Finite Volme Element Method Based On BB Dual Subdivision

The finite volume element method involves triang-ulation partition and dual partition on the region. The most popular dual subdivisions are barycenter subdivisions (BMB). and circumcenter dual subdivisions (CC). In addition, BB dual subdivisions by connecting barycenters directly are used in the field of computational fluid dynamic too. In fact, Considering the order of the convergence , there is the same velocity of convergence for CC dual subdivision , BMB dual subdivision and BB dual subdivision.In application ,we can choose any of the three methods for the practical problem. Based on BB dual subdivision , Li, yonghai and Li ronghua prove that the bilinrear is positively defined by precise calculation with weak restriction on subdivision. In this paper, we improve the proof of the positive definition of the bilinear form by putting strong condition on the subdivi-tion. New proof embodies the idea of control from macr-oscopy and wholeoAs for the construction of this article ,we firstly introduce three dualsubdivisions, , then come to introduce the finite volume element method for second order elliptic equation, And then prove that the bilinear form is positively defined and continuous. At last we get the convergence estimate.Firstly consider the second order elliptic boundaryvalue problem:Here Q is a polygonal region with boundary r f(x, y) is a known function on Q , /eL2(Q) , the coefficient a-a(x, y) is smooth enough, and there existsy>0, statisfyinga(jc, y)*y > for all(x, y)6QoLet Th be triangular subdivision of Q, r;be BB dual subdivision correspondlyo The trial function spaceUh is chosen as piecewise linear but entirely continuous finite element function space;The rest function space Vh is chosen as the piecewise constant function space respect to r;.And suppose u -Hl(Q)C\H2(Q), The trial function spaceuh is linear on KET. continuou on Q,t \s\Sf WVVM WlA'\St/w \SI W QQiand uh=0 T on the outside boundary)hThe test function spacevh ={vh\vh is a constmt on every dual element K^ET^ and vh is zero on the dual element respondto the boundary nodes. }The basises of the test function space are chosen asPI (xl,x2)EK*P, VP0GQA,(2) 0, otherwiseFor uHEUh j Let U'huh be the interpolation projection of uh onto test function space FA.that is to say,n;?4-The finite vlume element method for the problem(l)is to seek wAei/AsatisfyaQtk'V^-iffVp.)* VP0GQA, (4)whereK h PoJ 1 V dxdKP0(f,VPo)-fffdxdy,K'by(5) (6)The above formula is writed as(7)hereP0Eï¿¡lh(/.v,)the quadrilateral , which is composed of arbitrary two triangles A^i>,Pt and A/^P,Pt with common edge, is a h2quasi-parrallelogram element, that is to say that itsatisfiesIf supposeMPih- l2then we get that\Ch2 equivalent to/j -- + O(h), l2 -- + O(h)f suppose the triangular subdivisionsatisfy h2 quasi-parrallelogram condition, that is , the quadrilateral , which is composed of arbitrary twotriangles with common edge, is aFiglh2quasi-parrallelogram element. This condition is equivalent to k^-OQi), k2-O(h), k3-O(h).a(uh,n*huh)(10) du u . duhuh(P),Letduh dxduLduBydx du.duh ~dy~duh(?,-"*)(?;-??) Q3)Proposition 1.1 . I and|are equivalent;o-*andI . 1^ are equivalent to | . ||0 and | . \x respectively, it is to say that exist postitive constant CltC2,C3,C4 independent on Uh , satisfycilKIL. ? s IK It sC2lKllo. *. Vuteuh,Proposition 2 suppose the triangular subdivision Th satisfies quasi-parrallelogram condition , then TK(uh, Whuh) is positive defined (ViterA)Remark: Based on BB dual subdivision , Li, yonghaiand Li ronghua prove that the bilinrear is positively defined by precise calculation with weak restriction on subdivision. The estimate is conservative, which require the smallest angle of the trianglular subdivision is not too small. In this paper, the new proof is under the quasi-parrallelogram condition .With this condition , we prove that the bilinear is positively defined by trying to adopt the method of macroscopical control. The disturbing method is used in the order and new proof. It is so called disturbing method, that is to say, when the line between the barycenters of two neighboring triangles with common edge pass the midpoint of edge , the last three terms of (13) are zeros.Theorem 1 Under the condition of lemma 2, only if h is small enough, the quadric form a(uh,U'huh) is positivelydefined.Theorem 2 If (1)' s solution uSHl(Q)C\H2(Q),uh issolution of (2), then we have the estimate :...