Local error estimates for a finite element method for hyperbolic and convection-diffusion equations. (English) Zbl 0757.65104

Falk, Richard S.; Richter, Gerard R.

SIAM J. Numer. Anal. 29, No. 3, 730-754 (1992).

Local error estimates are derived for a finite element method for two- dimensional hyperbolic and convection dominated equations. An approach for establishing local error estimates for the continuous method of W. H. Reed and T. R. Hill [Triangular mesh methods for the neutron transport equation. Los Alamos Sci. Laboratory Rep. LA-UR-73-479 (1973)] is used that is also applicable to the discontinuous Galerkin method, thus underscoring the close interrelationships between the two methods.
A continuous piecewise polynomial approximation of degree \(\geq 2\) over a triangulation of \(\Omega\subset\mathbb{R}^ 2\) is generated. It is shown that crosswind propagation of the numerical solution is limited to a distance \(O(\sqrt{h}\log h^{-1})\), where \(h\) is the mesh size. A rather simple test function depending only on the crosswind variable \(t\) is applied for the analysis, what leads directly to stability results expressed in terms of this variable.

Reviewer: K.Zlateva (Russe)

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L45	Initial value problems for first-order hyperbolic systems

Keywords:

convection-diffusion equations; hyperbolic equations; convergence; Local error estimates; finite element method; discontinuous Galerkin method; stability