Finite Element Analysis in Fluid Mechanics

Anastasios Raptis (), Konstantina Kyriakoudi () and Michail A. Xenos ()
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Anastasios Raptis: University of Ioannina
Konstantina Kyriakoudi: University of Ioannina
Michail A. Xenos: University of Ioannina

A chapter in Mathematical Analysis and Applications, 2019, pp 481-510 from Springer

Abstract: Abstract In the last decades, the finite element method (FEM) in fluid mechanics applications has gained substantial momentum. FE analysis was initially introduced to solid mechanics. However, the progress in fluid mechanics problems was slower due to the non-linearities of the equations and inherent difficulties of the classical FEM to deal with instabilities in the solution of these problems. The main goal of this review is to analyze FEM and provide the theoretical basis of the approach mainly focusing on parabolic type of problems applied in fluid mechanics. Initially, we analyze the basics of FEM for the Stokes problem and we provide theorems for uniqueness and error estimates of the solution. We further discuss FE approaches for the solution of the advection–diffusion equation such as the stabilized FEM, the variational multiscale method, and the discontinuous Galerkin method. Finally, we extend the analysis on the non-linear Navier–Stokes equations and introduce recent FEM advancements.

Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-31339-5_18

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DOI: 10.1007/978-3-030-31339-5_18

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