Weak Galerkin finite element methods for parabolic equations. (English) Zbl 1307.65133
The authors propose a new weak Galerkin method for solving parabolic equations. The method allows the usage of totally discontinuous functions in approximation spaces and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and optimal-order error estimates in both \(H^1\) and \(L^2\) norms are obtained. The method is then tested by a couple of examples.
Reviewer: Weizhong Dai (Ruston)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
finite element methods; parabolic equations; weak Galerkin methods; numerical examples; error estimateReferences:
| [1] | V.Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational mathematics, Springer‐Verlag, New York, Inc., Secaucus, NJ, 2006. · Zbl 1105.65102 |
| [2] | K.Eriksson and C.Johnson, Adaptive finite element methods for parabolic problems IV: nonlinear problems, SIAM J Numer Anal32 ( 1995), 1729-1749. · Zbl 0835.65116 |
| [3] | S.Larsson, V.Thomée, and L. B.Wahlbin, Numerical solution of parabolic integrodifferential equations by the discontinuous Galerkin method, Math Comp67 ( 1998), 45-71. · Zbl 0896.65090 |
| [4] | X.Makridakii and I.Babus̆ka, On the stability of the discontinuous Galerkin method for the heat equation, SIAM J Num Anal34 ( 1997), 389-401. · Zbl 0873.65095 |
| [5] | P.Chatzipantelidis, R. D.Lazarov, and V.Thomée, Error estimates for a finite volume element method for parabolic equations in convex polygonal domains, Numer Methods Partial Differential Equations20 ( 2004), 650-674. · Zbl 1067.65092 |
| [6] | M.Yang and J.Liu, A quadratic finite volume element method for parabolic problems on quadrilateral meshes, IMA J Numer Anal31 ( 2011), 1038-1061. · Zbl 1241.65074 |
| [7] | J.Wang and X.Ye, A weak Galerkin finite element method for elliptic problems, J Comput Appl Math241 ( 2013), 103-115. · Zbl 1261.65121 |
| [8] | P.Raviart and J.Thomas, A mixed finite element method for second order elliptic problems, Mathematical Aspects of the Finite Element Method, I.Galligani and E.Magenes, editors, Lecture Notes in Math 606, Springer‐Verlag, New York, 1977. |
| [9] | J.Wang and X.Ye, A weak Galerkin mixed finite element method for second‐order elliptic problems, arXiv:1202.3655v2, 2012. |
| [10] | L.Mu, J.Wang, and X.Ye, Weak Galerkin finite element methods on polytopal meshes, arXiv:1204.3655v2, 2012. |
| [11] | J.Wang, Y.Wang, and X.Ye, Unified a posteriori error estimator for finite element methods for the Stokes equations, Int J Numer Anal Modeling, to appear. · Zbl 1452.76059 |
| [12] | L.Mu, J.Wang, and X.Ye, A new Galerkin finite element method for the Stokes equations, Preprint and private communication. |
| [13] | M. F.Wheeler, A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J Numer Anal10 ( 1973), 723-759. · Zbl 0232.35060 |
| [14] | L.Mu, J.Wang, Y.Wang, and X.Ye, A computational study of the weak Galerkin method for second‐order elliptic equations, Numer Algor ( 2012), DOI: 10.1007/s11075‐012‐9651‐1. · Zbl 1271.65140 |
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.
