Fast, reliable orbital evolutions of compact objects around massive black holes will be needed as input for gravitational wave search algorithms in the data stream generated by the planned Laser Interferometer Space Antenna (LISA). Currently, the state of the art is a time domain code by that computes the gravitational self-force on a point particle in an eccentric orbit around a Schwarzschild black hole. Existing time domain codes take up to a few days to compute just one point in parameter space. In a series of articles, we advocate the use of a frequency domain approach to the problem of gravitational self-force (GSF) with the ultimate goal of orbital evolution in mind. Here, we compute the GSF for a particle in a circular orbit in Schwarzschild spacetime. We solve the linearized Einstein equations for the metric perturbation in Lorenz gauge. Our frequency domain code reproduces the time-domain results for the GSF up to ～1000 times faster for small orbital radii. In forthcoming companion papers, we will generalize our frequency domain computations of the GSF to include bound (eccentric) orbits in Schwarzschild spacetimes, where we will employ the method of extended homogeneous solutions. We will eventually extend our methods to attempt a frequency domain computation of the GSF in Kerr spacetime.
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