Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations
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Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations Natalia Kopteva1 · Torsten Linß2
Received: 3 March 2015 / Accepted: 17 January 2017 © Springer Science+Business Media New York 2017
Abstract Linear and semilinear second-order parabolic equations are considered. For these equations, we give a posteriori error estimates in the maximum norm that improve upon recent results in the literature. In particular it is shown that logarithmic dependence on the time step size can be eliminated. Semidiscrete and fully discrete versions of the backward Euler and of the Crank-Nicolson methods are considered. For their full discretizations, we use elliptic reconstructions that are, respectively, piecewise-constant and piecewise-linear in time. Certain bounds for the Green’s function of the parabolic operator are also employed. Keywords Parabolic problems · Maximum-norm a posteriori error estimates · Backward Euler · Crank-Nicolson · Elliptic reconstructions Mathematics Subject Classifications (2010) 65M15 · 65M60
Dedicated to Prof. Hans-G¨org Roos on the occasion of his 65th birthday Communicated by: Karsten Urban Torsten Linß
[email protected] Natalia Kopteva [email protected] 1
Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
2
Fakult¨at f¨ur Mathematik und Informatik, FernUniversit¨at in Hagen, Universit¨atsstraße 11, 58095 Hagen, Germany
N. Kopteva, T. Linß
1 Introduction Residual-type a posteriori error estimates in the maximum norm for parabolic equations have been given in a number of works [2, 5–7, 11, 12]. It appears that the error constants in all known error estimators of this type, both for semidiscrete in time and fully discrete methods, for linear and semilinear equations, exhibit logarithmic dependence on the local time step. By contrast, numerical results suggest that such logarithmic factors are an artefact of the analysis (see, e.g., [13]). The aim of this paper is to eliminate the logarithmic dependence on the time step, and hence obtain maximum-norm a posteriori error estimates that are sharper and in line with numerical experiments. This purpose is achieved by refining our recent analysis [12] (where all estimators still involve logarithmic factors); in particular, a more careful treatment of certain residual terms is introduced that is tailored to specific time discretizations. Consider a semilinear parabolic equation in the form Ku := ∂t u + Lu + f (·, ·, u) = 0
in Q := × (0, T ],
(1a)
with a second-order linear elliptic operator L = L(t) in a spatial domain ⊂ Rn ¯ × [0, T ] × R → R, subject to the with Lipschitz boundary, and some function f : initial condition ¯ u(x, 0) = u0 (x) for x ∈ , (1b) and the Dirichlet boundary condition u(x, t) = 0 for (x, t) ∈ ∂ × [0, T ].
(1c)
We assume that the intial and boundary data satify the zero-th order compatibility ¯ Under standard condition, i.e. u0 |∂ = 0, and that u0 is H¨older continuous in . assumptions on f and L, problem (1) possesses a u
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