We establish a numerical solution to the time-dependent Schrödinger equation employing an adaptive, discontinuous spectral element basis that automatically adjusts to the requested precision. The explicit time evolution is accomplished by a band-limited, gradient-corrected, symplectic propagator and uses separated representations of operators for efficient computation in multiple dimensions. We illustrate the method calculating accurate bound and continuum transition probabilities along with the photoelectron spectra for H(1$s$), He${}^{+}$(1$s$), and Li${}^{2+}$(2$s$) in three dimensions and H${{}_2}^{+}$ in three and four dimensions under a two-cycle attosecond laser pulse with driving frequency of 36 eV and an intensity of $1\times {10}^{15}\text{W}/{\text{cm}}^2$.

• Received 6 July 2011
• Corrected 18 May 2012

DOI:https://doi.org/10.1103/PhysRevA.85.033403