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Analysis of two‐grid discretization scheme for semilinear hyperbolic equations by mixed finite element methods
Author(s) -
Wang Keyan,
Chen Yanping
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4831
Subject(s) - finite element method , mathematics , discretization , grid , nonlinear system , mixed finite element method , convergence (economics) , scheme (mathematics) , regular grid , hyperbolic partial differential equation , extended finite element method , mathematical optimization , mathematical analysis , partial differential equation , geometry , physics , quantum mechanics , economics , thermodynamics , economic growth
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.