Is laplace operator well-conditioned?

[yy.wayne]

Hello, 

I‘m kind of lost in the nature of laplace operator.

Poisson equations are easy to solve numerically, and it has

<grad, grad> type weak form.

However, in step-22 it says laplace operator is ill-conditioned

and expensive to invert with iterative method.

This question might be silly. Is laplcae operator a positive definite

system and easy to solve? Does it become ill-conditioned when

mesh size and mesh quality reduce?

[Wolfgang Bangerth]

> This question might be silly. Is laplcae operator a positive definite
> system and easy to solve? Does it become ill-conditioned when
> mesh size and mesh quality reduce?

"ill-conditioned" and "difficult to solve" are two different things. The
condition number of the discretized Laplace operator (i.e., the Laplace
matrix) grows like O(1/h^2), so it becomes quite large on fine meshes. But
linear systems with this matrix are relatively easy to solve if you use
multigrid methods.

Best
W.

--
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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[yy.wayne]

I get it now. Thanks for your rely.

Best,

Wayne