| 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 ?uid mechanics and electromagnetic field andother fields.The construction of Generalized Di?erence Method often has relationwith two grids ,one is original partition,the other is dual partition.On thetwo grids we define the trial function space and the test function space,thenumber of the latter is lower than the former,it's di?erent from the finiteelement method.Suppose ? is a bounded regional on the (x,y) plane with a smoothborder ?? .Consider the second order elliptic partial di?erential equationsof the first bo?undary value problems:the coe?cients of aij(x,y) (i,j = 1,2),q(x,y) is full smooth,and satisfy theelliptic condition.Select finite-dimensional space Uh appropriately for the trial functionspace ,and Vh for the test function space, and they have the same dimen-sion.The generalized Galerkin method is :find uh∈Uh ,such that and ?K?infers to the sum of linear integral on the boundary of dual unit K.if Uh = Vh ? U ,then (0-4) is the standard Galerkin method.For thegeneralized di?erence method we select Uh ? U the same with the finiteelement method,and select Vh = Uh for the Piecewise polynomial space in thelow.The first chapter of this paper introduces the linear format on triangularmesh and the isoparametric bilinear format on quadrilateral mesh of general-ized di?erence method,and introduces the result of the convergence results.The second chapter of this paper introduces the finite volumemethod(FVM) that proposed by F.Hermeline in 2000 years.Supposeλ,κare positive function and positive matrix .We solve the fol-lowing di?usion equation :The principle of FVM lies in the main following steps:1.Define two meshes on the domain?:a primary mesh and a dual mesh.2.Integrate the di?usion equation over each cell of both these meshes.3.Using Green formula,reduce the integral over one cell to the sum of the?uxes over each side of the cell.4.Introduce the values of the unknown function uh at the nodes of theprimary mesh and at those of the dual mesh as degrees of freedom for correctlyapproximating the ?uxes over the sides.follow these steps,we get the FVM format: Finally we introduce the result of the convergence results.We make a lot of examples of numerical experiments for the comparisonof GDM and FVM :The theoretical analysis and the numerical experiments prove that thetwo methods is first order accurate.The unknowns of GDM are only definedat the vertex of primary cell,it's nearly half of FVM's,so the price of calculationis small.And the price of calculation of FVM is big,but the results is exacteron various meshes,and the format is symmetric positive such that we can usethe conjugate gradient method,it is very convenient for computation.On veryspecial condition,the GDM is symmetric positive.Further we alter the FVMformat :by eliminating the unknowns at the centroid of primary cell and atthe midpoint of boundary side,we get the new FVM. Though the new FVMcause the precision had declined(it's still first order accurate),the price ofcalculation is great smaller than FVM's : it has dropped by some 50 percenton quadrilateral mesh and has dropped by some 30 percent on triangularmesh.The new FVM is a very good format.
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