CFD Online Logo CFD Online URL
Home > Forums > Main CFD Forum

Fast iterative methods (for Laplace equation)?

Register Blogs Members List Search Today's Posts Mark Forums Read

LinkBack Thread Tools Display Modes
Old   September 25, 2008, 18:29
Default Fast iterative methods (for Laplace equation)?
Posts: n/a
Does anyone know any iterative method for solving the Laplace equation (with the standard discretization) for which the number of iterations increases linearly with N? (N is the number of mesh points in one coordinate direction, i.e., N = 1/h)

Jacobi, Gauss-Seidel, SOR(unless with the optimal factor) are all O(N^2) method. I wonder if there is an O(N) method.

[multigrid is exceptional as it is O(1): the number of cycles is grid-independent!]
  Reply With Quote

Old   September 26, 2008, 05:24
Default Re: Fast iterative methods (for Laplace equation)?
Posts: n/a
The stationary iterations you mention require O(K) iterations where K is the condition number. Conjugate gradients requires O(\sqrt(K)). For second order elliptic BVP, K = O(N^2) where N is the number of points in each Cartesian direction. In this sense, unpreconditioned CG satisfies is O(N). But this is still terrible. If the coefficients are smooth enough and the operator is uniformly elliptic, multigrid preconditioning should perform very well. More robust alternatives which also maintain essentially linear cost are multi-level domain decompositions such as BDDC.

By the way, multigrid as a solver rarely beats a Krylov with multigrid preconditioning. Nonlinear multigrid (such as FAS) rarely beats a Newton-Krylov iteration with multigrid preconditioning. Grid sequencing is still useful to get a good initial Newton iterate on the fine mesh.
  Reply With Quote

Old   September 30, 2008, 16:28
Default Re: Fast iterative methods (for Laplace equation)?
Posts: n/a
  Reply With Quote


Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are On
Pingbacks are On
Refbacks are On

Similar Threads
Thread Thread Starter Forum Replies Last Post
Calculation of the Governing Equations Mihail CFX 7 September 7, 2014 06:27
Iterative solution to Poissons Equation Phiper Main CFD Forum 4 December 10, 2010 02:58
Constant velocity of the material Sas CFX 15 July 13, 2010 08:56
Iterative Methods for the Stokes problem Andreas Main CFD Forum 0 October 18, 2001 08:56
Iterative equation solvers in CFD Vitaly Bulgakov Main CFD Forum 32 March 1, 1999 12:11

All times are GMT -4. The time now is 06:44.