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Adaptive finite elements for flow problems with moving boundaries. I. Variational principles and a posteriori estimates. (English) Zbl 0583.76025

New self-adaptive finite element methods for the numerical analysis of transient flow problems with moving boundaries are presented. The general class of physical problems to which these methods apply include problems of the flow of viscous fluids through ducts with time-varying boundaries and the transient conduction of heat over domains which vary with time. These methods should be useful in the study of general fluid-structure interaction problems. The present exposition is the first in a series on this subject and is limited to adaptive schemes which use p-type (polynomial enrichment) strategies to improve the local accuracy of the approximation without refining the mesh or moving nodes. Specifically considered are space-time variational principles for flow problems with moving boundaries, a posteriori error estimates, p-versions of space time finite elements, with applications to a class of incompressible, viscous flow problems and heat conduction with time-varying boundaries. Both triangular and quadrilateral finite elements are employed, over which polynomial shape functions are defined by polynomials of degree p, and in the examples discussed here, \(1\leq p\leq 3\). The results of several numerical experiments on various model problems are given.
Reviewer: W.Ames

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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