The dynamics of multiphoton processes in $\mathrm{H}_{2}{}^{+}$ ions interacting with strong femtosecond fields is investigated by solving the time-dependent Schr\"odinger equation in the Born-Oppenheimer approximation. We use a spectral method with prolate spheroidal coordinates to represent the electronic wave functions. The theoretical approach is of spectral type in which the electronic and vibrational wave functions are represented with ${L}^{2}$ basis functions. In order to check the accuracy of these wave functions we have investigated the cases of the multiphoton ionization and dissociation of $\mathrm{H}_{2}{}^{+}$ at long wavelengths. First, we calculate the ionization probabilities and the electron spectra at $228\phantom{\rule{0.3em}{0ex}}\text{nm}$ and $400\phantom{\rule{0.3em}{0ex}}\text{nm}$, and we compare our results with other calculations when available. The effect of the initial dispersion of the internuclear distance $R$ in the equilibrium region is discussed in the context of short pulses. Then, we study the dissociation of $\mathrm{H}_{2}{}^{+}$ at $350\phantom{\rule{0.3em}{0ex}}\text{nm}$ and we compare our results with other recent calculations. Finally, the case of two-photon ionization of $\mathrm{H}_{2}{}^{+}$ at short wavelength is investigated, and we examine the electronic and vibrational spectra that have been obtained.

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