Stability of a computational algorithm

A partially resolving operator $L_m^h$, uniformly bounded in $h$ and $m$, describing the succession of steps in the computational algorithm for solving the equation

$$L^hu^h=f^h,$$

for example, a grid equation with step $h$ (cf. Closure of a computational algorithm). Stability of a computational algorithm guarantees weak influence of computational errors on the result of the calculation. However, the possibility is not excluded that the quantity $p(h)=\sup\|L_m^h\|$ grows comparatively slowly and a corresponding strengthening of the influence of computational errors for $h\to0$ remains practically admissible. The concept of stability of a computational algorithm has been made concrete in applications to grid-projection methods (cf. [4]) and in applications to iterative methods (cf. [6]). There are also other definitions of the stability of a computational algorithm (cf. e.g. [1], [3]).

References

[1]	I. [I. Babushka] Babuška, M. Práger, E. Vitásek, "Numerical processes in differential equations" , Wiley (1966)
[2]	N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[3]	M.K. Gavurin, "Lectures on computing methods" , Moscow (1971) (In Russian)
[4]	G.I. Marchuk, V.I. Agoshkov, "Introduction to grid-projection methods" , Moscow (1981) (In Russian)
[5]	A.A. Samarskii, A.V. Gulin, "Stability of difference schemes" , Moscow (1973) (In Russian)
[6]	A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian)