We describe an algorithm for mixed (class II) simulation of electron multiple elastic scattering using numerical differential cross-sections (DCS), which is applicable in a wide energy range, from ~100 eV to ~1 GeV. DCSs are calculated by partial-wave analysis, or from a suitable high-energy approximation, and tabulated on a grid of scattering angles and electron energies. The size of the required DCS table is substantially reduced by means of a change of variable that absorbs most of the energy dependence of the DCS. That is, the scattering angle h is replaced by a variable u, whose probability distribution function varies smoothly with the kinetic energy of the electron. A fast procedure to generate random values of u in restricted intervals is described. The algorithm for the simulation of electron transport in pure elastic scattering media (with energy-loss processes switched off) is obtained by combining this sampling procedure with a simple model for space displacements. The accuracy and stability of this algorithm is demonstrated by comparing results with those from detailed, event by event, simulations using the same DCSs. A complete transport code, including energy losses and the production of secondary radiations, is obtained by coupling the present elastic scattering simulation algorithm to the general-purpose Monte Carlo program PENELOPE. Simulated angular distributions of MeV electrons backscattered in aluminium and gold are in good agreement with experimental data.