Numerical method

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

${\displaystyle \left\{M_{n}\right\}_{n\in \mathbb {N} }=\left\{F_{n}(x_{n},y_{n})=0\right\}_{n\in \mathbb {N} },}$ 

with ${\displaystyle F_{n}:X_{n}\times Y_{n}\rightarrow \mathbb {R} }$ , ${\displaystyle x_{n}\in X_{n}}$  and ${\displaystyle y_{n}\in Y_{n}}$  for every ${\displaystyle n\in \mathbb {N} }$ . 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.^[1]

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

${\displaystyle \lim F_{n}(x,y+t)=F(x,y,t)=0,\quad \quad \forall (x,y,t)\in S.}$ 

When ${\displaystyle F_{n}=F,\forall n\in \mathbb {N} }$  on ${\displaystyle S}$  the method is said to be strictly consistent.^[1]

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

${\displaystyle {\begin{aligned}&\forall \varepsilon >0,\exists n_{0}(\varepsilon )>0,\exists \delta _{\varepsilon ,n_{0}}{\text{ such that}}\\&\forall n>n_{0},\forall \ell _{n}:\|\ell _{n}\|<\delta _{\varepsilon ,n_{0}}\Rightarrow \|y_{n}(x+\ell _{n})-y\|\leq \varepsilon .\end{aligned}}}$ 

One can easily prove that the point-wise convergence of ${\displaystyle \{y_{n}\}_{n\in \mathbb {N} }}$  to ${\displaystyle y}$  implies the convergence of the associated method is function.^[1]

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