The Weak Galerkin Finite Element Method For A Class Of The Incompressible Fluid

Fluid mechanics of porous media are widely used in astrophysics,weapon physics,natural transformation in scientific research,industrial production and engineering,and so on.As a link between theoretical analysis and experimental exploration,numerical simulation is an important research meanings.The study of numerical simulation plays an important role in understanding the general motion law and physical characteristics of porous media fluid.With the rapid development of computer,scientific calculation and post-processing techniques,the application of porous media fluids has been extended to nuclear physics,aerospace and engineering equipment,such as atmospheric movement,oceanic movement,exploitation and protection of assets in karst water deposits,etc.They urgently needs the continuous development and innovation of numerical simulation of fluid dynamics equations in porous media.Then,how to describe the law of fluid's movement.From 1687,the experiment of viscous flow had been made by Isac Newton.He found that the linear relationships between the resistance and gradient of velocity exist in almost all common fluid.These gave people an initial understanding of viscous flow.Then,Euler equation had been proposed in 1755.Later,after the efforts of many researchers,the ideal flow had been studied almost perfectly.However,there were a great difference of test result between ideal flow and actual flow,sometimes even opposite.Naiver and other specialists research the intermolecular forces in Euler equations.After that,George Gabriel Stokes use the viscosity coefficient to represent the intermolecular force.Finally,the fundamental equations of viscous fluid mechanics were established.At present,there are several numerical methods to solve fluid mechanics problems,such as,finite difference method[6,8,30,58,60,61,74],finite element method[10,25,26,69,80,101],finite volume method[16,17,18,33,36,95],discontinuous finite element[19,23,32,75,102],spectral analysis method[7,29]and weak Galerkin finite element method[13,15,26,41,44,45,68,76,83,92,103,104],and so on.However,there is a difficulty solving the flow problems.The low order element method?P₁\P₀or P₁\P₁?has the advantage of simple realization,high precision,and low computing cost.Its space doesn't satisfy inf-sup condition that the pressure functions nonphysical oscillation.In this paper,we use weak Galerkin finite element method to solve problems in fluid mechanics.Acceleration algorithm can further enhance the computational efficiency of numerical algorithm.The weak Galerkin finite element was proposed in 2011 to solve the second-order elliptic problems[81].Then,adding stabilizer[44,51,54,82]maintains the difference between classical partial differential operators and weak forms.The weak Galerkin finite element method is improved.The main idea of weak Galerkin finite element is that the classical differential operators are replaced by weak differential operators and weak functions represent classical functions.The advantage of weak Galerkin finite element method has two following points:1.The partition of domain can be arbitrary polygon or polyhedron.The mesh generation and numerical approximation are convenient and flexible.2.The form of weak functions is u_h={u₀,u_b},where u₀ and u_b denote the value of T and?T,respectively.The weak functions don't need smoothness.3.The scheme can be hybridized to reduce the degree of freedom of the whole discrete system in parallel.Following these properties,people have great interest in extension of weak Galerkin finite element method in other fields,such as,Brinkman equation[37,96],Darcy flow[15,41,86],Stokes equation[83,85,97,15,41,64],integro-differential equation[1,14,22,27,34,49,50,56,66,67,70,83,93],Biharmonic equation[46,53,99],random elliptic equation[38],parabolic problems[2,24,40,88,105],the stochastic partial differential problems[37,107],mixed weak Galerkin finite element method[82],and many other fields[5,12,35,39,47,59,72,79,84,90,94,97].In this paper,we use an effective technique-Schur complement to solve fluid problems.So the degree of the whole discrete system is reduced.This technique is successful to apply in[55,65,73,91,96,97]to analyze preprocessing.The main idea of Schur complement is that the boundary functions are replaced by internal functions.This paper has three parts.Weak Galerkin finite element method solves the incom-pressible Stokes equation in the first part.This method is a appropriate for polygonal mesh generation and easily constructs weak finite element space.So it is highly flexible and effi-cient.However,there are a great number of computation cost.We use Schur complement to reduce the degree of computation and maintain precision.The optimal convergent order of error of weak Galerkin finite element method are obtained.The Darcy-Stokes equation is solved by weak Galerkin finite element method in the latter part.In the second part,we show the Brinkman and Stokes equations which describes fluid flow in complex porous media.The permeability coefficient is highly variable,sometimes is large,sometimes is small.We derive on optimal error estimates in H¹and L²norm for velocity function and pressure function with semi-discrete and full-discrete weak Galerkin finite element schemes,respectively.The space of weak Galerkin finite element consists of polynomial-s of degree k-1 for pressure and polynomials of degree k?1 for the velocity.The velocity function on boundary of element T is k-degree polynomial.In the third part,we use weak Galerkin finite element to solve linear parabolic integro-differential equation with adding corresponding stabilizer.The semi-discrete and full-discrete weak Galerkin schemes are established,respectively.We derive the error estimates in H¹norm and L²norm.Several calculation results verify the correctness and validity of this method.