A new method for computing eigenmodes of a laser resonator by the use of finite element analysis (FEA) is presented. For this purpose, the scalar wave equation [Δ + k2]E(x,y,z) = 0 is transformed into a solvable 3D eigenvalue problem by separating out the propagation factor exp(-ikz) from the phasor amplitude E(x,y,z) of the time-harmonic electrical field. For standing wave resonators, the beam inside the cavity is represented by a two-wave ansatz. For cavities with parabolic optical elements the new approach has successfully been verified by the use of the Gaussian mode algorithm. For a DPSSL with a thermally lensing crystal inside the cavity the expected deviation between Gaussian approximation and numerical solution could be demonstrated clearly.