Application Of BPX Schemes To The Finite Volume Method

The Finite Volume Method,also called as Generalized Difference Method,was firstly put up by professor Li Yonghai in 1982.Its computational simplicity and preserving local conservation of certain physical quanti-ties,make it be widely used in computing fluid mechanics and electromagnetic field and other fields. Similarly to the finite element method,when the matrix induced from the FVM is preconditioned by BPX schemes ,we can get uniform boundary of condition number .Firstly,We introduce the construction of the FVM based on the case of two-dimension. Let Ω (?) R² be a bounded open convex polyhedral with boundary (?)Ω . Let A(x) = (a_ij(x)),x ∈ Ω,be a 2 × 2symmetric matrix .We assume that the matrix A(a;)satisfies the uniformly elliptic condition,that is,We define an elliptic partial differential operator of the second order byand for a given function f(x) ∈ L²(Ω)consider the boundary value problemLetT_n denote a family of perfect polyhedral primary partitions of Ω.We require that the partitions are quasi-uniform and regular. And we require a dual partition T* to be perfect and quasi-uniform.For a positive integer γ,we let P_γ be the space of polynomials of total degree at most γ —1. The trial space U_n is defined associated with the partitionThe test space V_n is defined associated with the dual partition T_n^* bySo called FVM is to find un € Un such thatan{un, vn) = (/, vn), Vvn e Vn (0-6)where]T] / un) â–  Vvndx - Y^ / (^Vun) ? nvnds (0-7)We define an interpolation projector IIn : Un t-$ Vn, then the system (0-6) is equivalent toan(un,Unwn) = (f,Ilnwn), VwneUn (0-8)Generally,we require that the bilinear form is bounded and positive defi-nite,that is, we assume there exist positive constants N, 70 and Mo such that for all \/n>N\an(un,Unwn)\ < Mo || un Idll wn ||i Vun,wneUn (0-9)andan(un, nnun) > r0 || un ||? V?n e t/n (0-10)Then,we introduce a multilevel preconditioning for the operator and matrix induced by the bilinear form an.We define operator An : Un —> Un such that for Vwn, wn eUn(Aun, wn) = an(uni Unwn) (0-11)We call An the FVM operator. Now,by the Riesz representation theo-rem,there exists Tn G ï¿¡/n,such that(fn,wn) = (f,nnwn) VwneUn (0-12)then (0-6) has an equivalent operator formAnun = Tn (0-13)A approach to precondition An is to find operators Cn, whose inverses C"1 are equivalent to An.Theorem 0.1 Suppose that there exist two positive constants 7 < F,and for Vn G N ,a self-adjoint positive definite operator Cn such that-l9,9) <\\ 9 \\l< T{C?gig), Vg e Un (0-14)Moreover suppose that the bilinear form satisfies hypotheses (0-9)^(0-10),then7o7where 70 and Mo are the positive constants appearing in (0-9) and (0-10) respectively.Theorem 0.2 Suppose that the bilinear form satisfies (0-9),(0-10)and that there exist positive constants 7 < F such that the self-adjoint positive definite operators Cn, Vn ï¿¡ N satisfies (0-14), then,KAn(CnAn) = 0(1),n -?â–  00. (0-15)Now we introduce the BPX preconditioners for FVM matrix.We denote by $n = {(j)p : P € Q,n} the nodal basis of Un and $? = {Xp â–  P € fin} the basis of Vn .Recall that D{n) = dimUn = dimVn.$oi convenience,we denote A(n) := {1, 2, ? ? ? , D(n)}, and order the points fin so that the basis $n and $n can be written by $n = {(/>* : i € A(n)}, $n = {xj : i e A(n)}.We introduce the FVM matrix An := [a(4>i,Xj) '? hj ï¿¡ A(n)j and fra := K/,%)):ieA(n)].With these notations,the variational equation (0-6) is equivalent to its matrix formAnun = fn (0-16)For VKnwe let Ejt denote the representation matrix of nodal basis $fc =(k,D(k)) °f Uk m term °f tne nodal basis *n = (<^n,l, ? ? * , 4>n,D{n)) off/n,i.e.,$fc = $nEfc (0-17)DefineC°:=^EfcRfcE[ (0-18)Jfc=0Since C° is non-singular,the finite volume method defined by (0-16) is equivalent to the linear systemC°Anun = C% (0-19)LetAn:=(C°)5An(C°)l, thenTheorem 0.3 Let C° be the matrix defined by (0-18) , then,Then,we introduce the generalized minimal residual algorithm (GM-RES)for solving nonsymmetric linear systems.We use GMRES to solving the preconditioned linear system induced from the FVM,and get the uniform convergence rate of the method.For two point boundary value problem,we(l)validate that the quadratic element trial spaces of Lagrangian type are nested,and show the condition numbers of matrixes before and after preconditioned by BPX schemes.(2)validate that the cubic element trial spaces of Hermite type are nested,and show the condition numbers of matrixes before and after preconditioned by BPX schemes and iterative numbers for solving the linear systems by GMRES.For Poisson problem,we(l)validate that the linear element trial spaces of Lagrangian type are nested,and show the condition numbers of matrixes before and after preconditioned by BPX schemes and iterative numbers for solving the linear systems by GMRES.(2) validate that the quadratic element trial spaces of Lagrangian type are nested,and show the condition numbers of matrixes before and after preconditioned by BPX schemes and iterative numbers for solving the linear systems by GMRES and Gauss-Seidel.At last,with the help of cell stiff matrix analysis method ,we write thebilinear form of the FEM asK= VPk - VMk— vMi - vMk= Up. - UMj > = UMi - UMj= vPj - vM.— VMi - vMjSo,z = Bu, z = Sv,then we have lK{uh,Vh) = Similarly,we write the bilinear form of FVM as a(uh,vh) =whereIK(uh,vh)=vPl+VM.Jl2duv1 V oxdy+dyThen ,we have lK{uh,Vh)SQ. (0-20)(0-21)(0-22)(0-23)(0-24)Let C — zog-(A A), and we find that ||C|| is not small, which shows the difference between the bilinear forms of the FVM and the FEM is not small.So the FVM is not approximation of the FEM.