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Adaptive Lagrange-Galerkin methods for unsteady convection-diffusion problems. (English) Zbl 0957.65085

The authors derive an a posteriori error bound for the Lagrange-Galerkin discretisation of an unsteady (linear) convection-diffusion problem, assuming only that the underlying space-time mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method. Moreover, based on this error bound, the authors designed an adaptive algorithm to ensure global control of the error with respect to a predetermined tolerance.

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
35K15 Initial value problems for second-order parabolic equations
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