Basic principles of virtual element methods. Math. Models Methods Appl. Sci. (2013) 23: 199-214.
[5]	On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO (1974) 8: 129-151.
[6]	A finite volume method based on the Crouzeix-Raviart element for elliptic PDE's in two dimensions. Numer. Math. (1999) 82: 409-432.
[7]	An HDG method for distributed control of convection diffusion PDEs. J. Comput. Appl. Math. (2018) 343: 643-661.
[8]	Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations. J. Sci. Comput. (2014) 59: 129-157.
[9]	Interior penalty discontinuous Galerkin methods for second order linear non-divergence form elliptic PDEs. J. Sci. Comput. (2018) 74: 1651-1676.
[10]	The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes. J. Comput. Phys. (2011) 230: 305-328.
[11]	Simplified weak Galerkin and new finite difference schemes for the Stokes equation. J. Comput. Appl. Math. (2019) 361: 176-206.
[12]	Y. Liu and J. Wang, A locking-free $ P_0 $ finite element method for linear elasticity equations on polytopal partitions, preprint, arXiv: 1911.08728, 2019.
[13]	L. Mu, J. Wang, Y. Wang and X. Ye, A weak Galerkin mixed finite element method for biharmonic equations, Numerical Solution of Partial Differential Equations: Theory, Algorithms, and their Applications, in: Springer Proceedings in Mathematics and Statistics, 45 (2013), 247-277. doi: 10.1007/978-1-4614-7172-1_13
[14]	A \begin{document}$C^0$\end{document}-weak Galerkin finite element method for the biharmonic equation. J. Sci. Comput. (2014) 59: 473-495.
[15]	P.-A. Raviart and J. M. Thomas, A Mixed Finite Element Method for Second Order Elliptic Problems, in: I. Galligani, E. Magenes (Eds.), Mathematical Aspects of the Finite Element Method, in: Lecture Notes in Math., vol. 606, Springer-Verlag, New York, 1977. Technical Report LA-UR-73-0479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.
[16]	M. Stynes, Sharp anisotropic interpolation error estimates for rectangular Raviart-Thomas elements, Math. Comp., 83 (2014), 2675-2689. doi: 10.1090/S0025-5718-2014-02826-3
[17]	A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation. J. Comput. Appl. Math. (2016) 307: 346-366.
[18]	A weak Galerkin finite element scheme for solving the stationary Stokes equations. J. Comput. Appl. Math. (2016) 302: 171-185.
[19]	A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. (2013) 241: 103-115.
[20]	A Weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comp. (2014) 83: 2101-2126.
[21]	X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method, Int. J. Numer. Anal. and Model., 17 (2020), 110-117. arXiv: 1904.03331.
[22]	X. Ye and S. Zhang, A conforming discontinuous Galerkin finite element method: Part Ⅱ, arXiv: 1907.01397.
[23]	Weak Galerkin finite element method for second order parabolic equations. Int. J. Numer. Anal. Model. (2016) 13: 525-544.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)