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Superconvergence of a 3D finite element method for stationary Stokes and Navier‐Stokes problems
Author(s) -
Matthies G.,
Skrzypacz P.,
Tobiska L.
Publication year - 2005
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20058
Subject(s) - superconvergence , mathematics , discretization , finite element method , piecewise , mathematical analysis , interpolation (computer graphics) , lagrange polynomial , poisson's equation , stokes problem , boundary value problem , boundary (topology) , partial differential equation , piecewise linear function , animation , physics , computer graphics (images) , computer science , polynomial , thermodynamics
For the Poisson equation on rectangular and brick meshes it is well known that the piecewise linear conforming finite element solution approximates the interpolant to a higher order than the solution itself. In this article, this type of supercloseness property is established for a special interpolant of the Q 2 − P 1 discelement applied to the 3D stationary Stokes and Navier‐Stokes problem, respectively. Moreover, applying a Q 3 − P 2 discpostprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that inhomogeneous boundary values can be approximated by the Lagrange Q 2 ‐interpolation without influencing the superconvergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost‐benefit analysis between the two third‐order methods, the post‐processed Q 2 − P 1 discdiscretization, and the Q 3 − P 2 discdiscretization is carried out. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005