The definition of asymptotic convergence?

Question

What is the difference between convergence and asymptotic convergence? Why say the convergence is asymptotic?

Answer 1

Typically convergence refers to the error decreasing with the fineness of the discretization; i.e. for finite difference/finite elements, this tends do deal with grid spacing $h$ and claims that error goes to $0$ as $h \rightarrow 0$.

Usually asymptotic convergence refers to a convergence behavior that is observed when $h$ is sufficiently small. One example is with finite differences: writing out a Taylor series

$u(x+h) = u(x) + u'(x)h + \frac{u''(x)}{2}h^2 + \ldots$

you can rearrange to get

$\left|\frac{u(x+h) - u(x)}{h} - u'(x)\right| = \left|\frac{u''(x)}{2}h + \frac{u'''(x)}{3!}h^2\ldots\right|$.

This gives us an exact expression for finite difference error. For $h$ small enough, $h^2 \ll h$, and the leading term $\frac{u''(x)}{2}h = O(h)$ will dominate the above expression. However, this is only true if $h$ is small enough, which implies that this is an asymptotic result.

An additional issue is that, when solving PDEs, sometimes there may be parameters which may influence the convergence of the method. For example, for the Helmholtz equation

$$\kappa^2 u + \Delta u = f$$

this parameter is the wavenumber $\kappa$, and for convection-diffusion

$$\nabla\cdot(\beta u) - \epsilon \Delta u = f$$

this parameter is $\epsilon$. Most discretizations achieve asymptotically (quasi)optimal behavior, that as $h$ is small enough (or as the discretization becomes fine enough), the solution will behave a certain way. However, this asymptotic limit for where $h$ is small enough can also depend on these parameters; asymptotic convergence is helpful, but it doesn't tell you everything.