An Adaptive Least-squares Mixed Finite Element Method for Pseudo-parabolic Integro-differential Equations

Authors: Zilong Feng, Hong Li, Yang Liu, Siriguleng He

Abstract:

In this article, an adaptive least-squares mixed finite element method is studied for pseudo-parabolic integro-differential equations. The solutions of least-squares mixed weak formulation and mixed finite element are proved. A posteriori error estimator is constructed based on the least-squares functional and the posteriori errors are obtained.

Keywords: Pseudo-parabolic integro-differential equation, least squares mixed finite element method, adaptive method, a posteriori error estimates.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1079476

References:

[1] R Ewing, R Lazarov, Y P Lin. Finite volume element approximations of nonlocal reactive flows in porous media, Numer. Methods Partial Differential Equations, 2000, 16: 285-311.
[2] Y Lin, V Thomee, L Wahlbin. Ritz-Volterra projection onto finite element spaces and application to integro-differential and related equations, SIAM J. Numer. Anal., 1995, 28: 1047-1070.
[3] A K Pani, G Fairweather. H1-Galerkin mixed finite element methods for parabolic partial integro-differential equations, IMA Journal of Numerical Analysis, 2002, 22: 231-252.
[4] Y Liu, H Li S He. Error estimates of H1-Galerkin mixed finite element methods for pseudo-hyperbolic partial integro-differential equation, Numerical Mathematics: A Journal of Chinese Universities, 2010, 32(1): 1-20
[5] D Y Shi, H H Wang. An H1-Galerkin nonconforming mixed finite element method for integro-differential equation of parabolic type, Journal of Mathematical Research and Exposition, 2009, 29(5): 871- 881.
[6] Z J Zhou, H Z Chen. H1-Galerkin mixed finite element method for pseudo-parabolic integro-differential equation, Journal of Shandong Normal University(Natural Science), 2005, 20(2): 3-7.
[7] H Li, Y Liu. Mixed discontinuous space-time finite element method for the fourth-order parabolic integro-differential equations, Mathematica Numerica Sinica, 2007, 29(4): 413-420.
[8] H Li, H Q Wang. The space-time discontinuous finite element method for a semi-linear parabolic integro-differential equation, Mathematica Numerica Sinica, 2006, 28(3): 293-308.
[9] H Guo, H X Rui. Least-squares Galerkin procedures for parabolic integro-differential equations, Appl. Math. Comput., 2004, 150: 749- 762.
[10] H Guo, H X Rui. Crank-Nicolson least-squares Galerkin procedures for parabolic integro-differential equations, Appl. Math. Comput., 2006, 180: 622-634.
[11] X Cui. Sobolev-Volterra projection and numerical analysis of finite element methods for integro-differential equations, Acta Mathematicae Applicatae Sinica, 2001, 24(3): 441-454
[12] T Zhang, C J Li. Superconvergence of finite element approximations to parabolic and hyperbolic integro-differential equations, Northeastern Math., 2001, (3): 279-288.
[13] Z D Luo, J Zhu, H J Wang. A nonlinear Galerkin/petrov-least squares mixed element method for the stationary Navier-Stokes equations, Applied Mathematics and Mechanics, 2002, 23(7): 697-706.
[14] D P Yang. Least-square mixed finite element methods for nonlinear parabolic problems, J. Comput. Math., 2002, 20(2): 153-164.
[15] Y P Chen, Y Q Huang, Z H Shen. Least-squares mixed finite element method for degenerate elliptic problems, Numer. Math. Sinica, 2001, 23: 87-94.
[16] Z Q Cai, K Johannes, S Gerhard. An adaptive least-squares mixed finite element method for the stress-displacement formulation of linear elasticity, Numer. Methods Partial Differential Equations, 2005, 21: 132- 148.
[17] H Y Duan, Q Lin . Mixed finite elements of least-squares type for elasticity, Computer Methods in Applied Mechanics and Engineering, 2005, 194(9-11), 1093-1112.
[18] H M Gu, H W Li. An adaptive least-squares mixed finite element method for nonlinear parabolic problems, Computational Mathematics and Modeling, 2009, 20(2): 192-206. .