Taming the Edwald sum in the computer simulation of charged systems
Journal of Computational Physics
A boundary integral approach to unstable solidification
Journal of Computational Physics
Fast potential theory. II: Layer potentials and discrete sums
Journal of Computational Physics
Locally corrected multidimensional quadrature rules for singular functions
SIAM Journal on Scientific Computing
Expokit: a software package for computing matrix exponentials
ACM Transactions on Mathematical Software (TOMS)
A fast 3D Poisson solver of arbitrary order accuracy
Journal of Computational Physics
A Method for Computing Nearly Singular Integrals
SIAM Journal on Numerical Analysis
Forward and inverse wave propagation using bandlimited functions and a fast reconstruction algorithm for electron microscopy
First-order system least squares (FOSLS) for coupled fluid-elastic problems
Journal of Computational Physics
A semi-Lagrangian contouring method for fluid simulation
ACM Transactions on Graphics (TOG)
Overdetermined Elliptic Systems
Foundations of Computational Mathematics
Locally corrected semi-Lagrangian methods for Stokes flow with moving elastic interfaces
Journal of Computational Physics
Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems
Journal of Computational Physics
A geometric nonuniform fast Fourier transform
Journal of Computational Physics
Spectrally accurate fast summation for periodic Stokes potentials
Journal of Computational Physics
Higher order numerical discretizations for exterior and biharmonic type PDEs
Journal of Computational and Applied Mathematics
Hi-index | 31.47 |
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.