Numerical Methods For Some Models In Fluid Mechanics

It is well known that many models in fluid mechanics are convection-dominateddiffusion problems, using the standard finite element method to model them, onecan only get an unstable scheme, which produces severely oscillating approximatesolutions. In order to get stable approximate solutions and model the practicalproblems better, in this paper, we consider some different stabilized finite elementsimulations for the parabolic, the elliptic and the two dimensional scdiment trans-port equations, which are convection-dominated diffusion problems.Chapter 1 is the Introduction.In Chapter 2, we consider a residual-free bubbles finite element simulation forthe parabolic convection-dominated diffusion equationUsing the standard finite element method to model the parabolic problem, onecan only get an unstable scheme since it is convection-dominated. To overcomethese defects of the standard method, we present a new stabilized finite element method by use of residual-free bubbles, which has been proposed for the ellipticproblems successfully. At last, the L²-error estimate was proved.In Chapter 3, we consider a residual-free bubbles-mixed finite element simula-tion for the elliptic convection-dominated diffusion equationThe existence and uniqueness of the discrete solution are proved, Finally L²-error,H¹-error estimate are derived.In Chapter 4, we consider a numerical simulation for the sediment transportequations, which is governed by the following systemWater continuity equation(?)Z/(?)t+(?)(Hu)/(?)x+(?)(Hv)/(?)y=0Water dynamic equations(?)u/(?)t+u(?)u/(?)x+v(?)u/(?)y+g(?)Z/(?)x+gu(u²+v²)^1/2/c²H-γtΔu=0,(?)v/(?)t+u(?)v/(?)x+v(?)v/(?)y+g(?)Z/(?)y+gv(u²+v²)^1/2/c²H-γtΔv=0,Silt transport equation(?)S/(?)t+u(?)S/(?)x+v(?)S/(?)y=C₀ΔS=αω/H(S-S_*),The equation of bottom topography changeγ′(?)Z₀/(?)t=αω(S-S_*),we simulate the Water continuity equation by a Finite Difference-StreamlineDiffusion method, The other equations are simulated by a characteristics finiteelement method, At last, by strict analysis, we obtain L²-error estimates for theunknown functions.