Finite element methods for symmetric hyperbolic equations
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A finite difference method for the solution of symmetric positive differential equations has already been developped (Katsanis ). The finite difference solutions where shown to converge at the rateO(ith1/2) ash 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 ), in order to take into account the boundary conditions.
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 eachn-simplex we obtain a rate of convergence ofO(hk) ash approaches zero,h being the supremum of the diameters of then-simplices.
KeywordsFinite Element Method Error Estimate Finite Difference Mathematical Method Parabolic Equation
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