Superconvergence Of One Dimension Fifth-order Finite Volume Method

The main work of this paper is that we construct the Lagrangian quintic element dimen-sional finite volume method for solving two-point boundary value problem.We choose quin-tic Lagrange interpolated function associated with the nodes as trial function,piecewise con-stant function as test function,and derivative superconvergent points as dual partition nodesso that a new kind of Lagrange quintic element finite volume element method is obtained forsolving two-point boundary value problems.It is proved that the method have optimal H1and L2error estimates.The superconvergence of numerical derivatives is discussed,and,thenumerical experiments show the results of theoretical analysis.Firstly,we discuss the numerical solutions of five quintic element Lagrange interpola-tion polynomial and its derivative superconvergent points.We use equidistant nodes in theinterval[h,h] to construct quintic element Lagrange interpolation function and the deriva-tive function at x3h0,then write T aylor expansion of u(h),u(3hhx0when taking equidistant nodes quintic element Lagrange interpolation polynomial deriva-tive superconvergent points.Secondly, we construct one-dimensional Lagrange quintic element finite volume ele-ment method for two point boundary value problem: where p angulation in the interval I=[a, b] is Th, assuming h=max{hi}, get the interval Ii=[x5(i1),x5i],(i=1,..., n) quintic Lagrange6equidistant nodes based on derivative supercon-vergence points of interpolation polynomial. The trial function space Uhcorresponding tothe partition Thas Lagrange fifth order finite element space, namely:1.uh C(I),u(a)=0;2.uhin each unit of Ii=[x5(i1),x5i] is quintic polynomial, which is composed of units andunit end point value of which determined by x5(i1),x5i4,x5i3,x5i2,x5i1,x5i. Uhis a5n di-mension subspace of U=H1(H1(I); U(a)=0}. We construct quin-tic Lagrange interpolation polynomials on the unit [1,1] as basis functions, then obtain thesubstrate space Uhcorresponding to the interpolation nodes x5i5,x5i4,x5i3,x5i2,x5i1,x5i.Finally,we get the solution format of the problem (2.1) of the quintic element finite volume elementmethod: for uh Uh, thereThen,we give the estimation error of one-dimensional Lagrange quintic interpolationfinite volume element method, including a priori estimate and H1norm error estimate,Proposition1When h fullly small, bilinear form a(uh, Πhuh) is positive definite, thereis a constantβ>0,independent of the subspace uhmakes that andProposition2Set the solution of problem (2.1) to element solution of finite volume format (2.4), there is H1norm error estimation: Fourth, we proved interpolation weak estimate, namely,and then we got the superconvergence as following,namely:Proposition3The solution of problem (2.1) u∈H17e(I)∩H(I),uhis quintic element solutionof finite volume format (2.4), there are:we use the first formula above to get the L2estimate:Finally,we use the numerical example, to confirm the results above.