The plane-wave pseudopotential (PWPP) method is used to perform density-functional-theory (DFT) calculations for two grain boundaries in aluminum studied previously using the embedded-atom method (EAM) and high-resolution transmission electron microscopy: (1) a \ensuremath{\Sigma}=11 (11\ifmmode\bar\else\textasciimacron\fi{}3)129\ifmmode^\circ\else\textdegree\fi{} symmetric tilt boundary and (2) a \ensuremath{\Sigma}=3 (112) twin boundary. Results are presented for the grain-boundary energies and the electron-density distributions in the boundary regions. The DFT results confirm that the EAM provides a good description of structural properties. However, the DFT grain-boundary energies are a factor of 2 larger than the EAM values, consistent with discrepancies found previously for (111) stacking faults in aluminum. In addition, two sources of instability are identified in the Teter-Payne-Allan preconditioned conjugate-gradients algorithm [Phys. Rev. B 40, 12 255 (1989)] used to perform the PWPP calculations. Simple techniques are proposed that eliminate these instabilities and their utility is demonstrated for these grain-boundary calculations.