 American Journal of Computational Mathematics, 2011, 1, 264-270 doi:10.4236/ajcm.2011.14032 Published Online December 2011 (http://www.SciRP.org/journal/ajcm) Copyright © 2011 SciRes. AJCM An Upwind Finite Volume Element Method for Nonlinear Convection Diffusion Problem Fuzheng Gao, Yirang Yuan, Ning Du School of Mathematics, Shandong University, Jinan, China E-mail: fzgao@sdu.edu.cn Received August 17, 2011; revised September 6, 2011; accepted Septembe r 15, 2011 Abstract A class of upwind finite volume element method based on tetrahedron partition is put forward for a nonlinear convection diffusion problem. Some techniques, such as calculus of variations, commutating operators and the a priori estimate, are adopted. The a priori error estimate in L2-norm and H1-norm is derived to determine the error between the approximate solution and the true solution. Keywords: Nonlinear, Convection-Diffusion, Tetrahedron Partition, Error Estimates 1. Introduction Consider the following nonlinear convection-diffusion problem: tuu ugut Fxx xJ (1) 0ut tJ xx (2)  00uu xxx3 (3) where is a bounded region with piecewise smooth boundary . R is a small positive constant and is a smooth vector function on ufFxx12u3fufu x xR , 00Fx . The finite volume element method (FVEM) is a dis-crete technique for partial differential equations, espe-cially for those arising from physical conservation laws, including mass, momentum and energy. This method has been introduced and analyzed by R. Li and his collabo-rators since 1980s, see  for details. The FVEM uses a volume integral formulation of the original problem and a finite partitioning set of covolumes to discretize the equations.The approximate solution is chosen out of a finite element spaces [1-3] The FVEM is widely used in computational fluid mechanics and heat transfer prob-lems [2-5]. It possesses the important and crucial prop-erty of inheriting the physical conservation laws of the original problem locally. Thus it can be expected to cap-ture shocks, or to study other physical phenomena more effectively. On the other hand, the convection-dominated diffusion problem has strong hyperbolic characteristics, and there-fore the numerical method is very difficult in mathemat-ics and mechanics. when the central difference method, though it has second-order accuracy, is used to solve the convection-dominated diffusion problem, it produces numerical diffusion and oscillation near the discontinu-ous domain, making numerical simulation failure. The case usually occurs when the finite element methods (FEM) and FVEM are used for solve the convection- dominated diffusion problem. For the two-phase plane incompressible displacement problem which is assumed to be -periodic, J. Douglas, Jr., and T.F.Russell have published some articles on the characteristic finite difference method and FEM to solve the convection-dominated diffusion problems and to overcome oscillation and faults likely to occur in the traditional method . Tabata and his collaborators have been studying upwind schemes based triangulation for convection-diffusion problem since 1977 [7-11]. Yuan, starting from the practical exploration and development of oil-gas resources, put forward the upwind finite dif-ference fractional steps methods for the two-phase three- dimensional compressible displacement problem . Most of the papers known concern on the FVEM for one- and two-dimensional linear partial differential equa-tions [1-4,13,14]. In recent years, M. Feistauer [15,16], by introducing lumping operator, constructed finite vo- lume-finite element method for nonlinear convection- diffusion problems. On the other hand, because the FEM costs great expense to solve the three-dimensional prob-lems, finite difference methods (FDM) are usually used to approximate the problems . These works inspire F. Z. GAO ET AL.265 us to look into the subject how to use the upwind FVEM to solve three-dimensional nonlinear convection-domi- nated diffusion problems. In this article, we continue to our work  and put forward an upwind FVEM for three-dimensional nonlinear convection-dominated dif-fusion problems based