A Novel Two-Grid Method for Semilinear Eliiptic Equations (Xu)

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SIAM Journal on Scientific Computing
Volume 15-1, January 1994, pp.  231-237
(C) 1994 by Society for Industrial and Applied Mathematics
All rights reserved


Title:            A Novel Two-Grid Method for Semilinear Elliptic 
                  Equations

Author:           Jinchao Xu

AMS Subject 
Classifications:  65M60, 65N15, 65N30

Key words:        elliptic boundary value problem, finite elements, 
                  two-grid

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ABSTRACT

A new finite element discretization technique based on two 
(coarse and fine) subspaces is presented for a semilinear 
elliptic boundary value problem.  The solution of a 
nonlinear system on the fine space is reduced to the 
solution of two small (one linear and one nonlinear) systems 
on the coarse space and  a linear system on the fine space. 
It is shown, both theoretically and numerically, that the 
coarse space can be extremely coarse and still achieve 
asymptotically optimal approximation. As a result, the 
numerical solution of such a nonlinear equation is not 
significantly more expensive than the solution of one single 
linearized equation.


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