Geodesic
Convergence of discretized attractors for parabolic equations on the line
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 14-41

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We show that, for a semilinear parabolic equation on the real line satisfying a dissipativity condition, global attractors of time-space discretizations converge (with respect to the Hausdorff semi-distance) to the attractor of the continuous system as the discretization steps tend to zero. The attractors considered correspond to pairs of function spaces (in the sense of Babin–Vishik) with weighted and locally uniform norms (taken from Mielke–Schneider) used for both the continuous and discrete systems. 
@article{ZNSL_2004_318_a1,
     author = {W.-J. Beyn and V. S. Kolezhuk and S. Yu. Pilyugin},
     title = {Convergence of discretized attractors for parabolic  equations on the line},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {14--41},
     publisher = {mathdoc},
     volume = {318},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a1/}
}
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W.-J. Beyn; V. S. Kolezhuk; S. Yu. Pilyugin. Convergence of discretized attractors for parabolic  equations on the line. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 14-41. http://geodesic.mathdoc.fr/item/ZNSL_2004_318_a1/