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Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions. (English) Zbl 1281.65121
The authors consider a semilinear second-order parabolic equation $$\partial_t u + {\mathcal L} u +f(x,t,u)= 0$$ with a linear operator $${\mathcal L}$$ and Dirichlet boundary conditions. The purpose of their work is to obtain computable a posteriori error estimates for fully discrete methods for the equation. First-order backward Euler, second-order Crank-Nicolson, and discontinuous Galerkin ($$dG(r)$$, $$r\geq 1$$, with Radau quadrature) methods are considered for the time discretization. Error estimates are given in maximum norm. Distinct feature of their analysis is the use of elliptic reconstructions that are piecewise-polynomial of degree $$p-1$$ in time, where $$p$$ is the order of the time discretization. Section 2 contains the introduction of Green’s function and a stability lemma that is crucial to the error analysis. In Section 3, the authors summarize the results of the semidiscrete methods (no spatial discretization) for the parabolic problem and elaborate on each method in the subsequent sections. A summary of the results for fully discrete methods is included in Section 8. Issues such as the computability of the estimates and elliptic reconstruction are also addressed in this section. Details of the fully discrete case of each method is covered in the next few sections. Results obtained in the article are also applied to a model equation with $$L =-\varepsilon^2\Delta u$$ in two different regimes: one with $$\varepsilon=1$$, and another with $$\varepsilon\ll 1$$ which yields a singularly perturbed reaction-diffusion equation whose solutions may exhibit sharp layer phenomena. The estimates in the latter case show an explicit dependence of the error on the parameter $$\varepsilon$$.

##### MSC:
 65M15 Error bounds 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 35K58 Semilinear parabolic equations 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 35K57 Reaction-diffusion equations 35B25 Singular perturbations in context of PDEs
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