Finite Volume Element Methods For Second Order Elliptic Equations

Finite volume element methods for second order elliptic equationsdomestic is called Finite Volume Method,it Was most early proposed byProfessor Ronghua Li in 1982.It's basic mentality is:it Will calculate theregion to divide into a series of not redundant control volume,and enablearound each mesh point to have a control volume,it will treat the solutionof the dierential equation to each integral of control volume , then wewill obtain group of separate equations.Unknown is on the mesh pointdependent variable value.The finite volume method basic mentality easyto understand that, and can obtain the direct physical interpretation.The construction of Generalized Dierence Method ofen 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 dierent from the finiteelement method.Suppose is a bounded regional on the (x,y) plane with a smoothborder .Consider the second order elliptic partial dierential equationsof the first boundary value problems:the coecients of aij(x,y) (i,j = 1,2),q(x,y) is full smooth,and satisfythe elliptic condition:there exists a constant r > 0,such thatsatisfies for any vector (ξi,ξj)∈R2和(x,y)∈. And suppose that f∈L2().Select finite-dimensional space Uh appropriately for the trial func- tion space ,and Vh for the test function space, and they have the samedimension.The generalized Galerkin method is :find uh∈Uh ,such thatif 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 selectVh = Uhfor the Piecewise polynomial space inthe low.Needs to explain on the two-dimensional problem is that in finitevolume method,there always has Vh U.Suitable selection di?erent Uhand Vh, will derive each di?erent computational methods.This article willderive each kind of limited volume form and discusses,through the di?erentstructure ofUh and Vh.The article will be able to introduce domestic and overseas regard-ing second order elliptic equation finite volume method,including one-dimensional second-dimensional dimension, second dimensions, third di-mensions, and can be clear about give their restraining analysis.Afterwardshas also given the introduction to the afterwards mix finite volume method.Our first chapter will firstly introduce the finite volume method for theone-dimensional second elliptic equation,in which including the di?erencescheme one-dimension, second-dimensions, third-dimension and restrain-ing step.At the same time we also have carried on the comparison to thefinite volume method and the finite element method, finally discoveredmay obtain the finite volume method restraining step estimate method us-ing the finite element method.Moreover obtains the finite volume methodrestraining step with the aid of the finite element method only to suit adimension, all does not suit to two dimensions and the high dimension,because regarding between the high dimension finite element method andfinite volume method both the weak formation is not the small amount.The second chapter of this article will be able to introduce the finitevolume method for two-dimensional second order elliptic equation.Whileintroduced on triangle grid finite volume method also introduced on the quadrangle grid finite volume method.But on the triangle grid finite vol-ume method includes one-dimension, two dimensions and three dimensionsituations, then had introduced its positive definiteness and the error es-timate.We have also given the explanation to coe?cient matrix symme-try(bilinearity form symmetry) of finite volume method for second orderelliptic equation.But we cannot draw the conclusion regarding the high di-mension,then regarding second order elliptic equations also only is regard-ing coe?cient matrix A is often the coe?cient, when coe?cient matrix forvariable coe?cient time,we also cannot obtain the conclusion.The third chapter of this article introduced the finite volume methoderror estimate,in which including L2 error estimate and Super convergence.In which mainly introduced the L2 error estimate of one-dimensionand third-dimension di?erence scheme,second-dimension situations also arestudying now.Also similarly introduced regarding the super convergence ofone-dimensional second order elliptic equation dimension and third dimen-sion di?erence scheme situation, The error estimate of two-dimensionalsecond order elliptic equation has also given showing.The fourth chapter of this article introduced the mix finite volumemethod,Mainly is two set of grid mix finite volume methods, includingthe triangle grid on mix finite volume method, on the rectangular networkmix finite volume method and on the general quadrangle network mix finitevolume method, and introduced their error estimate.