 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: t uu ugut Fxx xJ (1) 0ut tJ xx (2) 0 0uu xxx 3 (3) where is a bounded region with piecewise smooth boundary . R is a small positive constant and is a smooth vector function on ufFx x 12 u 3 fufu x x R , 00Fx . 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 [1] 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 [6]. 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 [12]. 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 [12]. 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 [17] 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 12 L -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 T 1 2 L -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): 1 C Continuity condition: 2 uL R x is Lipschitz continuous w.r.t. the second variableu. 2 C The vector function has 1-order con- tinuous partial derivative w.r.t. and u. uFx x Suppose the true solution of problem (1) possess cer- tain smooth and satisfy: R Regular condition: 212 uW LHLLH 2 . Before presenting the numerical scheme we introduce some notations. For simplicity we assume is the domain 000 LL YZ . Firstly, Let us consider a family of regular tetrahedron partition in the domain h T , which is a closure of . Let be maximum diameter of cell of h T. For a fixed tetrahe- dron partition h h TK, we define a closed tetrahedron set 1 ii K and node set 12 1 011 2 1 MM hhi ii iiM PP P i , where 0 is inner nodes set of and h boundary nodes set on . Let be all edges set. i 1 hi Ee M E Definition 2.1. Suppose that is a set of tetrahedron partition of , the set h is called regular, if there exists a positive constant 0 0 h TT hh T 1 independ- ent of , such that h 10 max 0 hKK KT hh h where h and are the diameter of and the maximum diameter of circumscribing sphere of tetrahe- dron , respectively. K K Definition 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 i jP is adjacent to . ii jh For a given tetrahedron partition h T with nodes PPP i Ph and edges i eE hP h , we construct two kind of dual partitions. First, we will construct the circumcen- ter dual partition of . Tih , let hi hi PKKTP is a vertex of . Let Q be a circumcenter of hi P . Connecting Q of the two face-adjacent tetrahedron cells which belong to hi P, then we can derive a polyhedron i which surrounds the node i. P Q are vertices of the polyhe- dron i which is called circumcenter dual partition corresponding to node i. i hPih is the circumcenter dual partition of h T. Denote by the midpoint of and its adjacent node P TK P ij P i P P. The other dual partition as follows. h , let k eE hk h eKKT and k is a edge of e . Denote by 1 k and 2 k the vertices of the edge k and P Pe Q the circumcenter of the hk e . Suppose k e is a polyhedron whose vertices are 1 k P2 k P Qs. k e is called dual cell for edge . k e 1 k hek K T is the other dual partition to . h T Let h be the node set of dual partition. For h Q , let Q be tetrahedron cell which includes . Q Let and Q be volumes of dual cell and tetrahedron cell Q , respectively. Let be the di- ameter of tetrahedron cell Q h . As follows, we assume that the partition family h is regular, i.e., there exist positive constants 12 T CC independent of , such that the following condition (A2) satisfies: h 33 12 33 12 h p Qh Ch KCh P ChKCh Q (4) Suppose that a trial function space 1 0 h UH, whose basis functions are 1 1 ii P possessing the form 01 23 yz based on [15], and h T 0 iih PP . Test function space 2 h VL is a piecewise constant function space corresponding to the dual partition h T , whose basis functions are h PP . Copyright © 2011 SciRes. AJCM  F. Z. GAO ET AL. 266 1 0otherwise P PK P and 0h PP For the following analysis, we introduce two interpo- lation operators. Suppose that and are inter- polation operators from h h 1 0 to and , respec- tively, satisfying h Uh V 1 1 M hi i uuPP i (5) Ph hKT uuPP (6) Multiplying both sides of (1) by , integrating on dual partition cell v i P, using Green formula, and sum- ming with respect to , we have i h 1 0 t uvauvbuvgv vH d (7) where d PP ii ih KK P auvuvu vs x (8) d ih PKK PP ii buvv vs Fx Fd (9) Converting into [1] F 0d u uu FxFxuu (10) Let 0 0 max 0()d max 0()d u ij ij u ij ij uuu uu xFx xFx u uu (11) where ij is the unit outward normal vector of ij i P . For we introduce bilinear form hhhh uUvV i ih hhhhi ij Pj ijij hiijij hj buvvP Pu PPu P (12) where ij is the area of . ij So far, we can obtain the semi-discrete upwind finite volume element scheme: Find such that hh uU ,hthh hhh hhh uvauvbuvg uv x (13) where d P i ih hhh h K P au vuvs Let tTN , denote by nn tntuut n 12 nn hh uutn N, 11nnn thh h uuu t . If approximate solution h is known, then can be found by the following full-discrete upwind finite volume element scheme. 1n h uU n h u 11 1 nn n hthhhhhhh n hh uv auv buv gu v 1 x (14) 3. Auxiliary Lemmas Define the discrete norm and the discrete semi-norm [1] as follows. 22 2 00i Ph i hhh hi hKT uu uP P K (15) 2 2 21 11 E k M hhkhkk hk uuPuPeK e (16) 22 101 hhh hh uuu 2 h (17) obviously, the discrete norm and the discrete semi-norm are equivalent to the continuous norm and the full-norm on , respectively. h Lemma 1. Suppose all cells Q U of the partition h satisfy conditions (A2), h T T is a circumcenter dual parti- tion. hh h uu U , there exist positive constants 0 C independent of h such that 2 1 hhhhhh auuuuU (18) 011 hhh hhhhh auC uuU uuu (19) h hhhhhh hh auu auuuU u (20) Remarks: 1) From Lemma 1, we know that is symmet- rical and positive definite in . a h U 2) Let 1 2 1 hhhh uauu , then 1 is equiva- lent to 1 in . h U Lemma 2. Let 1 2 0 hhhh uuu h , 0 is equi- valent to 0 in . h U The proof of lemma 2 can be completed by computing integral on cell Q , directly. Theorem 1. (Trace Theorem) [20]. Suppose that has a piecewise Lipschitz boundary, and that is a real number in range p 1p . Then there exists a constant , such that C 1 11 11 pp p pp p LLW vCvv vW Lemma 3. For h small enough, suppose P is a random point in dual partition cell i , ij ij PP K , Copyright © 2011 SciRes. AJCM  F. Z. GAO ET AL.267 then 2 12 dPP ij ii i KK j uPusChuu x (21) Proof. From Hölder inequality, we can get that 1 2 2 d d i ij i j j ij uPus Chu Pus x x 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 h uu U there exists a positive constant , such that C , hhh hhh uuuu (22) 0. hhhh h uuCuu 0 (23) Proof. From the properties of the functions in , for each partition cell h U h T, we know that h u has the following expression. 