# Fast iterative methods (for Laplace equation)?

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

 September 25, 2008, 18:29 Fast iterative methods (for Laplace equation)? #1 pzhang Guest   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!]

 September 26, 2008, 05:24 Re: Fast iterative methods (for Laplace equation)? #2 Jed Guest   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.

 September 30, 2008, 16:28 Re: Fast iterative methods (for Laplace equation)? #3 pzhang Guest   Posts: n/a Thanks!

 Thread Tools Display Modes Linear Mode

 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 OffTrackbacks are On Pingbacks are On Refbacks are On Forum Rules

 Similar Threads Thread Thread Starter Forum Replies Last Post Mihail CFX 7 September 7, 2014 06:27 Phiper Main CFD Forum 4 December 10, 2010 02:58 Sas CFX 15 July 13, 2010 08:56 Andreas Main CFD Forum 0 October 18, 2001 08:56 Vitaly Bulgakov Main CFD Forum 32 March 1, 1999 12:11

All times are GMT -4. The time now is 08:07.