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This book focuses on the finite element method influid flows. It is targeted at researchers, from those just starting out up topractitioners with some experience. Part I is devoted to the beginners who arealready familiar with elementary calculus. Precise concepts of the finiteelement method remitted in the field of analysis of fluid flow are stated,starting with spring structures, which are most suitable to show the conceptsof superposition/assembling. Pipeline system and potential flow sections showthe linear problem. The advection-diffusion section presents the time-dependentproblem; mixed interpolation is explained using creeping flows, and elementarycomputer programs by FORTRAN are included. Part II provides information on recentcomputational methods and their applications to practical problems. Theories ofStreamline-Upwind/Petrov-Galerkin (SUPG) formulation, characteristicformulation, and Arbitrary Lagrangian-Eulerian (ALE) formulation and others arepresented with practicalresults solved by those methods.