Abstract
In this paper, we present a two-grid scheme for a semilinear parabolic integro-differential equation using a new mixed finite element method. The gradient for the method belongs to the square integrable space instead of the classical H(div; Ω) space. The velocity and the pressure are approximated by the P02–P1 pair which satisfies the inf-sup condition. Firstly, we solve an original nonlinear problem on the coarse grid in our two-grid scheme. Then, to linearize the discretized equations, we use Newton iteration on the fine grid twice. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = O(H6|lnH|2). As a result, solving such a large class of nonlinear equations will not be much more difficult than the solution of one linearized equation. Finally, a numerical experiment is provided to verify theoretical results of the two-grid method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, New York: Springer-Verlag, 1991.
Bi, C. and Ginting, V., Two-Grid Finite Volume Element Method for Linear and Nonlinear Elliptic Problems, Num. Math., 2007, vol. 108, iss. 2, pp. 177–198.
Bi, C. and Ginting, V., Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems, J. Sci. Comput., 2011, vol. 49, no. 3, pp. 311–331.
Chen, L. and Chen, Y., Two-Grid Method for Nonlinear Reaction-Diffusion Equations by Mixed Finite Element Methods, J. Sci. Comput., 2011, vol. 49, no. 3, pp. 383–401.
Chen, Y., Huang, Y., and Yu, D., A Two-Grid Method for Expanded Mixed Finite-Element Solution of Semilinear Reaction-Diffusion Equations, Int. J. Num. Meth. Eng., 2003, vol. 57, iss. 2, pp. 193–209.
Chen, Y., Liu, H., and Liu, S., Analysis of Two-Grid Methods for Reaction-Diffusion Equations by Expanded Mixed Finite Element Methods, Int. J. Num. Meth. Eng., 2007, vol. 69, pp. 408–422.
Cannon, J.R. and Lin, Y., A Priori L 2 Error Estimates for Finite-Element Methods for Nonlinear Diffusion Equations with Memory, SIAM J. Num. An., 1990, vol. 27, pp. 595–607.
Chen, S.C. and Chen, H.R., New Mixed Element Schemes for a Second-Order Elliptic Problem, Math. Num. Sin., 2010, vol. 32, no. 2, pp. 213–218.
Ciarlet, P.G., The Finite Element Method for Elliptic Problems, Amsterdam: North-Holland, 1978.
Dawson, C., Wheeler, M.F., and Woodward, C.S., A Two-Grid Finite Difference Scheme for Nonlinear Parabolic Equations, SIAM J. Num. An., 1998, vol. 35, pp. 435–452.
Dawson, C.N. and Wheeler, M.F., Two-Grid Methods for Mixed Finite Element Approximations of Nonlinear Parabolic Equations, Cont. Math., 1994, vol. 180, pp. 191–203.
Douglas, J. and Roberts, J.E., Global Estimates for Mixed Methods for Second-Order Elliptic Equations, Math. Comp., 1985, vol. 44, pp. 39–52.
Ewing, R.E., Lin, Y.P., Sun, T., Wang, J.P., and Zhang, S.H., Sharp L 2-Error Estimates and Superconvergence of Mixed Finite Element Methods for Non-Fickian Flows in Porous Media, SIAM J. Num. An., 2002, vol. 40, no. 4, pp. 1538–1560.
Yanik, E.G. and Fairweather, G., Finite Element Methods for Parabolic and Hyperbolic Partial Integro-Differential Equations, Nonlin. An., 1988, vol. 12, pp. 785–809.
Grisvard P., Elliptic Problems in Nonsmooth Domains, Boston: Pitman Advanced Pub., 1985.
Huang, Y., Shi, Z.H., Tang, T., and Xue, W., A Multilevel Successive Iteration Method for Nonlinear Elliptic Problems, Math. Comp., 2004, vol. 73, no. 246, pp. 525–539.
Huang, Y. and Xue, W., Convergence of Finite Element Approximations and Multilevel Linearization for Ginzburg–Landau Model of d-Wave Superconductors, Adv. Comp. Math., 2002, vol. 17, pp. 309–330.
Lin, Y., Galerkin Methods for Nonlinear Parabolic Integrodifferential Equations with Nonlinear Boundary Conditions, SIAM J. Num. An., 1990, vol. 27, pp. 608–621.
Pani, A.K., An H 1-Galerkin Mixed Finite Element Method for Parabolic Partial Differential Equations, SIAM J. Num. An., 1998, vol. 35, no. 2, pp. 712–727.
Pani, A.K. and Fairweather, G., H 1-Galerkin Mixed Finite Element Method for Parabolic Partial Integro-Differential Equations, IMA J. Num. An., 2002, vol. 22, pp. 231–252.
Russell, T.F., Time Stepping along Characteristics with Incomplete Iteration for a Galerkin Approximation of Miscible Displacement in Porous Media, SIAM J. Num. An., 1985, vol. 22, no. 5, pp. 970–1013.
Raviart, P.A. and Thomas, J.M., A Mixed Finite Element Method for Second Order Elliptic Problems, in Mathematical Aspects of Finite Element Methods, Berlin: Springer, 1977, pp. 292–315 (Lect. Notes Math., vol. 606).
Shi, F., Yu, J.P., and Li, K.T., A New Stabilized Mixed Finite-Element Method for Poisson Equation Based on Two Local Gauss Integrations for Linear Element Pair, Int. J. Comp. Math., 2011, vol. 88, pp. 2293–2305.
Sinha, R.K., Ewing, R.E., and Lazarov, R.D., Mixed Finite Element Approximations of Parabolic Integro-Differential Equations with Nonsmooth Initial Data, SIAM J. Num. An., 2009, vol. 47, no. 5, pp. 3269–3292.
Thomée, V. and Zhang, N.Y., Error Estimates for Semidiscrete Finite Element Methods for Parabolic Integro-Differential Equations, Math. Comp., 1989, vol. 53, pp. 121–139.
Wu, L. and Allen, M.B., A Two-Grid Method for Mixed Finite-Element Solution of Reaction-Diffusion Equations, Num. Methods Partial Diff. Eqs., 1999, vol. 15, pp. 317–332.
Xu, J., A Novel Two-Grid Method for Semilinear Elliptic Equations, SIAM J. Sci. Comp., 1994, vol. 15, pp. 231–237.
Xu, J., Two-Grid Discretization Techniques for Linear and Nonlinear PDEs, SIAM J. Num. An., 1996, vol. 33, no. 5, pp. 1759–1777.
Xu, J. and Zhou, A., A Two-Grid Discretization Scheme for Eigenvalue Problems, Math. Comp., 2001, vol. 70, pp. 17–25.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Russian Text © The Author(s), 2019, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2019, Vol. 22, No. 2, pp. 167–185.
Rights and permissions
About this article
Cite this article
Liu, C., Hou, T. Two-Grid Methods for a New Mixed Finite Element Approximation of Semilinear Parabolic Integro-Differential Equations. Numer. Analys. Appl. 12, 137–154 (2019). https://doi.org/10.1134/S1995423919020046
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995423919020046