OPTIMAL L^{2}-ERROR ESTIMATES FOR EXPANDED MIXED FINITE ELEMENT METHODS OF SEMILINEAR SOBOLEV EQUATIONS

- Journal title : Journal of the Korean Mathematical Society
- Volume 51, Issue 3, 2014, pp.545-565
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2014.51.3.545

Title & Authors

OPTIMAL L^{2}-ERROR ESTIMATES FOR EXPANDED MIXED FINITE ELEMENT METHODS OF SEMILINEAR SOBOLEV EQUATIONS

Ohm, Mi Ray; Lee, Hyun Young; Shin, Jun Yong;

Ohm, Mi Ray; Lee, Hyun Young; Shin, Jun Yong;

Abstract

In this paper we derive a priori error estimates for expanded mixed finite element formulations of semilinear Sobolev equations. This formulation expands the standard mixed formulation in the sense that three variables, the scalar unknown, the gradient and the flux are explicitly treated. Based on this method we construct finite element semidiscrete approximations and fully discrete approximations of the semilinear Sobolev equations. We prove the existence of semidiscrete approximations of u, and and obtain the optimal order error estimates in the norm. And also we construct the fully discrete approximations and analyze the optimal convergence of the approximations in norm. Finally we also provide the computational results.

Keywords

semilinear Sobolev equations;expanded mixed finite element method;semidiscrete approximations;fully discrete approximations;computational results;

Language

English

References

1.

T. Arbogast, M. F. Wheeler, and I. Yotov, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM. J. Numer. Anal. 34 (1997), no. 2, 828-852.

2.

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 724-760.

3.

D. N. Arnold, J. Jr. Douglas, and V. Thomee, Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comp. 36 (1981), no. 153, 53-63.

4.

G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appl. Math. Mech. 24 (1960), no. 5, 1286-1303.

5.

R. W. Carroll and R. E. Showalter, Singular and degenerate Cauchy problems, Mathematics in Sciences and Engineering, Vol. 127, Academic Press, New York, 1976.

6.

P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew Math. Phys. 19 (1968), no. 4, 614-627.

7.

Z. Chen, BDM mixed methods for a nonlinear elliptic problem, J. Comput. Appl. Math. 53 (1994), no. 2, 207-223.

8.

Z. Chen, Expanded mixed finite element methods for linear second-order elliptic problems I, RAIRO. Model. Math. Anal. Mumer. 32 (1998), no. 4, 479-499.

9.

Z. Chen, Expanded mixed finite element methods for quasilinear second-order elliptic problems II, RAIRO. Model. Math. Anal. Numer 32 (1998), no. 4, 501-520.

10.

Y. Chen, Y. Huang, and D. Yu, A two-grid method for expanded mixed finite-element solution of semilinear reaction-diffusion equations, Internat. J. Numer. Methods. Engrg 57 (2003), no. 2, 193-209.

11.

Y. Chena and L. Li, $L^p$ error estimates of two-grid schemes of expanded mixed finite element methods, Appl. Math. Comp. 209 (2009), no. 2, 197-205.

12.

P. L. Davis, A quasilinear parabolic and a related third order problem, J. Math. Anal. Appl. 40 (1972), no. 2, 327-335.

13.

J. Douglas and J. E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39-52.

14.

R. Duran, Error analysis in $L^p$ , $1{\leq}p{\leq}{\infty}$ for mixed finite element mehtods for linear and quasi-linear elliptic problems, RAIRO Mode. Math. Anal. Numer. 22 (1988), no. 3, 371-387.

15.

R. E. Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J. Math. Anal. 6 (1975), no. 2, 283-294.

16.

R. E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM J. Numer. Anal. 15 (1978), no. 6, 1125-1150.

17.

F. Gao, J. Qiu and Q. Zhang, Local discontinuous Galerkin finite element method and error estimates for one class of Sobolev equation J. Sci. Comput. 41 (2009), no. 3, 436-460.

18.

D. Kim and E.-J. Park, A posteriori error estimator for expanded mixed hybrid methods, Numer. Methods Partial Differential Equations 23 (2007), no. 2, 330-349.

19.

Y. Lin, Galerkin methods for nonlinear Sobolev equations, Aequationes Math. 40 (1990), no. 1, 54-66.

20.

Y. Lin and T. Zhang, Finite element methods for nonlinear Sobolev equations with nonlinear boundary conditions, J. Math. Anal. Appl. 165 (1992), no. 1, 180-191.

21.

M. T. Nakao, Error estimates of a Galerkin method for some nonlinear Sobolev equations in one space dimension, Numer. Math. 47 (1985), no. 1, 139-157.

22.

23.

M. R. Ohm, H. Y. Lee, and J. Y. Shin, Error analysis of a mixed finite element approxi-mation of the semilinear Sobolev equations, J. Appl. Math. Comput. 40 (2012), no. 1-2, 95-110.

24.

P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Proc. Conf. on Mathemaical Aspects of Finite Element Methods, pp. 292-315. Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977.

25.

D. Shi and Y. Zhang, High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations, Appl. Math. Comput. 218 (2011), no. 7, 3176-3186.

26.

T. Sun and D. Yang, A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations, Appl. Math. Comput. 200 (2008), no. 1, 147-159.

27.

T. Sun and D. Yang, Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations, Numer. Methods Partial Differential Equations 24 (2008), no. 3, 879-896.

28.

T. W. Ting, A cooling process according to two-temperature theory of heat conduction, J. Math. Anal. Appl. 45 (1974), 289-303.

29.

M. F. Wheeler, K. R. Roberson, and A. Chilakapati, Three-dimensional bioremediation modeling in heterogeneous porous media, Computational methods in water resources IX, Vol. 2, Mathematical modeling in water resources, T. F. Russell, R. E. Ewing, C. A. Brebbia, W. G. Gray and G. F. Pindar, editors, Computational Mechanics Publications, 299-315, Southampton, UK, 1992.