A splitting extrapolation for solving nonlinear elliptic equations with d-quadratic finite elements

Authors:
Yong Cao;Xiaoming He;Tao Lü
Affiliations:
Department of Mechanical Engineering and Automation, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen, Guangdong 518055, PR China;Department of Mathematics, Virginia Tech, MCB 460, Blacksburg, VA 24061, USA;Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
Venue:
Journal of Computational Physics
Year:
2009

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Abstract

Nonlinear elliptic partial differential equations are important to many large scale engineering and science problems. For this kind of equations, this article discusses a splitting extrapolation which possesses a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than Richardson extrapolation. According to the problems, some domain decompositions are constructed and some independent mesh parameters are designed. Multi-parameter asymptotic expansions are proved for the errors of approximations. Based on the expansions, splitting extrapolation formulas are developed to compute approximations with high order of accuracy on a globally fine grid. Because these formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems.