Cubic Element And Double Cubic Element Finite Volume Methods Of Elliptic Equations

The finite volume method(FVM)is an important numerical method for solving partial differential equations.Because of its ability to maintain the conservation with local integral in physical quantities,this method has been widely used in fluid mechanics,geology and other fields.This paper focuses on the convergence laws of one-dimensional three dimensional and two-dimensional double three dimensional finite volume methods.On the basis of theory(one dimension)and numerical experiments(one and two dimensions),the convergence order of L2 norm and superconvergence of the corresponding format are given when the dual subdivision satisfies symmetry change.Firstly,consider the two point boundary value problem:Where I =[a,b],p?C1(I),pmax? p(x)? pmin>0,f? L2(I).Subdivision of interval I =[a,b]into Th.Choosing a = x0<x1<……<xn = b,[Xi-1,Xi]represents any unit.The trial function space Uh is Lagrange type three dimensional finite element space corresponding to Th.The corresponding computing node(Lagrange interpolation node)is taken as the three bisection point of each cell(including endpoints),which written xi-k/3(k=1,2).The dual dissection Th*is corresponding to Th.a = x0<xa*<x1/2*<x1-a*<x1<……<xi<xi+a*<xi+1/2*<xi+1-a*<xi+1<……<xn= b.Where xi+a*=a(xi+1-xi)+xi,xi+1-a*=(1-a)(xi+1-xi)+xi,xi+1/2*=1/2(xi+1+xi),i= 1,2,…,n.And 0<a<-is the parameter used to adjust the position of the dual dissection.The test function space Vh is piecewise constant function space corresponding to the split Th*.As for the computing node xj,xj-1/3,xj-1/3,(j = 1,…,n)on the trial function space Uh,the corresponding dual elements are(?)and(?).The cubic element finite volume method for two point boundary value problems is:find uh?Uh,such that ah(Uh,vh)= {f,vh),(?)vh?Vh the corresponding unit form is.Where(?)are the characteristic functions on the corresponding node Xj,xj-1/3,xj-2/3 of dual element.{?i(x),?j-1/3(x),?j-2/3(x):1?j?n} as a group of base test function space.We define ?h*:Uh ?Vh as interpolation operator from trial function space to test function space(for a detailed definition,see Section 2.3),So the formula is equivalent to So we get the stability and H1 error estimation of the formatTheorem 1.(Stability)When 0<a<1/3 and h is sufficiently small,a(uh,vh)is stability,that is,there exists a positive constant ?>0 independent of Uh,such thatTheorem 2.(H1 error estimation)Suppose u? H4(I)and uh are the solutions of the problem(1.1)and the cubic element finite volume,then the following estimateholds when 0<a<1/3 and h is sufficiently small?u-uh?1?Ch3|u|4.We extend the orthogonality condition[13]in one-dimensional constraints.It is proved that all three dimensional finite volume schemes with symmetric dual structure have the best convergence order according to L2 norm.Theorem 3.(L2 error estimation)Suppose u ? HE1(I)? H5(I)and uh are the solutions of the problem(1.1)and the cubic element finite volume,then the followingestimate holds when 0<a<1/3 and h is sufficiently small 3?u-uh??Ch4|u|5.We verify the above theoretical results by numerical experiments,at the same time,the following conclusions are obtained.Conclusion 1:On the uniform mesh generation Th,dual partition with thechange of a symmetry.The mean value and exact solution derivative of the numerical solution of the element end point and midpoint(without boundary element)has the following superconvergence results.Conclusion 2:On the uniform mesh generation Th.The global superclose property is only established when the dual partition node is taken as the stress point.?uI-uh?=O(h4)Where uI is u piecewise three times Lagrange interpolation.Then,consider the two dimensional Poisson problem:Where ?= {(x,y)|a?x?b,c?y?d?,(?)? is the boundary of ?.Th is a uniform rectangular mesh generation on Q.The trial function space Uh is Lagrange double three dimensional finite element space corresponding to Th.For each rectangular element,each edge is divided as one dimension according to a.Connecting the dual partition nodes on the opposite edges,we get the dual partition in each rectangular cell.The dual dissection Th*is built which is corresponding to Th.The test function space Vh is piecewise constant function space corresponding to the split Th*.The double cubic element finite volume method for two dimensional Poisson problem is:find uh?Uh,such that ah(uh,Vh)=(f,vh),(?)vh?Vh.the corresponding unit form is.Where N is all computing nodes,?P is characteristic function on the dual unit KP*of P.We have obtained the following conclusions through numerical experiments.Conclusion 1:when dual partition with the change of(?)symmetry,the optimal convergence order of error estimates for H1 and L2 norm can be achieved.?u-uh?1=O(h3),?u-uh ? = O(h4).Conclusion 2:On the uniform mesh generation Th,dual partition with the change of ? symmetry.The mean value of the numerical solution gradient and the gradient of the exact solution at the symmetric point(the vertex,the middle point and the center point of each unit)has the following superconvergence results.Conclusion 3:On the uniform mesh generation Th.The global superclose property is only established when the dual partition node is taken as the stress point.?uI-uh?=O(h4).?Where uI is u piecewise double three times Lagrange interpolation.