Cubic Finite Volume Element Method For One-dimensional Parabolic Equations Based On Optimal Stress Points

In this paper,a new kind of Lagrange-type finite volume element method to solve the one-dimensional elliptic equations is presented firstly, and trial space is the Lagrange-type finite element space whose interpolation nodes is the root of Four Lobatto polynomial. The points of dual unit is the supercon-vergence points of the derivative of interpolation polynomial.The piecewise constant function space corresponding to the dual partition are chosen as the test space Vh. It is proved that the method has optimal H1 and L2 error esti-mates. The superconvergence of numerical gradients at optimal stress points is discussed.Using this method of thinking,a new kind of Lagrange-type fi-nite volume element method to solve the one-dimensional parabolic equations is constructed.It is proved that the method has the best convergence order estimates under the semi-discrete and fully discrete format.First, we consider one-dimensional parabolic equations of mixed prob-lems:其中I= [a,b],p∈C1(I),p(x)≥Pmin>0,f∈L2(I). Simply,suppose The trial space Uh is defined as: And{φi(x),φi-(?)2(x),φi-(?)1(x)；1≤i≤n}are lagrange basis functions,ui= uh(xi,t),ui-(?)2=uh(xi-(?)2,t),ui-(?)1=uh(xi-(?)1,t).Suppose In the interval[xj-1,xj],the optimal stress points xj-e1=xj-e1hj,xj-e2=xj-e2hj,xj-e3=xj-e3hj, Taking a=x0