High-Order, Fast-Direct Methods for Separable Elliptic Equations

The Rayleigh–Ritz–Galerkin method with tensor product B-splines yields high-order discretiz-ations for elliptic partial differential equations. For smooth problems the resulting linear system of equations is both smaller and denser than the corresponding systems for lower order discretizations. However, several fast direct methods are known for solving these low order systems when the partial differential equation is separable. In this paper we show how to extend the matrix decomposition technique to yield a fast direct method for high-order, finite-element discretizations.