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Stability, analyticity, and almost best approximation in maximum norm for parabolic finite element equations. (English) Zbl 0932.65103
Semidiscrete solutions in quasi-uniform finite element spaces of order \(O(h^r)\) of the initial boundary value problem with Neumann boundary conditions are considered for a second-order parabolic differential equation with time-independent coefficients in a bounded domain in \(\mathbb{R}^N\). It is shown that the semigroup on \(L_\infty\), defined by the semidiscrete solution of the homogeneous equation, is bounded and analytic uniformly in \(h\). It is also shown that the semidiscrete solution of the inhomogeneous equation is bounded in the space-time \(L_\infty\)-norm, modulo a logarithmic factor for \(r=2\). A corresponding almost best approximation property is also presented.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
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