Can someone give me some reference(book/article) in which the computational cost of sparse direct methods is compared with sparse iterative methods for solving the linear system of equations of the form:
$$ Ax=b $$ with $A \in C^{NxN}$ and $x,b \in C^{Nx1}$, where the linear system is arising from the discretization of PDEs, for example, by finite element methods.
What I know is that for dense matrices the typical computational cost of direct methods is $O(N^3)$. I have read somewhere on internet that for sparse direct methods the computational cost is from $O(N^{1.5})$ to $Nlog(N)$., but what is the source of this claim. My experience of working with Umfpack is that it is much more expensive than $Nlog(N)$. Which direct solver has this computational cost and for which type of problem?