on tetrahedron partition and its dual partition of . Some techniques, such as calculus of variations, commutating operator and the a priori error estimate, are adopted. The a priori error estimate in -norm and 12LH-norm is derived to determine the error between the approximate solution and the true solution. The remainder of this paper is organized as follows. In Section 2, we put forward the upwind FVEM for prob-lem (1). In this section, we introduce notations, construct tetrahedron mesh partition h of and its dual parti-tion. Some auxiliary lemmas and the a priori error esti-mate in -norm and T12LH-norm of the scheme are shown In Section 3 and Section 4, respectively. In Sec-tion 5, some concluding remarks are presented. Throughout this paper we use C (without or with sub- script) to denote a generic constant independent of dis-crete parameters. We also adopt the standard notations of Sobolev spaces and norms and semi-norms as in [18, 19]. 2. The UFVE Method Suppose problem (1) satisfy condition (A1): 1C Continuity condition:  2guL R x is Lipschitz continuous w.r.t. the second variableu. 2C The vector function has 1-order con-tinuous partial derivative w.r.t. and u. uFxxSuppose the true solution of problem (1) possess cer-tain smooth and satisfy: R Regular condition: 212uW LHLLH 2. Before presenting the numerical scheme we introduce some notations. For simplicity we assume  is the domain 000LLLXYZ  . Firstly, Let us consider a family of regular tetrahedron partition in the domain hT, which is a closure of . Let be maximum diameter of cell of hT. For a fixed tetrahe-dron partition hhTK, we define a closed tetrahedron set 1KNiiK and node set  12101121MMMhhi iiiiMPP P i, where 0 is inner nodes set of and h boundary nodes set on . Let be all edges set. i1hiEe M EDefinition 2.1. Suppose that is a set of tetrahedron partition of , the set h is called regular, if there exists a positive constant 00hTT hhT1 independ-ent of , such that h10max 0hKKKT hhh  where Kh and K are the diameter of and the maximum diameter of circumscribing sphere of tetrahe-dron , respectively. KKDefinition 2.2. The two tetrahedron cells are called face- adjacent if they have one common face, while edge-ad- jacent if they have one common edge. Definition 2.3. The two nodes are called adjacent if they form one edge which belongs to h. Denote by EijjP is adjacent to . ii jhFor a given tetrahedron partition hT with nodes PPPiPh and edges ieEhPh, we construct two kind of dual partitions. First, we will construct the circumcen-ter dual partition of . Tih , let hi hiPKKTP is a vertex of K. Let jQ be a circumcenter of hiKP . Connecting jQ of the two face-adjacent tetrahedron cells which belong to hiP, then we can derive a polyhedron iPK which surrounds the node i. PjQ are vertices of the polyhe-dron iPK which is called circumcenter dual partition corresponding to node i. ihPih is the circumcenter dual partition of hT. Denote by the midpoint of and its adjacent node PTKPijPiPjP. The other dual partition as follows. h , let keEhk heKKT and k is a edge of eK. Denote by 1k and 2k the vertices of the edge k and P PejQ the circumcenter of the hkKe . Suppose keK is a polyhedron whose vertices are 1kP2kPjQs. keK is called dual cell for edge . ke1EkMhekKT is the other dual partition to . hTLet h be the node set of dual partition. For hQ, let QK be tetrahedron cell which includes . QLet PK and QK be volumes of dual cell PK and tetrahedron cell QK, respectively. Let be the di-ameter of tetrahedron cell QhK. As follows, we assume that the partition family h is regular, i.e., there exist positive constants 12TCC independent of , such that the following condition (A2) satisfies: h33123312hpQhCh KCh PChKCh Q (4) Suppose that a trial function space 10hUH, whose basis functions are 11MiiP possessing the form 01 23xyz   based on , and hT0iihPP. Test function space 2hVL is a piecewise constant function space corresponding to the dual partition hT, whose basis functions are hPP . Copyright © 2011 SciRes. AJCM F. Z. GAO ET AL. 266 10otherwisePPKP and 0hPP For the following analysis, we introduce two interpo-lation operators. Suppose