Two-grid Raviart-Thomas mixed finite element methods combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations

• Published in: Advances in computational mathematics Vol. 46; no. 2
• Main Authors: Hou, Tianliang, Chen, Luoping, Yang, Yueting, Yang, Yin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2020; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Convergence; Crank-Nicholson method; Finite element method; Grid method; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Nonlinear equations; Visualization; A priori error estimates; 49J20; Crank-Nicolson scheme; Two-grid; Raviart-Thomas mixed finite element; 65N30; Nonlinear parabolic equations
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-020-09777-z

In this paper, we discuss a priori error estimates of two-grid mixed finite element methods for a class of nonlinear parabolic equations. The lowest order Raviart-Thomas mixed finite element and Crank-Nicolson scheme are used for the spatial and temporal discretization. First, we derive the optimal a priori error estimates for all variables. Second, we present a two-grid scheme and analyze its convergence. It is shown that if the two mesh sizes satisfy h = H 2 , then the two-grid method achieves the same convergence property as the Raviart-Thomas mixed finite element method. Finally, we give a numerical example to verify the theoretical results.