| The Finite Volume Method,also called as Generalized Difference Method,was firstly put up by professor Li Qonghua in 1982.Its computational simplicity and preserving local conservation of certain physical quantities,make it be widely used in computing fluid mechanics and electromagneticfield and other fields.Firstly,We introduce the construction of the FVM based on the case of elliptic equation. We consider an elliptic boundary value problem of the form:Seek a functionu :Ω(?)R2→R such thatwithΩa bounded,convex,polygonal domain in R2 ,f∈L2(Ω),a(x,y) is bounded inΩ,or there is 0h ofΩ.Let hk be the diameter of the rectangle K∈Th and h =maxk∈Th hk. With each side e∈Ehin we associate a region Ke,consisting of the two rectangles of Th that have e as a common side,Q as its barycenter,let Th(Ke) be the set of the rectangles of K and denote by me the middle point of a side e∈Eh.Node is defined by the middle point of a side.The mean value is associated with the node and will be used as degrees of freedom.It's also non-conforming.Denote the sides of the rectangles of K are ei,i= 1,2,3,4,rotated bilinear functional basis is defined byΦei(x,y) = a + bx + cy + d(x2—y2) on the four sides,let the mean value equal to one on its side and equal to zero on the other sides. Next,we construct the dual partition Th* and its associated test function space.Divide each rectangle of the primal partition into four subtriangles.The dual grid is defined as a union of quadrilaterals,each of which is made up of two subtriangles.The same way to define the functional basis is rotated bilinear nonconforming form.Define the associate test function space Vh as the space of certain piecewise constant vector functions. So called FVM is to find uh∈Uh,such thatas(uh, vh) = (f, vh), (?)vh∈Vh (0-3)whereThen,we prove the bilinear form is positive definite.Lemma 0.1 let uh be the solution of (0-3) andΠh*uh be the projection of uh on Vh,then as(uh,Πh*uh) is positive definite,i.e.there exists a positiveα>0,such that:Next,we will estimate the equation (0-3) in the H1-norm.Theorem 0.1 Let u be the solution of (0-1),(0-2) and uh the solution of (0-3),a(x,y) is bounded inΩ,then there is a constant C ,independent of h,such thatThen we will estimate the equation (0-3) in the L2-norm.Theorem 0.2 Let u be the solution of (0-1),(0-2) and uh the solution of (0-3),then there exists a constant C,independent of h,such that:Especially,let a(x, y) equals to one,then general elliptic equation becomes Poisson equation as follow:LetTh denote a family of perfect polyhedral primary partitions of Q.We require that the partitions are quasi-uniform and regular. And we require a dual partition Th* to be perfect and quasi-uniform. The trial function space and test function space are defined by general elliptic equation. So called FVM is to find uh∈Uh,such thatwhereLemma 0.2 let uh the solution of (0-7),a(uh,vh) the bilinear form of finite element,as(uh,vh) the bilinear form of finite volume method,then there exists a constantα> 0,as(uh,vh) is positive definite,i.e.:as(uh,vh)≥At last we will introduce H1-norm and L2-norm estimation of error of equation (0-7)..Theorem 0.3 let u the solution of (0-5),(0-6),uh the solution of (0-7),then there exists a constant C,independent of h,such that:Theorem 0.4 let u the solution of (0-5), (0-6),u-h the solution of (0-7),then there exists a constant C,independent of h,such that:... |
