Geodesic
Estimates of the distance to the exact solution of parabolic problems based on local Poincaré type inequalities
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 44, Tome 425 (2014), pp. 7-34
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The goal of the paper is to derive two-sided bounds of the distance between the exact solution of the evolutionary reaction-diffusion problem with mixed Dirichlet–Robin boundary conditions and any function in the admissible energy space. The derivation is based upon special transformations of the integral identity, that defines the generalized solution. In order to obtain estimates with easily computable local constants we exploit classical Poincaré inequalities and Poincaré type inequalities for functions with zero mean boundary traces. The corresponding constants are estimated in [10] and [8]. Bounds of the distance to the exact solution contain only these constants associated with subdomains. It is proved that the bounds are equivalent to the energy norm of the error. 
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S. Matculevich; S. Repin. Estimates of the distance to the exact solution of parabolic problems based on local Poincaré type inequalities. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 44, Tome 425 (2014), pp. 7-34. http://geodesic.mathdoc.fr/item/ZNSL_2014_425_a1/

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