Weak Galerkin finite element methods for Sobolev equation. (English) Zbl 1357.65175
Summary: We present some numerical schemes based on the weak Galerkin finite element method for one class of Sobolev equations, in which differential operators are approximated by weak forms through the usual integration by parts. In particular, the numerical method allows the use of discontinuous finite element functions and arbitrary shape of element. The proposed schemes will be proved to have good numerical stability and high order accuracy when time variable is continuous. Also an optimal error estimate is obtained for the fully discrete scheme. Finally, some numerical results are given to verify our analysis for the scheme.
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 |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
Sobolev equation; weak Galerkin; weak gradient; discrete weak gradient; error estimate; stability; numerical resultReferences:
| [1] | Esen, A.; Kutluay, S., Application of a lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput., 174, 833-845 (2006) · Zbl 1090.65114 |
| [2] | Arnold, D. N.; Douglas, J.; Thomée, V., Superconvergence of finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comp., 27, 737-743 (1981) |
| [3] | Ewing, R. E., Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation, IAM J. Numer. Anal., 15, 1125-1150 (1978) · Zbl 0399.65083 |
| [4] | Wahlbin, L., Error estimates for a Galerkin method for a class of model equations for long waves, Numer. Math., 23, 289-303 (1975) · Zbl 0283.65052 |
| [5] | Omrani, K., The convergence of the fully discrete Galerkin approximations for the Benjamin-Bona-Mahony (BBM) equation, Appl. Math. Comput., 180, 614-621 (2006) · Zbl 1103.65101 |
| [6] | Guo, L.; Chen, H., \(H_1\)-Galerkin mixed finite element method for the regularized long wave equation, Computing, 77, 205-221 (2006) · Zbl 1098.65096 |
| [7] | Pani, A. K.; Fairweather, G., \(H_1\)-Galerkin mixed finite element methods for parabolic partial integro-differential equations, IMA J. Numer. Anal., 22, 231-252 (2002) · Zbl 1008.65101 |
| [8] | Avilez-Valente, P.; Seabra-Santos, F. J., A Petrov-Galerkin finite element scheme for the regularized long wave equation, Comput. Mech., 34, 256-270 (2004) · Zbl 1091.76031 |
| [9] | Dogan, A., Numerical solution of regularized long wave equation using Petrov-Galerkin method, Commun. Numer. Methods Eng., 17, 485-494 (2001) · Zbl 0985.65121 |
| [10] | Gao, F.; Qiu, J.; Zhang, Q., Local discontinuous Galerkin finite element method and error estimates for one class of Sobolev equation, J. Sci. Comput., 41, 436-460 (2009) · Zbl 1203.65181 |
| [11] | Zhang, Q.; Gao, F., A fully-discrete local discontinuous Galerkin method for convection-dominated Sobolev equation, J. Sci. Comput., 51, 107-134 (2012) · Zbl 1244.65145 |
| [12] | Gao, F.; Rui, H., A split least-squares characteristic mixed finite element method for Sobolev equations with convection term, Math. Comput. Simulation, 80, 341-351 (2009) · Zbl 1180.65126 |
| [13] | Gao, F.; Qiao, F.; Rui, H., Numerical simulation of the modified regularized long wave equation by split least-squares mixed finite element method, Math. Comput. Simulation, 109, 64-73 (2015) · Zbl 1540.65373 |
| [14] | Gao, F.; Wang, X., A modified weak Galerkin finite element method for Sobolev equation, J. Comput. Math., 33, 307-322 (2015) · Zbl 1340.65270 |
| [15] | Wang, J.; Ye, X., A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241, 103-115 (2013) · Zbl 1261.65121 |
| [16] | Gao, F.; Mu, L., On \(L^2\) error estimates for weak Galerkin finite element methods for parabolic problems, J. Comput. Math., 32, 195-204 (2014) · Zbl 1313.65246 |
| [17] | Wang, J.; Ye, X., A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., 83, 2101-2126 (2014) · Zbl 1308.65202 |
| [18] | Mu, L.; Wang, J.; Wei, G.; Ye, X.; Zhao, S., Weak Galerkin method for the elliptic interface problem, J. Comput. Phys., 250, 106-125 (2013) · Zbl 1349.65472 |
| [19] | Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comp., 52, 411-435 (1989) · Zbl 0662.65083 |
| [20] | Mu, L.; Wang, X.; Wang, Y., Shape regularity conditions for polygonal/polyhedral meshes, examplified in a discontinuous Galerkin discretization, Numer. Methods Partial Differential Equations, 31, 308-325 (2015) · Zbl 1338.65250 |
| [21] | Mu, L.; Wang, J.; Ye, X., A weak Galerkin finite element method with polynomial reduction, J. Comput. Appl. Math., 285, 45-58 (2015) · Zbl 1315.65099 |
| [22] | Wheeler, M. F., A priori error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10, 723-759 (1973) · Zbl 0232.35060 |
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.
