Cubic Element Finite Volume Method For Two-point Boundary Value Problem

The Finite Volume Method, also called as Generalized Di?erence Method,was firstly put up by professor Li Ronghua in 1982. Its computational sim-plicity and preserving local conservation of certain physical quantities,makeit be widely used in computing ?uid mechanics and electromagnetic field andother fields.Firstly, We introduce the construction of the Lagrange cubic elementfinite volume methods based on the case of second order elliptic equation. Weconsider an elliptic boundary value problem of the form:where,p∈C¹(I),p(x)≥p_min > 0,f∈L²(I).Discretize the interval I = (a,b) into a grid Th with nodesSelect the trial function space Uh as the cubic element space of Lagrangiantype with respect to Th. So any function uh in Uh satisfies the followingconditions:(i) uh∈C(I),u_h(a) = 0;(ii) uh is a cubic polynomial at each element Ii = [xi?1,xi] and it is determineduniquely by its values at the two endpoints and the two one-third of thepoints of the element.Next, we place a dual grid Th? with nodal pointsThe test function space corresponding to is taken as the piecewise constantfunction space, which is a 3n-dimensional subspace spanned by the basis func- tions of the nodes x_ithe nodes x_i-2/3and the ones of x_i-1/3Next we derive the cubic element di?erence scheme corresponding to theproblem (0-1): Find u_h∈U_h, such thatwhere Next we show the equivalence of the norms:Theorem 0.1 Within Uh, the norms |·|0,h and |·|_1,h are equivalent to|·|0 and |·|1 respectively, namely,there exist positive constants C₁,C₂,C₃ andC₄ independent of Uh such thatthe positive definiteness of the operator aTheorem 0.2 For suffciently small h, is positive definite, that is, there exists a positive constantα> 0 such thatFinally, we show estimates in H1 of the solution:Theorem 0.3 Suppose u∈H⁴(I) and uh are the solutions of the prob-lem (0-1) and the cubic element difference scheme (0-2) respectively, then thefollowing estimate holds:...