Sorry that I didn't state the equation clear.
I'm solving a time-dependent wave equation, and dt is the time step. For
backward Euler
time integration, I get M+dt^2*K = (v,u) + (\delta T)^2*(\nabla v, \nabla
u). The time step
dt does not scale with mesh size h, so the condition number is not bounded.
在2024年10月4日星期五 UTC+8 23:52:08<Wolfgang Bangerth> 写道：
>
> > Yes, I'm solving a transient wave equation. Should the behavior
> > between M + dt*K and M + dt^2*K be similar when solving the linear
> > system? I can only understand them as positive helmholtz equation,
> > the latter more dominated by mass matrix.
> The condition number of K is h^{-2}, so if you choose dt=h, for the wave
> equation you should generally obtain a matrix
> M + dt^2 K
> whose condition number is bounded as you refine the mesh. For the heat
> equation, you get
> M + dt K
> which will (if you choose dt=h) have a condition number that grows like
> h^{-1}.
>
> As a consequence, I'm a *bit* surprised that your iterations grow by so
> much, but it's been a long time since I've solved the wave equation and
> so I don't recall whether what you see is expected or not.
>
> Best
> W.
>
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