Two-grid methods for expanded mixed finite element approximations of semi-linear parabolic integro-differential equations

• Published in: Applied numerical mathematics Vol. 132; pp. 163 - 181
• Main Authors: Hou, Tianliang, Chen, Luoping, Yang, Yin
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2018
• Subjects: A priori error estimates; Expanded mixed finite element method; Semi-linear parabolic integro-differential equations; Superconvergence; Two-grid scheme; Semi-linear parabolic integro-differential equations; Superconvergence; A priori error estimates; Expanded mixed finite element method; Two-grid scheme
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2018.06.001

In this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart–Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz–Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h=H2, the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method.