Abstract: We constructed a finite volume scheme of fifth order Hermite type with high accuracy for twopoint boundary value problems, choosing trial and test spaces as the fifthorder finite element space of Hermite type and the piecewise linear function space respectively. We didn’t use higher derivatives as interpolation conditions, which is the same as the thirdorder finite element of Hermite type, but the scheme obtained had higher accuracy. The optimal convergence rates in H-1 and L-2 norms are proved, and the convergence order in L-2 norm is one order higher than that in H-1 norm. The numerical examples confirm the theoretical results.

Key words: finite volume element method, fifthorder element of Hermite type, dual partition, twopoint boundary value problem