High Order Finite Volume Element Method For One-Dimensional Problems

The finite volume element method, also called as generalized difference method, was firstly put up by professor Ronghua Li in 1982. Due to its compu-tational simplicity and preserving local conservation of certain physical quan-tities, is has been widely used in computing fluid mechanics,solid mechanic-s, electromagnetic field and other fields. At present the schemes and the theo-retical results of the finite volume element method are mostly ahout low order element, but that for the high order element are less.Firstly, we introduce four classes of high order finite volume methods for one-dimensional elliptic problems.two of them based on a local L2 projec-tion,others of them based on a special projection of Ch0 onto piecewise constants.LetΩ=(xL,xR) be a bounded interval and consider the following two-point boundary value problem: whereα,β,γ,f are smooth, real-valued functions defined onΩ,and Integrating (1a) over a subintervalω(ωL,ωR) we obtain which, whenωR=xR,is written as If uh satisfies(2a)on a boundary subinterval (ωL,ωR),then we shall say thatωis a boundary control volume of typeⅠ.Similarly.if uh satisfies(2b) on a boundary subinterval (ωL,ωR),then we shall say thatωis a boundary control volume of typeⅡ.The basic ingredient in the definition of our numerical methods is a linear oprator (?)h:phri→L2(Ω),satisfying the following assumptions:A discrete variational formulation of(1a)-(1b)is the defined as fol-lows:for h∈(0,1)we seek uh∈Shr=Phr∩H(Ω)such that where the bilinear form Bh:Hh2×Phr→R is defined asHere we formulate a family of locally conservative methods on Shr which based on a local L2 projection of Pr onto pr-2 for r≥2,or on a local L2 projection of pr onto pr-1 for r≥2. Proposition 1 Let r≥2, and(?)h: be defined: then(1)(3)holds withα=r-1 and(4)is satisfied withσ=0. (2)The method(5)is a locally conservative method on Shr with overlapping control volumes {Ijh}j=1Jh and {(xj-1h,xj+1h)}j=1Jh-1,where IJhh is a boudary control volume of both typesⅠandⅡ.(3)The corresponding approximation uh belongs to Shr (?)C1(Ω)and satisfies a homogeneous Neumann boudary condition at xR. Proposition 2 Let m∈N,r=2m,and(?)h:Phr→Phr-1 be defined by: Then(1) (3)holds withα=r and(4)is satisfied withσ=1.(2)The method(5)is a locally conservative method on Shr with control vol-umes {Ijh}=j=1Jh,where IJhh is a boundary control volumes of typeⅠ.Aswe define a special proj ection of the piecewise continuous functions of Ch0 onto a space consisting of piecewise constant functions and use it to derive finite volume methods on Shr for(r≥2).Let s≥2,(?)={(?)j}j=0s (?) R be the nodes of a partition of[0,1].i.e., (?)0=0,(?)s=1,(?)s-1<(?)s,j=1,…,s。Proposition 3 Let be the nodes of a partition of [0,1],and then: (1)(?)h satisfies(3)withα=1 and(4)with,σ=0.(2)The methods(5)is a finite volume method with control volumes where (xJh-1+(?)-2(?)) is a boundary control volume of both typesⅠandⅡ.As(3)The corresponding approximation uh belongs to Shr (?) C1(Ω)and satisfies a homogeneous Neumann boundary condition at xR.Proposition 4 Let m∈N,r=2m,(?)={(?)}i=0r be the nodes of a partition of [0,1],and Then(1)the (?)h satisfies(3)withα=1 and(4)withσ=1.(2)the method(5)is a locally conservative method with control volumes where IJhh is a boundary control volume of typeⅠ.AsMoreover,if r∈{2,4,6} and then the method(5)is a finite volume method with overlapping control volumes: .where the lattee intervals are boundary control volumes of typeⅡand IJhh is a boundary control volume of typeⅠ. suppose Q={(x,y)|a