Two-Grid method for nonlinear parabolic equations by expanded mixed finite element methods

• Published in: Numerical methods for partial differential equations Vol. 29; no. 4; pp. 1238 - 1256
• Main Authors: Chen, Yanping, Chen, Luoping, Zhang, Xiaochun
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.07.2013; Wiley Subscription Services, Inc
• Subjects: Algorithms; Compressibility; Error analysis; Finite element method; Mathematical analysis; Mathematical models; Newton iterations; Nonlinear equations; nonlinear parabolic equations; Nonlinearity; optimal approximation; two-grid methods
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.21753

In this article, we develop a two‐grid algorithm for nonlinear reaction diffusion equation (with nonlinear compressibility coefficient) discretized by expanded mixed finite element method. The key point is to use two‐grid scheme to linearize the nonlinear term in the equations. The main procedure of the algorithm is solving a small‐scaled nonlinear equations on the coarse grid and dealing with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Error estimation to the expanded mixed finite element solution is analyzed in detail. We also show that two‐grid solution achieves the same accuracy as long as the mesh sizes satisfy H = O(h1/2). Two numerical experiments are given to verify the effectiveness of the algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013