# Finite element methods for symmetric hyperbolic equations

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## 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(ith*^{1/2}) 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 Rayleigh 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.

*V*

_{ h }of

*H*

^{1}(Ω), 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:

*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.

## Keywords

Finite Element Method Error Estimate Finite Difference Mathematical Method Parabolic Equation## Preview

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## References

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