Efficient solution of elliptic systems 

Linear constant-coefficient elliptic systems of 
partial differential equations occur frequently
in computational science, and challenge
standard solution methodologies.  We present 
new ``locally-corrected spectral boundary integral
methods'' for their accurate discretization 
and fast solution.

Arbitrary elliptic systems such as Poisson, Yukawa, Helmholtz, 
Maxwell, Stokes and elasticity equations are transformed
to an overdetermined first-order form amenable to
unified solution. A simple well-conditioned
boundary integral equation, for solutions satisfying 
arbitrary boundary conditions posed on complex interfaces, 
is derived from first principles. A fast Ewald summation
formula for the periodic fundamental solution is derived by
Fourier analysis, linear algebra, and local asymptotic 
expansion. Ewald summation evaluates box, volume and layer
potentials efficiently, and separates the boundary integral 
equation into a low-rank system with regular spectral structure,
followed by a simple local correction formula.