Multi-domain spectral method for self-force calculations

Second-order self-force calculations will be critical for modelling extreme-mass-ratio inspirals, and they are now known to have high accuracy even for binaries with mass ratios $\sim 1:10$. Many of the challenges facing these calculations are related to slow convergence of spherical-harmonic (or spheroidal harmonic) mode sums in a region containing the small companion. In this paper, we begin to develop a multi-domain framework that can evade those problems. Building on recent work by Osburn and Nishimura, in the problematic region of spacetime we use a puncture scheme and decompose the punctured field equations into a basis of Fourier and azimuthal $m$ modes, avoiding a harmonic decomposition in the $\theta$ direction. Outside the problematic region, we allow for a complete spherical- or spheroidal-harmonic decomposition. As a demonstration, we implement this framework in the simple context of a scalar charge in circular orbit around a Schwarzschild black hole. Our implementation utilizes several recent advances: a spectral method in each region, hyperboloidal compactification, and an extremely high-order puncture.