The search for stable finite element methods for simple relativistic systems

Abstract

Three finite element methods are applied to the describing static, spherically symmetric fluid bodies in general relativity. The usual Ritz method with linear splines is unconditionally unstable, although the instability is mild. As most Ritz and Galerkin methods will produce similar unstable methods a weighted residual method with cubic Lagrange splines is tried. This again proves unstable, and it seems that this approximation is inherently inappropriate for hyperbolic equations. However, the cubic Hermite spline approximation, with some modification, proves to be both A-stable and S-Stable, and is accurate to order h 3.