An Adaptive Least-Squares Mixed Finite Element Method for Fourth Order Parabolic Problems

Ning Chen, Haiming Gu

Department of Mathematics, Qingdao University of Science and Technology, Qingdao, China.

DOI: 10.4236/am.2013.44092                

Abstract

A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth order parabolic problems analyzed and developed in this paper. The Ciarlet-Raviart mixed finite element space is used to approximate. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated. The convergence of the adaptive least-squares mixed finite element method is proved.

Keywords

Adaptive Method; Least-Squares Mixed Finite Element Method; Fourth Order Parabolic Problems; Least-Squares Functional; A Posteriori Error

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. K. Aziz, R. B. Kellogg and A. B. Stephens, “Least Squares Methods for Elliptic Systems,” Mathematics of Computation, Vol. 44, 1985, pp. 53-70. doi:10.1090/S0025-5718-1985-0771030-5
[2]	H. M. Gu, D. P. Yang, S. L. Sui and X. M. Liu, “Least Squares Mixed Finite Element Method for a Class of Stokes Equation,” Applied Mathematics and Mechanics, Vol. 21, No. 5, 2000, pp. 557-566. doi:10.1007/BF02459037
[3]	H.-M. Gu and X.-L. Xu, “The Least-Squares Mixed Finite Element Methods for a Degenerate Elliptic Problem,” Mathematica Applicata, Vol. 15, No. 1, 2002, pp. 118-122.
[4]	Z. Q. Cai, B. Lee and P. Wang, “Least-Squares Methods for Incompressible Newtonian Fluid Flow: Linear Stationary Problems,” SIAM Journal on Numerical Analysis, Vol. 42, No. 2, 2004, pp. 843-859. doi:10.1137/S0036142903422673
[5]	M. Maischak and E. P. Stephan, “A Least Squares Coupling Method with Finite Elements and Boundary Elements for Transmission Problems,” Computers & Mathematics with Applications, Vol. 48, No. 7-8, 2004, pp. 995-1016. doi:10.1016/j.camwa.2004.10.002
[6]	H. M. Gu and D. P. Yang, “Least-Squares Mixed Finite Element Method for Sobolev Equations,” Indian Journal of Pure and Applied Mathematics, Vol. 31, No. 5, 2000, pp. 505-517.
[7]	M.-Y. Kim E.-J. Park and J. Park, “Mixed Finite Element Domain Decomposition for Nonlinear Parabolic Problems,” Computers & Mathematics with Applications, Vol. 40, No. 8-9, 2000, pp. 1061-1070. doi:10.1016/S0898-1221(00)85016-6
[8]	Z. Q. Cai, J. Korsawe and G. Starke, “An Adaptive Least Squares Mixed Finite Element Method for the Stress Displacement Formulation of Linear Elasticity,” Numerical Methods for Partial Differential Equations, Vol. 21, No. 1, 2005, pp. 132-148. doi:10.1002/num.20029
[9]	Z. D. Luo, “Theoretical Bases for Mixed Finite Element Methods and Application,” Science Press, Beijing, 2006.
[10]	S.-Y. Yang, “Analysis of a Least Squares Finite Element Method for the Circular Arch Problem,” Applied Mathematics and Computation, Vol. 114, No. 2-3, 2000, pp. 263-278. doi:10.1016/S0096-3003(99)00122-8
[11]	T. Tang and J. C. Xu, “Adaptive Computations: Theory and Algorithms,” Science Press, Beijing, 2007.
[12]	A. Agouzal, “A Posteriori Error Estimators for Nonconforming Approximation,” Mathematical Modelling and Numerical Analysis, Vol. 5, No. 1, 2008, pp. 77-85.
[13]	X. Li, M. S. Shephard and M. W. Beall, “3D Anisotropic Mesh Adaptation Using Mesh Modifications,” Submitted to Computer Methods in Applied Mechanics and Engineering, 2003.