Experimental convergence order

The term experimental order of convergence (English: experimental order of convergence , EOC ) is understood in the numerical analysis an estimate of the rate of convergence of a sequence. In order to calculate this, the limit value is assumed to be known.

This tool is often used to validate finite element and discontinuous Galerkin methods .

definition

Let be three consecutive terms and the sequence limit. The experimental convergence order is then ${\displaystyle x_{k},x_{k+1},x_{k+2}}$ ${\displaystyle x}$

${\displaystyle \mathrm {EOC} ={\frac {\log {\frac {\|x_{k+1}-x\|}{\|x_{k+2}-x\|}}}{\log {\frac {\|x_{k}-x\|}{\|x_{k+1}-x\|}}}}}$

where is a suitable norm . ${\displaystyle \|\cdot \|}$

motivation

Let be the already known limit of the sequence . The sequence converges with speed if there is a constant that satisfies the inequality ${\displaystyle x}$ ${\displaystyle (x_{k})_{k\in \mathbb {N} }}$ ${\displaystyle \alpha }$ ${\displaystyle c>0}$

${\displaystyle \|x_{k+1}-x\|\leq c\|x_{k}-x\|^{\alpha }}$

Fulfills. For the sake of simplicity, it is now assumed that the convergence can occur exactly

${\displaystyle \|x_{k+1}-x\|=c\|x_{k}-x\|^{\alpha }}$

to be discribed. This formulation then also applies to the next term in the sequence

${\displaystyle \|x_{k+2}-x\|=c\|x_{k+1}-x\|^{\alpha }}$

Division of the two equations yields

${\displaystyle {\frac {\|x_{k+1}-x\|}{\|x_{k+2}-x\|}}=\left({\frac {\|x_{k}-x\|}{\|x_{k+1}-x\|}}\right)^{\alpha }}$

So it applies

${\displaystyle \alpha =\log _{\frac {\|x_{k}-x\|}{\|x_{k+1}-x\|}}{\frac {\|x_{k+1}-x\|}{\|x_{k+2}-x\|}}}$

where denotes the logarithm to the base . A conversion of the logarithm to the base results in the definition of . ${\displaystyle \log _{\frac {\|x_{k}-x\|}{\|x_{k+1}-x\|}}}$ ${\displaystyle {\frac {\|x_{k}-x\|}{\|x_{k+1}-x\|}}}$ ${\displaystyle 10}$ ${\displaystyle \mathrm {EOC} }$

Application: Numerical solutions of differential equations

Let be numerical solutions of a method that approximately solves ( partial ) differential equations . There are different values ​​of a discretization parameter that describes the resolution of the discretization. In the one-dimensional case it is usually the length of the largest interval. In the higher-dimensional case, an analog measure is used for the fineness of the grating, for example the largest inscribed diameter in two dimensions . Let be the limit of the procedure for . Then the experimental order of convergence is dependent on and through ${\displaystyle u_{h},u_{h^{\prime }}}$ ${\displaystyle h,h^{\prime }}$ ${\displaystyle h}$ ${\displaystyle u}$ ${\displaystyle h\rightarrow 0}$ ${\displaystyle \mathrm {EOC} }$ ${\displaystyle h}$ ${\displaystyle h^{\prime }}$

${\displaystyle \mathrm {EOC} (h,h^{\prime })={\frac {\log {\frac {\|u_{h}-u\|}{\|u_{h^{\prime }}-u\|}}}{\log {\frac {h}{h^{\prime }}}}}}$

given. This case can be explained by an a priori error estimator of the form

${\displaystyle \|u_{h}-u\|\leq ch^{\alpha }}$

motivate with constants . As before, simplicity is also used here ${\displaystyle c,\alpha >0}$

${\displaystyle \|u_{h}-u\|=ch^{\alpha }}$

accepted. This applies to both discretization and . Dividing the two equations gives ${\displaystyle h}$ ${\displaystyle h^{\prime }}$

${\displaystyle {\frac {\|u_{h}-u\|}{\|u_{h^{\prime }}-u\|}}=\left({\frac {h}{h^{\prime }}}\right)^{\alpha }}$.

So it applies

${\displaystyle \alpha =\log _{\frac {h}{h^{\prime }}}{\frac {\|u_{h}-u\|}{\|u_{h^{\prime }}-u\|}}}$,

which after converting the logarithm to base 10 gives the formula for . ${\displaystyle \mathrm {EOC} (h,h^{\prime })}$

Relation to the true convergence order

No convergence can be demonstrated using the EOC, as this is assumed. If there is a convergent method, it cannot generally be said whether the actual convergence rate is overestimated or underestimated by the EOC.

Individual evidence

1. ↑ G. Sacrifice, Numerical Mathematics for Beginners , 2001, p. 304.