Editing Rate of convergence (section)

=== Definitions ===
A sequence of discretized approximations <math>(y_k)</math> of some continuous-domain function <math>S</math> that converges to this target, together with a corresponding sequence of discretization scale parameters <math>(h_k)</math> that converge to 0, is said to have asymptotic ''order of convergence'' <math>q</math> and asymptotic ''rate of convergence'' <math>\mu</math> if

<math display="block">\lim _{k \rightarrow \infty} \frac{\left|y_k - S\right|}{h_k^{q}}=\mu,</math>

for some positive constants <math>\mu</math> and <math>q</math> and using <math>|x|</math> to stand for an appropriate [[distance metric]] on the [[Function space|space of solutions]], most often either the [[uniform norm]], the [[Taxicab geometry|absolute difference]], or the [[Euclidean distance]]. Discretization scale parameters may be spacings of a [[regular grid]] in space or in time, the inverse of the number of points of a grid in one dimension, an average or maximum distance between points in a [[polygon mesh]], the single-dimension spacings of an irregular [[sparse grid]], or a characteristic quantum of energy or momentum in a [[Quantum mechanics|quantum mechanical]] [[Basis set (chemistry)|basis set]].

When all the discretizations are generated using a single common method, it is common to discuss the asymptotic rate and order of convergence for the method itself rather than any particular discrete sequences of discretized solutions. In these cases one considers a single abstract discretized solution <math>y_h</math> generated using the method with a scale parameter <math>h</math> and then the method is said to have asymptotic ''order of convergence'' <math>q</math> and asymptotic ''rate of convergence'' <math>\mu</math> if

<math display="block">\lim _{h \rightarrow 0} \frac{\left|y_h - S\right|}{h^{q}}=\mu,</math>

again for some positive constants <math>\mu</math> and <math>q</math> and an appropriate metric <math>|x|.</math> This implies that the error of a discretization asymptotically scales like the discretization's scale parameter to the <math>q</math> power, or <math display="inline">\left|y_h - S \right| = O(h^{q})</math> using [[Big O notation|asymptotic big O notation]]. More precisely, it implies the leading order error is <math>\mu h^{q},</math> which can be expressed using [[Small o notation|asymptotic small o notation]] as<math display="inline">\left|y_h - S\right| = \mu h^{q} + o(h^{q}).</math>

In some cases multiple rates and orders for the same method but with different choices of scale parameter may be important, for instance for [[Finite difference method|finite difference methods]] based on multidimensional grids where the different dimensions have different grid spacings or for [[Finite element method|finite element methods]] based on polygon meshes where choosing either average distance between mesh points or maximum distance between mesh points as scale parameters may imply different orders of convergence. In some especially technical contexts, discretization methods' asymptotic rates and orders of convergence will be characterized by several scale parameters at once with the value of each scale parameter possibly affecting the asymptotic rate and order of convergence of the method with respect to the other scale parameters.