Numerical method

Let $F(x,y)=0$ be a well-posed problem, i.e. $F:X\; \backslash times\; Y\; \backslash rightarrow\; \backslash mathbb\{R\}$ is a real or complex functional relationship, defined on the Cartesian product of an input data set $X$ and an output data set $Y$, such that exists a locally lipschitz function $g:X\; \backslash rightarrow\; Y$ called resolvent, which has the property that for every root $(x,y)$ of $F$, $y=g(x)$. We define numerical method for the approximation of $F(x,y)=0$, the sequence of problems

: $\backslash left\; \backslash \{\; M\_n\; \backslash right\; \backslash \}\_\{n\; \backslash in\; \backslash mathbb\{N\}\}\; =\; \backslash left\; \backslash \{\; F\_n(x\_n,y\_n)=0\; \backslash right\; \backslash \}\_\{n\; \backslash in\; \backslash mathbb\{N\}\},$

with $F\_n:X\_n\; \backslash times\; Y\_n\; \backslash rightarrow\; \backslash mathbb\{R\}$, $x\_n\; \backslash in\; X\_n$ and $y\_n\; \backslash in\; Y\_n$ for every $n\; \backslash in\; \backslash mathbb\{N\}$. The problems of which the method consists need not be well-posed. If they are, the method is said to be stable or well-posed.{{cite book

| last = Quarteroni, Sacco, Saleri

| title = Numerical Mathematics

| publisher = Springer

| location = Milano

| year = 2000

| page = 33

| url = http://www.techmat.vgtu.lt/~inga/Files/Quarteroni-SkaitMetod.pdf

| access-date = 2016-09-27

| archive-url = https://web.archive.org/web/20171114040621/http://www.techmat.vgtu.lt/~inga/Files/Quarteroni-SkaitMetod.pdf

| archive-date = 2017-11-14

| url-status = dead

}}

Necessary conditions for a numerical method to effectively approximate $F(x,y)=0$ are that $x\_n\; \backslash rightarrow\; x$ and that $F\_n$ behaves like $F$ when $n\; \backslash rightarrow\; \backslash infty$. So, a numerical method is called consistent if and only if the sequence of functions $\backslash left\; \backslash \{\; F\_n\; \backslash right\; \backslash \}\_\{n\; \backslash in\; \backslash mathbb\{N\}\}$ pointwise converges to $F$ on the set $S$ of its solutions:

: $$

\lim F_n(x,y+t) = F(x,y,t) = 0, \quad \quad \forall (x,y,t) \in S.

When $F\_n=F,\; \backslash forall\; n\; \backslash in\; \backslash mathbb\{N\}$ on $S$ the method is said to be strictly consistent.

Denote by $\backslash ell\_n$ a sequence of admissible perturbations of $x\; \backslash in\; X$ for some numerical method $M$ (i.e. $x+\backslash ell\_n\; \backslash in\; X\_n\; \backslash forall\; n\; \backslash in\; \backslash mathbb\{N\}$) and with $y\_n(x+\backslash ell\_n)\; \backslash in\; Y\_n$ the value such that $F\_n(x+\backslash ell\_n,y\_n(x+\backslash ell\_n))\; =\; 0$. A condition which the method has to satisfy to be a meaningful tool for solving the problem $F(x,y)=0$ is convergence:

: $$

\begin{align}

&\forall \varepsilon > 0, \exist n_0(\varepsilon) > 0, \exist \delta_{\varepsilon, n_0} \text{ such that} \\

&\forall n > n_0, \forall \ell_n : \| \ell_n \| < \delta_{\varepsilon,n_0} \Rightarrow \| y_n(x+\ell_n) - y \| \leq \varepsilon.

\end{align}

One can easily prove that the point-wise convergence of $\backslash \{y\_n\backslash \}\; \_\{n\; \backslash in\; \backslash mathbb\{N\}\}$ to $y$ implies the convergence of the associated method.

• Numerical methods for ordinary differential equations
• Numerical methods for partial differential equations

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Category:Numerical analysis