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\(L_{\infty}\)-boundedness of the finite element Galerkin operator for parabolic problems. (English) Zbl 0533.65071

The initial-boundary value problem for the evolution equation (the heat equation) \(\dot u-\Delta u=f\) in \(\Omega\times(0,T]\), \(\Omega \subseteq R^ N\) (\(N\leq 3)\) with boundary conditions \(u=0\) on \(\partial \Omega \times(0,T]\) and \(u_{t=0}=u_ 0\) in \(\Omega\) is considered. The main result of this paper is: In \(N=1,2,3\) space dimensions and for finite elements of order 4 or higher, i.e. at least cubics, the mapping \(u\to u_ h\) (where h is the mesh size of the approximation) is bounded in \(L_{\infty}\): \[ \| u_ h\|_{L_{\infty}(0,T,L_{\infty}(\Omega))} \leq C\| u\|_{L_{\infty}(0,T,L_{\infty}(\Omega))}. \]
Reviewer: J.Lovíšek

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35K05 Heat equation
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
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References:

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