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Finite Element Analysis in Fluid Mechanics

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Mathematical Analysis and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 154))

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, Ioannina, Greece

    Anastasios Raptis, Konstantina Kyriakoudi & Michail A. Xenos

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  1. Anastasios Raptis
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  2. Konstantina Kyriakoudi
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  1. Department of Mathematics, Zografou Campus, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

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Raptis, A., Kyriakoudi, K., Xenos, M.A. (2019). Finite Element Analysis in Fluid Mechanics. In: Rassias, T., Pardalos, P. (eds) Mathematical Analysis and Applications. Springer Optimization and Its Applications, vol 154. Springer, Cham. https://doi.org/10.1007/978-3-030-31339-5_18

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