A Class Of Finite Volume Method Based On New Dual Partitions For Two-point Boundary Value Problems

In this paper, a class of Lagrangian finite volume element method based on new dual partitions for two-point boundary value problems is presented. We find an orthogonal conditions, with which we can fix dual partitions. The advantage of the orthogonal conditions is that it is a sufficient condition of ensuring corresponding FVM schemes have optimal convergence orders of L2 norms. For trial function space which is Lagrangian finite element space of order m, the dual units fixed by the orthogonal conditions has m+1 orders of L2 norms. What’s more, the error estimate of L2 norms is proved simply in this case.For two-point boundary value problems: where I=[a, b], p(x)>Pmin>0, r(x)>0, p,r∈C1(I), f ∈L2(I). We divide I into n units. We define x; as node of original unit, j=1,2,…,n. The trial function space Uh is taken as the Lagrangian finite element space of order m. We define ∏h* as interpolation operator from trial function space to test function space and Pk(I) as polynomial space on unit I in which the order of the functions is less than k. The dual units are fixed by the orthogonal conditions. Let nodes of trial units is bilateral symmetry and meet ∫xi-1h(g-∏hg*g)dx=0, (?)h∈Pm-1(Ij), (?)∈P1(Ij) on original unit Ij= [xj-1,xj],j=1,2, …, n. We define xj* as node of dual units, j*=0,1,2, …, m·n. The test function space Vh is defined as the piece-wise constant function space. The finite volume method for two-point boundary value problems is: find uh∈Uh, such that a(uh,vh)= (f,vh),(?)vh∈Vh. WhereThen the dual partitions have optimal convergence order of L2 norms, namely for trial function space which is Lagrangian finite element space of order m, the dual units fixed by the orthogonal conditions have m+l orders of L2 norms. The specific error estimate is where u is the solution of the two-point boundary value problems, uh is the solution of FVM schemes.It is proved in this article and numerical experiments are preformed in witch trial function space is Lagrangian finite element space of order one to order five. The results are in accord with the theory in this article.