Finite element methods for symmetric hyperbolic equations

Abstract

Abstract. A finite difference method for the solution of symmetric positive differential equations has already been developped (Katsanis [4]). The finite difference solutions where shown to converge at the rate O (h 89 as h approaches zero, h being the maximum distance between two adjacent mesh points. Here we try to get a better rate of convergence, using a Rayteigh Ritz Galerkin method.We first give a "weak" formulation of the equations, slightly different from the usual one (Friedrichs [3]), in order to take into account the boundary conditions.We define a finite dimensional subspace V h of H 1 (~), in which we look for an approximate solution u h. We show that when the exact solution u is smooth enough, we get the error estimate: where I. t denotes the Euclidean norm in R P. Thus, as is the case for elliptic or parabolic equations, the problem of estimating the error is reduced to questions in approximation theory. When those results are applied to finite element methods, with polynomial approximations of degree =<k over each n-simplex we obtain a rate of convergence of O (h k) as h approaches zero, h being the supremum of the diameters of the n-simplices,

Error Estimates for the Numerical Identification of a Variable Coefficient

“…An approximation scheme using Galerkin's method is proposed in Frind and Pinder [4], although no error analysis is given, and a finite difference scheme is proposed and analyzed in Richter [7], Further references for various approaches to this problem can be found in the paper of Yoon and Yeh [9], We also note that although the approximate problem (Ph ) is based on viewing the underlying equation (1) as an elliptic equation for the pressure u(x), some of the analysis will be based on viewing (1) as a hyperbolic equation for the transmissivity a. In this regard, the work of Lesaint [5] has been useful.…”

Explicit Finite Element Methods for Linear Hyperbolic Systems

“…An example of (1)- (2) is the wave equation in two space dimensions: Of the many previous £nite element treatments of the general problem (1)- (2) and its related non-transient counterpart (e.g., [1], [5], [7], [8]), we know of none which is explicit, i.e., develops an approximate solution in an element by element fashion. This is our focus, in the setting of arbitrarily large m and N .…”

A Finite Difference Method for Symmetric Positive Differential Equations

“…Furthermore, it also gives a simple and unified numerical treatment for these problems (see e.g. [1,7,9]). Otherwise, if a numerical method applies directly to a PDE of mixed type, the treatment of the interface on which the PDE changes type is in general very difficult to handle.…”

“…Several numerical methods have been developed for Friedrichs's systems [7,9,10]. Friedrichs [5] was the first to propose a finite difference procedure for the numerical solutions of symmetric positive systems in rectangular regions.…”