that and are inter-polation operators from hh10H to and , respec-tively, satisfying hUhV11MhiiuuPP i (5) PhhKTuuPP  (6) Multiplying both sides of (1) by , integrating on dual partition cell viPKP, using Green formula, and sum-ming with respect to , we have ih10tuvauvbuvgv vH d (7) where dPPiiih KKPauvuvu vs x (8) dihPKKPPiibuvv vs  Fx Fd (9) Converting into  F 0duuuFxFxuu (10) Let 00max 0()dmax 0()duij ijuij ijuuuuuxFxxFxuuu (11) where ij is the unit outward normal vector of ij iPK . For we introduce bilinear form hhhhuUvV  iihhhhhi ijPjijij hiijij hjbuvvPPu PPu P   (12) where ij is the area of . ijSo far, we can obtain the semi-discrete upwind finite volume element scheme: Find such that hhuU,hthh hhh hhhuvauvbuvg uv x (13) where dPiihhhh hKPau vuvs Let tTN , denote by nntntuut n12nnhhuutn N, 11nnnthh huuu t . If approximate solution h is known, then can be found by the following full-discrete upwind finite volume element scheme. 1nhuUnhu111nn nhthhhhhhhnhhuv auv buvgu v 1 x (14) 3. Auxiliary Lemmas Define the discrete norm and the discrete semi-norm  as follows. 22200iPhihhh hihKTuu uPPK (15)  222111EkMhhkhkkhkuuPuPeKe (16) 22101hhhhhuuu2h (17) obviously, the discrete norm and the discrete semi-norm are equivalent to the continuous norm and the full-norm on , respectively. hLemma 1. Suppose all cells QUK of the partition h satisfy conditions (A2), hTT is a circumcenter dual parti-tion. hh huu U , there exist positive constants 0C independent of h such that 21hhhhhhauuuuU  (18) 011hhhhhhhhauC uuUuuu  (19) hhhhhhh hhauu auuuUu  (20) Remarks: 1) From Lemma 1, we know that is symmet-rical and positive definite in . ahU2) Let 121hhhhuauu , then 1 is equiva- lent to 1 in . hULemma 2. Let 120hhhhuuu h, 0 is equi- valent to 0 in . hUThe proof of lemma 2 can be completed by computing integral on cell QK, directly. Theorem 1. (Trace Theorem) . Suppose that  has a piecewise Lipschitz boundary, and that is a real number in range p1p. Then there exists a constant , such that C  111 11ppppppLLWvCvv vW   Lemma 3. For h small enough, suppose P is a random point in dual partition cell iPK, ijij PPKK , Copyright © 2011 SciRes. AJCM F. Z. GAO ET AL.267 then  212dPPij iiiKKjuPusChuu  x (21) Proof. From Hölder inequality, we can get that   122ddiijijjij uPusChu Pusxx Using Taylor expansion, trace theory in which we choose and Hölder inequality, we can complete the proof of lemma 3. 2pLemma 4. For hh huu U  there exists a positive constant , such that C,hhh hhhuuuu  (22) 0.hhhh huuCuu 0 (23) Proof. From the properties of the functions in , for each partition cell hUhKT, we know that hKu has the following expression. 0000111 122223333,,,,,,,, ,,,,,,, ,hiiiiiiiiKiii iiii iuxyzt uxyztuxyztuxyztuxyzt (24) where 012316lllllaxbyczdlii iiVe and is the volume of tetrahedron i.e., Ve 0123iiiiPPPP000111222333111161iiiiiiiiiiiixyzxyzVexyzxyz 0123Pl iiiili, whose coordinates are llliiixyzPP , are four vertices of tetrahedron cell 0123iiiiwhich belongs to h. 012 3l are the volume coor-dinates which are corresponding to tetrahedron cell . For , PPT0123iiiiPPPPlliiii0i11112202023333111122202033333111111111iiiiiiiiiiiiiiiiiiiiiiiiiiiiiyz xzayzbxzyz xz12iixyxycxydxyzzxyzxy   Analogously, we can define the remaining coefficients 123lll labcd liii . Further, 0123dddhhhh hlKTliiii hKKPluuuP uxyz   For simplifying numerical integral, we divide the po- lyhedron integral domain 0iPKK into six tetrahedron integral domains 001012012300101301230020120123002 023 0123003 013 012312345tetrahedrontetrahedrontetrahedrontetrahedrontetrahedroniiiiiiiiiiiii iiiiiiiiiiiiiiiiiiii iii iiiiiiiiiiiiiiVPPPVPPPVPPPVPPPVPPPPPPPP003023 01236tetrahedron iii iii iiiiVPPPP where 01ii is the midpoint of segment 01ii while and 012 3iiii are circumcenters of triangular surface 012iii and tetrahedron 0123iiii, respectively. Analogously, we can define the remaining points. PPPP012iiiPPPPPPPPNoting the Equality (24), we have that 00 122 33ddd ddd[]ddd .PPiiPihhKKK KKii iiKKii iiu xyzuxyzuPtuPtuPtuPt xyz1  For simplicity, we will omit the variable in func-tion tuxyzt01ii. From volume coordinate formula, not-ing 231ii, we can derive  000 1122 3301261dddddd733348PijhKKKii iiVjii iiiiiuxyzuP uPuPuPxyzKuPuP uPuP3i   Further, 7333373333 7348333 7hhhh KTKuu  where 0123hihihihiuP uPuP uP and 0123Thi hihi hiuP uPuP uP From the above equality, we can complete the proof of Copyright © 2011 SciRes. AJCM F. Z. GAO ET AL. 268 hLemma 4 easily. 