Postprocessing the Galerkin Method: The Finite-Element Case

Abstract

Abstract.A postprocessing technique, developed earlier for spectral methods, is extended here to Galerkin finite-element methods for dissipative evolution partial differential equations. The postprocessing amounts to solving 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). The error analysis of the new method is performed in L 2 and in L ∞ norms. , an inexpensive novel technique to increase the accuracy and computational efficiency of Fourier spectral methods was developed. In this paper, we present a general technique to improve the convergence rate of Galerkin methods which applies to finite-element methods (and in the particular case of Fourier-Galerkin methods it coincides with that in [21]; see also [37]). We do this with no direct address to [13].Let Ω ⊂ R d be a domain with a smooth boundary. We consider dissipative PDEs (see, for instance, [8], [24], [25], [41]) which can be written as evolution systems of the form

The Postprocessed Mixed Finite-Element Method for the Navier–Stokes Equations: Refined Error Bounds

“…Numerical experiments in [6], [16], [19], [17], [26], [28] have repeatedly shown, for the different discretizations considered, that the increase in accuracy and convergence rate predicted by the theory is also seen in practice (provided errors arising from the time discretization are kept sufficiently small). Nevertheless, in the present paper we give an explanation of this fact; that is, the gain in (spatial) accuracy in the postprocessing step takes place independently of errors arising from the temporal discretization being present or not.…”

Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

“…In this section we use an efficient numerical method, the postprocessing Galerkin method, which was introduced in [38], [39], [40], [65], to improve the error estimates (3.47) and (4.57).…”

Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

“…Lakkis and Nochetto [21] used ad-hoc geometric energy norms to derive conditional a posteriori estimates for quasilinear equations such as the mean curvature flow of graphs. De Frutos and Novo [11] proved a posteriori error estimates of the p-version of space discrete schemes for parabolic equations; a similar function to the elliptic reconstruction and its improved approximation properties is used by García-Archilla and Titi [17]. Finally, for applications of suitable reconstructions to time discretizations of various type, we refer to [3,23].…”