Advertisement

Numerische Mathematik

, Volume 21, Issue 3, pp 244–255 | Cite as

Finite element methods for symmetric hyperbolic equations

  • P. Lesaint
Article

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 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 [3]), in order to take into account the boundary conditions.

We define a finite dimensional subspaceV h ofH1(Ω), in which we look for an approximate solutionu h . We show that when the exact solutionu is smooth enough, we get the error estimate:
$$\left| {u - u_h } \right|L^2 (\Omega ) \leqq C\mathop {\inf }\limits_{v_h \in V_h } \left\{ {\left\| {u - v_h } \right\|H^1 (\Omega ) + \mathop {\sup }\limits_{w_h \in V_h } \frac{{\int\limits_\Gamma {\left| {u - v_h } \right|\left| {w_h } \right|d\Gamma } }}{{\left| {w_h } \right|L^2 (\Omega )}}} \right\}$$
where |·| denotes the Euclidean norm inR 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 eachn-simplex we obtain a rate of convergence ofO(hk) ash approaches zero,h being the supremum of the diameters of then-simplices.

Keywords

Finite Element Method Error Estimate Finite Difference Mathematical Method Parabolic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ciarlet, P. G., Raviart, P.-A.: General Lagrange and Hermite interpolation inR n with applications to finite element methods. Arch. Rat. Mech. Anal. Vol.46, 177–199 (1972)Google Scholar
  2. 2.
    Ciarlet, P. G., Raviart, P.-A.: Interpolation theory over curved elements, with applications to finite element methods. Computer Methods in Applied Mechanics and Engineering1, 217–249 (1972)Google Scholar
  3. 3.
    Friedrichs, K. O.: Symmetric positive linear differential equations. Comm. Pure Appl. Math.11, 333–418 (1958)Google Scholar
  4. 4.
    Katsanis, Th.: Numerical solution of symmetric positive differential equations. Math. Comp.22, 763–783 (1968)Google Scholar
  5. 5.
    Lax, P. D., Philipps, R. S.: Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math.13, 427–455 (1960)Google Scholar
  6. 6.
    Nécas, J.: Les méthodes directes en théorie des équations elliptiques. Masson et Cie Editeurs 1967Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • P. Lesaint
    • 1
  1. 1.Centre d'Etudes de LimeilVilleneuve-Saint-GeorgesFrance

Personalised recommendations