4. Convergence Analysis Now we consider the error estimates of the approximate solution. Let nn nnnnnnhhhhhuu uuuue   Choosing in (7), then we have 1ntt111 1nnnntutvau vbuvguv x(25) Subtracting (14) from (25), we obtain that  1111111nnhthhhhnnhhhnnhh hhnnhhev aevrv avbu vbu vguguv   xxnt (26) where . 11nnht htruu 1nnChoosing hh in Equality (26), denote by 12 and 1234TT the left and right hand side terms of Equality (26), respectively. We will analyze the six terms successively. h hvee TTWWFor , from the definition of 1W0, we have that 2110012nnhhWeet2 (27) Rewriting as 2W112111121 2223nn nnhhhhhnn nnhhhhhhnn nnhhhhhhWae eeeaeeae eaeeaeeWW W   (28) From (20) of Lemma 1, we can get the estimate to as follows. 23W23 0W (29) From (27)-(29), we have 2212 012211 101122124nnhhnn nhh htWWe ettee e   21nhe (30) For each terms of the right hand side of (26). Using interpolation theory, triangulation inequality and lemma 4, we know that 222 22111000 2nnnnhhtt tTCeetu huSimilarly, we can bound as 2T11221nnnhhTChuee Further, making use of triangulation inequality and important inequality, we have that 2212211nn nhhTCee hu212 (32) From the Lipschitz property of gux in condition 2C, making use of triangle inequality, important ine-quality and Lemma 4, we have 2241 1420nnnhhTChue e 20 (33) Combining (34),(35) with (36), we know that 22 2 21 2141231121nn nnhhTCeehu hu h (34) Combining (31), (32), (33) with (34) and applying Sobolev space embedding theory, we know that the of (26) satisfies RHS22 21100222 121221nn nhh ttnntRHSCeetuhhu u20 (35) From (30) and (35), using inverse estimate we know 22 221101 01222 211001222 12122122 2141nnn nhhh hnnn nnhhh httnnttteee etee Ceetuhhu u210   Further, we get that 22 211010221110222222 1112022221nnnnhhh hnnn nhhh hnnntt ttteee etee Cteetu hhuu2120 (36) Summing from 1 to with respect to in the above inequality, we can obtain that Nn41 (31) 22 221101 012221100111222222111202112221NNN Nhhh hNNnn nnhhhhnnNNnnntt tnntteee eteeCt eeCttuCthh uu  (37) Noting the equivalence of 0 and 1 with 0 Copyright © 2011 SciRes. AJCM F. Z. GAO ET AL.269 and 1, respectively. Using the inverse estimate, we have that there exist three positive constants 012 such that 22 22202100101 h  Further, (37) may be rewritten as 22201 201022210000222 121220221NNnhhnNNnnhttnnNnntntheteCteCtt uCth h uu     (38) Choosing in such way that th200 120th, further, (38) can be rewritten as ntt2201022210000222 12 1220 1NNnhhnNNnhnnNnntneteCteCttuCth h uu  (42) where 012. Using discrete Gronwall’s lemma, we know that 1 22010222222 1112020NNnhhnNnnntt tneteCttuh huu  (43) Noting that , combining finite element space interpolation theory, we can obtain the resulting error estimates to the approximate solution as follows. Nt T10220hhTH THLLuu uuOh t  (44) where, 200120supsup .nTX TXLLXntTNnXNtTnvvvvt   Therefore we have the following theory. Theorem 2. Suppose that the solution to the problem (1) is sufficiently smooth. When and are small enough and satisfy the relationship . The ini-tial value is chosen as interpolation of , then the Equation (44) holds. h tuh0tO0hu5. Conclusions In this paper, we continued our work  and presented a class of upwind FVEM based on tetrahedron partition for a three dimensional nonlinear convection diffusion equation, analyzed and derived error estimate in - norm and 2L1H-norm for the method. In the ongoing work, we will discuss how to derive optimal error estimate in -norm and how to code and present numerical results to demonstrate the performance. 2L6. Acknowledgements The research was partially supported by the Scientific Research Award Fund for Excellent Middle-Aged and Young Scientists of Shandong Province (grant no. BS- 2009HZ015), and NSFC (grant no. 10801092). 7. References  R. H. Li, Z. Y. Chen and W. 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