  We show that a pseudospectral representation of the wavefunction using
multiple spatial domains of variable size yields a highly accurate, yet
efficient method to solve the time-dependent Schr\"odinger equation. The
overall spatial domain is split into non-overlapping intervals whose size is
chosen according to the local de Broglie wavelength. A multi-domain weak
formulation of the Schr\"odinger equation is obtained by representing the
wavefunction by Lagrange polynomials with compact support in each domain,
discretized at the Legendre-Gauss-Lobatto points. The resulting Hamiltonian is
sparse, allowing for efficient diagonalization and storage. Accurate time
evolution is carried out by the Chebychev propagator, involving only sparse
matrix-vector multiplications. Our approach combines the efficiency of mapped
grid methods with the accuracy of spectral representations based on Gaussian
quadrature rules and the stability and convergence properties of polynomial
propagators. We apply this method to high-harmonic generation and examine the
role of the initial state for the harmonic yield near the cutoff.
