| This paper discusses high -order irregular-Hermite finite volume methods for one-dimension elliptic problems .Then we introduce differential formats of the one-dimensional four times and five times irregular-Hermite finite volume methods . At the same time we also discussed the difference scheme of the two finite volume methods are positive and the theoretical results of H1 error estimates , and gave the corresponding proof. Finally, the results of the numerical example are given.First we introduce four-times irregular-Hermite finite volume method.Difference scheme is as follows: find uh∈Uh, satisfy thatorThe introduction of equivalent discrete norm, theTheorem1. In Uh ,|·|1,h and |·|1 are equivalence, Namely, there exists normal number C1, C2 independent of Uh, such that,Then,the definiteness of a(uh,Πh*uh) is positive.Theorem2. SupposedΠh*u is uh the interpolation projection to the test function space Vh, when h small enough,a(uh,Πh*uh) is positive, namely,there exists normal numberβ1 > 0 independent of Uh, so thatTheorem3. Suppose u∈H5(I) is the ture solution ,uh is the numerical solution of difference scheme, there error estimate is as follows:Next we introduce four-times irregular-Hermite finite volume method. Difference scheme is as follows: find uh∈(?)h, satisfy thatThe introduction of equivalent discrete norm, theTheorem4. In the space (?)h ,|·|1,h and |·|1 are equivalence, Namely, there exists normal number C1, C2 independent of (?)h, so that,Then,the definiteness of a(uh,Πh*uh) is positive.Theorem5. SupposedΠh*u is uh the interpolation projection to the test function space (?)h, when h small enough,a(uh,Πh*uh) is positive, namely,there exists normal numberβ2 > 0 independent of (?)h, so thatTheorem6. Suppose u∈H6(I), uh is the ture solution ,uh is the numerical solution of difference scheme, there error estimate is as follows:From the discussion of this paper about four-times and five-times irregular-Hermite fimite volume methods,as well as the results of numerical examples, availablethe following conclusions: for the difference scheme of high-oder irregular-Hermite finite volume method, L2 error estimate and L∞error estimate are not always higher than H1 error estimate, and sometimes can only be achieved with the band.
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