GLOBAL AND INTERIOR POINTWISE BEST APPROXIMATION RESULTS FOR THE GRADIENT OF GALERKIN SOLUTIONS FOR PARABOLIC PROBLEMS

In this paper we establish a best approximation property of fully discrete Galerkin solutions of second-order parabolic problems on convex polygonal and polyhedral domains in the L∞(I; W1,∞(Ω)) norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous...

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Veröffentlicht in:SIAM journal on numerical analysis 2017-01, Vol.55 (4), p.2025-2049
Hauptverfasser: LEYKEKHMAN, DMITRIY, VEXLER, BORIS
Format: Artikel
Sprache:eng
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Zusammenfassung:In this paper we establish a best approximation property of fully discrete Galerkin solutions of second-order parabolic problems on convex polygonal and polyhedral domains in the L∞(I; W1,∞(Ω)) norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors [D. Leykekhman and B. Vexler, Numer. Math., 135 (2017), pp. 923-952]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish an interior best approximation property that shows more local dependence of the error at a point.
ISSN:0036-1429
1095-7170
DOI:10.1137/16M1080252