A Finite Element Method For The Stokes Problem

Fluid dynamics is wildly used in the discipline of environtology?hydraulics?oceanography?meteorology.The famous Navier—Stokes equation which describes the conservation of momentum of viscous incompressible fluid was proposed in nineteenth century.Due to the nonlinear term in it,the equation can only be solved analytically under specific condition.But Navier—Stokes equation can be simplified into some solvable equations according to the real situation.Droplets' motion in the viscous fluid,for example,can be described by the Stokes equation because of the small Reynolds number.Analyzing the numerical solution of Stokes equation,not only can reveal partial characteristic of fluid mechanics,but also provide experience when deal with Navier—Stokes equation in the future.In order to solve Stokes equation,We come up with HDM(Hybrid Discontinuous Mixed)method which is based on HDG(Hybrid Discontinuous Galerkin)method and mixed finite element method.HDM method uses H(div)space to approximate the space of velocity,and corresponding Lagrange polynomial space to approximate the space of pressure,just as the way that mixed finite element method does.But the different part with respect to mixed finite method is,the velocity space and pressure space in HDM method are discontinuous between every two elements like the discontinuous finite element method,which makes the HDM method satisfy the property of local conservation.It is well known that local conservation is essential to the stability of numerical solution of transport problems.Besides,we come up with a new data structure MMF(Matrix Multiplication Form)to store the basis functions for HDM and other finite element method.The basis function can be scalar basis function or vector basis function belonging to H(div)space.Based on MMF,the computation of local stiffness matrix can be converted to BLAS 3 computation without any loops.Through the comparison of MATLAB codes,we can tell that the efficiency of code using MMF data structure is improving linearly along the refining mesh.As the efficiency goes higher,MMF can also simplify the complexity of developing FEM codes.Nowadays,computer architecture is becoming more and more sophisticated,and the implement of finite element method can't keep pace with the development of hardware and software.With the help of MMF data structure,the computation and optimization of local stiffness matrix are wholly accomplished by BLAS,which means the code developers don't have to modify the code according to new hardware or platform when the hardware is altered,all they need to do is changing the original BLAS to the corresponding one.Firstly,related research at home and abroad will be reviewed;Then we will explicate the demonstration of three conservation equations in fluid dynamics,and derive the Stokes equation under some assumptions;After that,we will introduce the HDM method and derive the corresponding weak form;Finally,we will introduce how to use MMF to store the scalar and vectorial basis function.Besides,we will use numerical experiment to verify the correctness of HDM method.