Analysis of two‐grid discretization scheme for semilinear hyperbolic equations by mixed finite element methods. (23rd April 2018)

• Record Type: Journal Article
• Authors: Wang, Keyan; Chen, Yanping
• Is Part Of: Mathematical methods in the applied sciences. Volume 41:Number 9(2018)
• Page Start: 3370
• Page End: 3391
• Publication Date: 2018-04-23
• Subjects: error estimate -- hyperbolic equations -- mixed finite element method -- two‐grid method; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4831
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, the full discrete scheme of mixed finite element approximation is introduced for semilinear hyperbolic equations. To solve the nonlinear problem efficiently, two two‐grid algorithms are developed and analyzed. In this approach, the nonlinear system is solved on a coarse mesh with width H, and the linear system is solved on a fine mesh with width h ≪ H . Error estimates and convergence results of two‐grid method are derived in detail. It is shown that if we choose H = O ( h 1 3 ) in the first algorithm and H = O ( h 1 4 ) in the second algorithm, the two‐grid algorithms can achieve the same accuracy of the mixed finite element solutions. Finally, the numerical examples also show that the two‐grid method is much more efficient than solving the nonlinear mixed finite element system directly.