Locally-corrected spectral methods and overdetermined elliptic systems

We present fast locally-corrected spectral methods for linear constant-coefficient elliptic systems of partial differential equations in d-dimensional periodic geometry. First, arbitrary second-order elliptic systems are converted to overdetermined first-order systems. Overdetermination preserves ellipticity, while first-order systems eliminate mixed derivatives, resolve convection–diffusion conflicts, and simplify derivative computations. Second, a periodic fundamental solution is derived by Fourier analysis and mollified for rapid convergence, independent of the regularity of the elliptic problem. Third, a new Ewald summation technique for first-order elliptic systems locally corrects the mollified solution to achieve high-order accuracy. We also discuss second-kind boundary integral equations based on single layer potentials formed with the mollified and corrected fundamental solution, which form a useful toolkit for solving general elliptic boundary value problems in general domains. The resulting spectral methods provide highly accurate solutions and derivatives for periodic problems.