# What is the meaning of “preasymptotic” and “superconvergent”?

Precisely the title of the question.

I have encountered these terms in two areas: conjugate gradient method, and adaptive finite elements.

Preasymptotic refers to the following concept: A priori error estimates only say that as the mesh size $h\rightarrow 0$, we have that (for example) $\|e\|\le Ch^2$. But this is an asymptotic statement: it is not an equality that holds for all $h$ but one typically only sees the quadratic decay whenever $h$ is small enough. In other words, in numerical experiments, if you plot the error as a function of the mesh size in a log-log plot, you get a nice straight line with slope 2 for small enough mesh sizes. But, for large mesh sizes, the data points are more scattered and do not follow the exact relationship. This, the range of large $h$ where the decay has not yet set in, is called the preasymptotic range (range as in "a part of the possible range of mesh sizes $h$").

The reason this preasymptotic range exists is that the exact error relationship is a nonlinear function of the mesh size. For example, if the exact error were of the form $\|e\|=c_1h^2 + c_2h^3 + c_3h^4$, then for small enough $h$, all but the first term are essentially irrelevant. However, for large $h$, the other terms may contribute.

• You probably only want one of the two answers you just posted. – Jed Brown Mar 24 '13 at 17:01
• I don't know. The OP had two questions. I decided to give two answers so that people can comment on each of them individually. – Wolfgang Bangerth Mar 26 '13 at 2:58
• Fair enough, though I would still tend to combine simply because it fits the format of the site (e.g., there can be only one "accepted" answer). – Jed Brown Mar 26 '13 at 3:37

The meanings of those terms depend on context.

Superconvergence is usually used to mean you are converging faster than the "optimal" rate, and occasionally this sort of weirdly fast convergence can be proven rigorously. One example in DG is that hyperbolic problems generally have an optimal estimate that decays like $h^{n+1/2}$ but some recent superconvergence papers shows that you can actually recover $h^{2n+1}$ in some limited cases.

Preasymptotic means: some numerical results you have demonstrated don't fit the analysis for a small parameter choice, but past some cut-off point it does. Sometimes I have seen authors use this to explain why their numerical results don't fit their analysis.

Pre-asymptoticity in conjugate gradient method: Asymptotic behavior sometimes assumes certain conditions. For example, given $h$ the meshsize or similar, and $h\to 0$: $$\|f_h - g_h \| \leq c h.$$ but what if $$\|f_h - g_h \| \leq c k h^2$$ if we assume $kh = O(1)$ then $f_h$ is asymptotic to $g_h$, this is asympototic analysis. If we do not assume that condition, for example $k$ may be increasing wave number, then this is pre-asymptotic analysis.

In preconditioned conjugate gradient(PCG) method, the asymptotic convergence rate for large sparse SPD matrices depends on the spectral condition number of the PCG iteration matrix. Normally we just suppose this condition number is fixed, the pre-asymptotic analysis for conjugate gradient method also takes into account the behavior of this condition number in the analysis of the convergence rate.

Superconvergence in adaptive finite elements: suppose $u$ be the true solution to the usual $H^1_0(\Omega)$ variational problem $$\int_{\Omega} \nabla u \cdot \nabla v = \int_{\Omega} f\,v$$ and $u_h$ be the finite elements solution. Adaptive finite elements method(AFEM) essentially want to use an estimator $\eta$ to approximate the error $\|\nabla(u-u_h)\|$ locally and globally, such that we could perform the adaptive mesh refinement.

The superconvergence says something like this, for a postprocessed computable quantity $\sigma$: $$\|\nabla u - \sigma\| \leq c h^{\alpha} \|\nabla u - \nabla u_h\|.$$ As you can see, when the meshsize is small, $\|\nabla u - \sigma\|$ is much smaller than the true error, convergent at a much faster rate than the approximation error $\|\nabla u - \nabla u_h\|$, and this is what superconvergence means in AFEM.

Now simply by triangle inequality: $$\Big| \|\nabla u - \sigma\| - \|\sigma- \nabla u_h\| \Big|\leq \|\nabla u - \nabla u_h\| \leq \|\nabla u - \sigma\| + \|\sigma- \nabla u_h\|$$ Since $\|\nabla u - \sigma\|$ is negligible when the meshsize is small, whenever superconvergence is true, we could use the quantity $\|\sigma- \nabla u_h\|$ to be a computable error estimator to estimate the non-computable approximation error $\|\nabla u - \nabla u_h\|$.

Superconvergent refers to the concept that sometimes convergence is faster than one would usually expect. As an example, we know that for the Laplace equation, pointwise convergence happens as $\|u-u_h\|_{L_\infty} \le Ch^2 |\log h|$ when using linear elements. But, on sufficiently regular meshes, it can be shown that at certain points (on quadrilateral cells the $2\times 2$ Gauss integration points) the error is actually of order $h^3$. In other words, at some points of the mesh, the error is significantly smaller than at all the others.

As hinted at in one of the other answers to this question, this can be exploited. If we know, for example, that the error is smaller at certain points than others, then we should interpolate a higher order polynomial through these points and we can hope to get a more accurate solution than what the finite element method provided us initially. This process is called a "recovery procedure". It can be used to get a more accurate solution, or a better approximation of the gradient. In either case, the difference between the original finite element solution and the "recovered" one (or their respective gradients) can be used to approximate the error between the original finite element solution and the exact solution.