A rotationally symmetric inhomogeneously loaded open hemispherical resonator is analyzed using the finite-difference (FD) frequency-domain method. State-of-the art or new techniques are proposed to achieve high accuracy and efficiency of computations. These include applying the Galerkin method followed by an inexact shift-and-inverse Lanczos technique with an approximate starting eigenvector for selective computation of a single desired high-order mode, and compensating numerical dispersion error with the modified FD formulas. As a result, the final value of the computed resonant frequency for a desired mode agrees within 0.1% with the measured value and the computations involving a few 100000 unknowns are carried out in minutes on a personal computer.