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An optimal estimate for the local discontinuous Galerkin method. (English) Zbl 0946.65072
Cockburn, Bernardo (ed.) et al., Discontinuous Galerkin methods. Theory, computation and applications. 1st international symposium on DGM, Newport, RI, USA, May 24-26, 1999. Berlin: Springer. Lect. Notes Comput. Sci. Eng. 11, 285-290 (2000).
Summary: \(L^2\) error estimates for the local discontinuous Galerkin (LDG) method have been theoretically proven for linear convection diffusion problems and periodic boundary conditions. It has been proven that when polynomials of degree \(k\) are used, the LDG method has a suboptimal order of convergence \(k\). However, numerical experiments show that under a suitable choice of the numerical flux, higher order of convergence can be achieved.
In this paper, we consider Dirichlet boundary conditions and we show that the LDG method has an optimal order of convergence \(k+1\).
For the entire collection see [Zbl 0935.00043].

MSC:
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
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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