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Postprocessing the Galerkin method: The finite-element case. (English) Zbl 0952.65078

A postprocessing technique, developed earlier for spectral methods, is extended to Galerkin finite-element methods for dissipative evolution partial differential equations. The postprocessing amounts to solve a linear elliptic problem on a finer grid (or higher-order space) once the time integration on the coarser mesh is completed. This technique increases the convergence rate of the finite-element method to which it is applied, and this is done at almost no additional computational cost. The numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied (say, quadratic finite elements) as well as standard methods of similar order of convergence as the postprocessed one (say, cubic finite elements). An error analysis of the new method is performed in \(L^{2}\) and in \(L^{\infty } \) norms.

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
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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