The xLab Series of Mathematica-based high precision 2-D elliptic PDE solvers

摘要

A basis for solutions to two-dimensional elliptic partial differential equations is formed from closed-form solutions suggested by each segment of a region's boundary in isolation. Each component solution attains harmonic boundary conditions on the associated segment from which it decays into the region. Harmonic matching techniques are used to compute the combinations of closed-form solutions that deliver harmonic values on a prescribed segment and zero boundary values on all other segments. These combinations permit the coefficients for an arbitrary set of boundary conditions to be quickly determined from their Fourier expansions on the various segments. Navier's equation of equilibrium elasticity and Mathematica's big-number arithmetic are used as a particularly uncompromising development platform. The usual efficiencies arise when machine numbers are used. The regions may contain cracks and circular and elliptic holes and may either be enclosed by a polygonal boundary or attain a constant state of stress at infinity. The notebook interface caters to users with little experience with Mathematica.