Finite Volume Element Methods For The Stokes Equations

In this text, we introduce the finite volume element method and the finiteelement method for the stokes equations,which have the compensating terms.The Stokes equation is stationary in two dimensions,as follows:where is a bounded open convex polyhedral with boundary .μiscoe?cient of viscosity ,whereμ= 1. u is the velocity, f is the body force,andp is pressure.Firstly,we give the definition of Kh ,which is the original partitions.In thefinite volume element method,we still give the definition of K_h^* inΩ,which isthe dual partitions.Then we describe the definition of the test function spaceand the definition of the trial function space,that is:The definitions of the test function space U_h^* and P_h^* associated with originalpartition K_h^* are The finite element method,which we solve the equation (0.0.1)-(0.0.3) withusing,is (uh,ph)∈(Uh,Ph) such thatAs in [1], we define a linear map Th : Uh→U_h^*,which satisfiswhere {χj} is the basis function of the test function space U_h^*, and is also thecharacteristic function of U_h^*.Similarly,we define a linear map T_h : Ph→P_h^*,which satisfiswhere {χj} is the basis function of the test function space P_h^* ,χjand is alsothe characteristic function of U_h^*.Then the finite element volume method,which we solve the equation(0.0.1)-(0.0.3) with using,is: Find (uh,ph)∈(Uh,Ph) such thatIn this text we take square area: [0,1]×[0,1] for example,introduce thefinite volume element method and the finite element method in detail,and dothe experiments.(I) Finite element Method on TrianglesK_h is a regular ,quasi-uniform triangulation ofΩinto a union of trian-gles.We consider the trial function space U_h and P_h,as followTake (0.0.12)(0.0.13) into(0.0.8),on triangles,we can get the finite elementmethod with compensating term.The method and the three following methods are similar.(II) Finite volume element Method on TrianglesKh is a regular ,quasi-uniform triangulation ofΩinto a union of triangles.Connecting the triangle's barycenter to the midpoint on each of the edges ofK, we get the K_h^*. Then the trial function space Uh and Ph iswhere P1(K) represents the space of linear functions on set K.The test function spaces U_h^* of velocity and P_h^* of pressure are:where P1(K) represents the space of linear functions on set K.(III) Finite element Method on RectanglesLet Kh be a regular ,quasi-uniform partition of ? into a union of rectan-gles. Then the trial function space Uh and Ph iswhere P1(K) represents the space of linear functions on set K.(IV) Finite volume element Method on RectanglesKh,Let Kh be a regular ,quasi-uniform partition of ? into a union ofrectangles. Connecting the rectangle's center point, we get the K_h^*. The trialfunction space Uh and Ph associated with dual partition Kh as follow:And the test function spaces Vh of velocity and Qh of pressure are: where P₁(K) represents the space of linear functions on set K.Because of these four methods of calculation, we get are non-singular,positive definite, and sub-blocks of symmetric coe?cient matrix, so they are allthe saddle-point problems; because the equations have a non-zero right-handside items, we can get only one solution.Among the experiment results of thesemethods, the degrees of convergence are similar except the postprocessing ofvelocity. The postprocessing of velocity on rectangles is less half degree thanit on triangles.