Conforming And Nonconforming Finite Volume Methods Applied To The Stokes Equations

The Finite Volume Method,also called as Generalized Di?erence Method,was firstly put up by professor Ronghua Li in 1982. Its computational sim-plicity and preserving local conservation of certain physical quantities, makeit be widely used in computing fluid mechanics and electromagnetic field andother fields.Firstly, We introduce the construction of the FVM based on the caseof two-dimension. LetΩ? R2 be a bounded open convex polyhedral withboundary ?Ω. We consider the stationary Stokes problem in two dimensions:where the symbols△,▽, and▽·denote the Laplacian, gradient, and diver-gence operators, respectively, and f(x) is the unit external volumetric forceacting on the ?uid at x∈Ω. We assumeν≡1 here.First of all, we introduce the original partitions Th and dual partitionsTh inΩ. Then we construct the trial function space associated with originalpartition Th and the test function space associated with dual partition Th ? .For a positive integer r, we let Pr be the space of polynomials of total degreeat most r-1. Then define the trial function space Uh associated with originalpartition Th as follow:For conforming FVM Define the test function space V h associated with dual partition T_h^? as follow:For pressure P, we define the trial function space Ph associated with originalpartition . HereΩˉdenotes the polygon area which is close totheΩ.We derive the finite volume approximation solution of (0-1)-(0-3), whichis (uh,ph)∈U_h×P_h such thatAs in [16], we define a linear mapγh : Uh→V h. Then we define thefollowing bilinear forms:Then the finite volume approximation problem of (0-7)-(0-8) becomes:Find (uh,ph)∈Uh×Ph such that We can prove that the bilinear form B is equal to b. So we call this methodas Saddle Point Problem.After introducing the FVM simply, now we choose ? ? R2 as unit squarearea: [0,1]×[0,1]. Then we describe the conforming and nonconforming FVMand their experiment results in detail.(I) Steady Conforming FVM on TrianglesLet original partition T_2h be a union of triangles of ?, divide each triangleτ∈T_2h into four triangles by joining the midsides to form a refined partitionTh. Define the trial function space Uh associated with original partition Th asfollow:Define the trial function space P2h associated with original partition T_2h asfollow:Let the dual partition Th ? be a convex hull by connecting the barycenters ofthe triangles in Th. Then define the test function space V h associated withdual partition Th ? as follow:whereReplace the spaces in (0-7)-(0-8) by these spaces we can get the steadyconforming FVM on triangles, we don't give them here(the three followingmethods is the same as here).(II) Steady Conforming FVM on RectanglesLet original partition T_2h be a union of rectangles of ?, divide each rectangleτ∈T_2h into four rectangles by joining the midsides to form a refined partition T_h. Define the trial function space U_h associated with original partition T_h asfollow:Define the trial function space P2h associated with original partition T2h asfollow:Let the dual partition Th ? be a convex hull by connecting the center of therectangles in Th. The test function space V h associated with dual partitionTh ? as follow:where(III) Nonconforming FVM on TrianglesLet ? = J_j=1be a quasi-regular triangular partition of ? with diam(?)≤h.Denote the boundary edge of Kj byγj = ?? ?Kj, the interface betweenelements Kj and Kk byγjk =γkj = ?Kj ?Kk, and the center ofγj andγjkbyξj andξjk, respectively. Letvˉ(ξjk) = |γ1jk|γjk v|?Kjds.Define the dual partition Th ? of Th to be the union of the quadrilaterals forboth rectangular or triangular elements. Each quadrilateral in dual partitionTh ? is made up of two subtriangles, which share a common edge. Define thetrial function space Uh associated with original partition Th, the trial functionspace Ph associated with original partition Th and the test function space V hassociated with dual partition Th ? as follows: (IV) Nonconforming FVM on RectanglesLet ? = Jj=1be a quasi-regular rectangular partition of ? with diam(?)≤h.Other symbols are the same as (III).Define the trial function space Uh associated with original partition Th asfollow:where Q?1(Kj) denotes the space of functions of the form a+bx1+cx2+d(x12?x22). The other spaces'definition are the same as (III). We don't give themhere.These four methods are steady. The coe?cient matrices derived fromthese four methods are positive definite and block symmetrical, and the ranksof them is full, i.e. nonsingular. Then the systems have one and only solutions.As the increasing of n, the errors are decreasing. We also show the 3-dimensiongraphics of these methods. The graphics are concerned about velocity andpressure, respectively, including numerical solutions and true solutions. Thegraphics are more and more smooth as the increasing of n. Especially forvelocities'graphics, they are looked like"saddles". It's one of the reasonsthat we called this FVM as Saddle Point Problem.