Geometric foundations of numerical algorithms and symmetry

This paper outlines a new general construction, named "multi-space", that f orms the proper geometrical foundation for the numerical analysis of differ ential equations - in direct analogy with the role played by jet space as t he basic object underlying the geometry of differential equations. Applicat ion of the theory of moving frames leads to a general framework for constru cting symmetry-preserving numerical approximations to differential invarian ts and invariant differential equations.