Die Anwendung der Iterierten Defektkorrektur auf Variationsmethoden für elliptische Randwertprobleme

Abstract

--Zusammenfassung The Application of Iterated Defect Correction to Variational Methods for Elliptic Boundary ValueProblems. We construct a method which makes it possible to apply the idea of iterated defect correction to finite element methods. The construction is motivated heuristically. We believe that the significance of our method lies in the possibility to write "metaalgorithms" for existing finite element program packages which entail a substantial improvement of the accuracy of these program packages. The efficiency of the method is demonstrated in a number of examples. Wir geben ein Verfahren an, das es erm6glicht, den Gedanken der Iterierten Defektkorrektur auf Methoden der Finiten Elemente anzuwenden. Das Verfahren wird heuristisch motiviert. Wir glauben, dab die Bedeutung unseres Verfahrens in der M6glichkeit liegt, ffir bestehende Finite-ElementeProgrammpakete ,,Metaalgorithmen" zu scbreiben, die eine betr/ichtliche Steigerung der Genauigkeit dieser Programmpakete bewirken. Die Effizienz des Verfahrens wird an einigen Beispielen demonstriert.

Modified characteristics mixed defect-correction finite element method for the time-dependent Navier–Stokes problems

“…Due to its good efficiency, there are many works devoted to this method, e.g. the convection-diffusion equation, [24] adaptive refinement for the convection-diffusion problems, [25] adaptive defect-correction methods for the viscous incompressible flow, [26] two-parameter defect-correction method for computation of the steady-state viscoelastic fluid flow, [27] variational methods for the elliptic boundary value problems, [28] defect-correction parameter-uniform numerical method for a singularly perturbed convection-diffusion problem, [29] the convection-dominated flow, [30] finite volume local defect-correction method for solving the transport equation, [31] the singular initial value problems, [32] the time-dependent Navier-Stokes equations, [33][34][35] the stationary Navier-Stokes equation, [36] second-order defect-correction scheme, [37] finite element eigenvalues with applications to quantum chemistry, [38] and so on. In [28], a method which makes it possible to apply the idea of iterated defect correction to finite element methods was given.…”

Convergence Analysis for the Iterated Defect Correction Scheme of Finite Element Methods on Rectangle Grids

“…We consider the iterated defection correction scheme of finite element methods proposed in [7] for the following second order elliptic equation…”

A defect-correction method for unsteady conduction convection problems I: spatial discretization

“…In this method, the nonlinear equations with an added artificial viscosity term on a finite element grid are solved and these solutions are corrected on the same grid using a linearized defect-correction technique. Due to its good efficiency, there are many works devoted to this method, e.g., convection-diffusion equation [6], adaptive refinement for convection-diffusion problems [2], adaptive defect correction methods for viscous incompressible flow [7], two-parameter defect-correction method for computation of steady-state viscoelastic fluid flow [5], variational methods for elliptic boundary value problems [8], defect-correction parameter-uniform numerical method for a singularly perturbed convection-diffusion problem [12], convection-dominated flow [18], finite volume local defect correction method for solving the transport equation [23], singular initial value problems [22], the time-dependent Navier-Stokes equations [24,25,33], the stationary Navier-Stokes equation [26], second order defect correction scheme [31], finite element eigenvalues with applications to quantum chemistry [37], common structural principle of the defect-correction technique [38] and so on. In [8], a method which makes it possible to apply the idea of iterated defect correction to finite element methods was given.…”