Finite Element Method For The Stationary Darcy-Brinkman Equations

The Darcy-Brinkman equation is a complex multiphysics problem with coupling velocity,temperature,pressure and concentration.This model describes the double diffusion phenomenon of heat and mass transfer in saturated porous media which is used in modern industrial and technological applications,such as: solid oxide fuel cell cooling,oil reservoir recovery,pollutant migration and treatment in groundwater,etc.It is of practical significance to study the numerical calculation method of the model.The DarcyBrinkman equation is a multi-variable coupled nonlinear system,so there are many difficulties in approximating and analyzing the equation.This thesis studies the finite element method of the stationary Darcy-Brinkman equation.First,an appropriate auxiliary equation and compression operator are constructed based on the basic assumptions of the region and boundary,and the fixed point theorem is used to prove the existence and uniqueness of the solution of the stationary Darcy-Brinkman equation.Then an estimate of the regularity of the weak solution is given.Next,the nonlinear system generated by the discretization of the finite element method is solved by the Newton iteration method.The stability and error estimation of the numerical scheme under different norms for velocity,temperature,concentration and pressure are analyzed which reveal the relationship between the error,the discrete grid and the parameters of the model.Finally,the article gives relevant examples to verify the convergence order and effectiveness of the numerical method.