โ Scribed by John Strain
No coin nor oath required. For personal study only.
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.
๐ SIMILAR VOLUMES
In this paper we propose preconditioners for spectral element methods for elliptic and parabolic problems. These preconditioners are constructed using separation of variables and are easy to invert. Moreover they are spectrally equivalent to the quadratic forms which they are used to approximate.
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