Euler–Poisson–Darboux Equation

Abstract

In this work we deal primarily with the derivation of various convergence estimates for some semidiscrete and fully discrete procedures which might be used in the approximation of exact solutions of initial-boundary value problems with homogeneous Dirichlet boundary conditions for the Euler-Poisson- Darboux equation. These procedures include the ordinary Galerkin method based on conforming finite element subspaces as well as certain methods which do not require such restrictions. Although the equation is of hyperbolic type, the results are somewhat analogous to those known for parabolic equations. This is due to the presence of a limited smoothing property. This paper contains L2 estimates, maximum norm estimates, negative norm estimates, interior estimates of difference quotients and superconvergence estimates of the error. Most of the proofs are based on different modifications of an energy method, some of them intrinsically depend on the Weinstein recursion formulae and on known results for the associated elliptic problem. Several of these estimates are obtained for positive time under weak assumptions on the initial data.