0000111 1 22223333 ,,,,,,,, , ,,,,,, , hiiiiiiii K iii iiii i uxyzt uxyztuxyzt uxyztuxyzt (24) where 0123 1 6 lllll axbyczdlii ii Ve and is the volume of tetrahedron i.e., Ve 0123 iiii PPPP 00 0 11 1 22 2 33 3 1 1 1 1 6 1 ii i ii i ii i ii i yz yz Ve yz yz 0123 Pl iiii l i, whose coordinates are ll l iii yz PP , are four vertices of tetrahedron cell 0123 iiii which belongs to h. 012 3l are the volume coor- dinates which are corresponding to tetrahedron cell . For , PP T 0123 iiii PPPPl liiii 0 i 11 11 22 0202 33 33 11 11 222 020 333 3 3 11 11 11 1 1 1 ii ii ii iiii ii ii ii ii iii iii iii i i yz xz ayzbxz yz xz 1 2 i i yxy cxydxy z z yz xy Analogously, we can define the remaining coefficients 123lll l abcd liii . Further, 0123 ddd h hhh hl KTliiii h KK P l uuuP uxy z For simplifying numerical integral, we divide the po- lyhedron integral domain 0 i P K into six tetrahedron integral domains 0010120123 0010130123 0020120123 002 023 0123 003 013 0123 1 2 3 4 5 tetrahedron tetrahedron tetrahedron tetrahedron tetrahedron iiiiiiiiii iii iiiiiii iiiiiiiiii iii iii iiii iiiiiiiiii VPPP VPPP VPPP VPPP VPPP P P P P P 003023 0123 6tetrahedron iii iii iiii VPPPP where 01 ii is the midpoint of segment 01 ii while and 012 3 iiii are circumcenters of triangular surface 012 iii and tetrahedron 0123 iiii , respectively. Analogously, we can define the remaining points. P P PP 012 iii PPPPPPPP Noting the Equality (24), we have that 00 1 22 33 ddd ddd [ ]ddd . PP ii P i hhK KK KK ii ii KK ii ii u xyzuxyz uPtuPt uPtuPt xyz 1 For simplicity, we will omit the variable in func- tion t uxyzt 01 ii . From volume coordinate formula, not- ing 23 1 ii , we can derive 0 00 11 22 33 012 6 1 ddd ddd 7333 48 P i j hK KK ii ii V j ii ii iii uxyz uP uP uPuPxyz KuPuP uPuP 3 i Further, 7333 3733 33 73 48 333 7 h hhh KT K uu where 0123 hihihihi uP uPuP uP and 0123 T hi hihi hi uP 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 nnnnnn hhhhh uu uuuue Choosing in (7), then we have 1n tt 111 1nnnn t utvau vbuvguv x(25) Subtracting (14) from (25), we obtain that 11 1 11 11 nn hthhhh nn hhh nn hh hh nn hh ev aev rv av bu vbu v uguv xx n t (26) where . 11nn ht ht ruu 1nn Choosing hh in Equality (26), denote by 12 and 1234 TT the left and right hand side terms of Equality (26), respectively. We will analyze the six terms successively. h h vee TTWW For , from the definition of 1 W0 , we have that 2 1 100 1 2 nn hh Wee t 2 (27) Rewriting as 2 W 11 2 11 11 21 2223 nn nn hhhhh nn nn hhhhhh nn nn hhhhhh Wae eee aeeae e aeeaee WW W (28) From (20) of Lemma 1, we can get the estimate to as follows. 23 W 23 0W (29) From (27)-(29), we have 22 12 01 22 11 1 01 1 22 1 24 nn hh nn n hh h t WWe e t t ee e 2 1 n h e (30) For each terms of the right hand side of (26). Using interpolation theory, triangulation inequality and lemma 4, we know that 222 2 2 11 1000 2 nnnn hhtt t TCeetu hu Similarly, we can bound as 2 T 11 221 nnn hh TChuee Further, making use of triangulation inequality and important inequality, we have that 22 12 211 nn n hh TCee hu 2 1 2 (32) From the Lipschitz property of ux in condition 2 C, making use of triangle inequality, important ine- quality and Lemma 4, we have 22 41 1 420 nnn hh TChue e 2 0 (33) Combining (34),(35) with (36), we know that 22 2 2 1 21412 31121 nn nn hh TCeehu hu h (34) Combining (31), (32), (33) with (34) and applying Sobolev space embedding theory, we know that the of (26) satisfies RHS 22 2 11 00 2 22 121 221 nn n hh tt nn t RHSCeetu hhu u 2 0 (35) From (30) and (35), using inverse estimate we know 22 22 11 01 01 222 2 11 00 1 2 22 121 22 1 22 2 1 4 1 nnn n hhh h nnn nn hhh htt nn t tt eee e t ee Ceetu hhu u 2 1 0 Further, we get that 22 2 11 010 22 11 10 2 222 22 1 11 2 02 22 2 1 nnnn hhh h nnn n hhh h n nn tt t tt eee e tee Ctee tu hhuu 2 1 2 0 (36) Summing from 1 to with respect to in the above inequality, we can obtain that Nn 41 (31) 22 22 11 01 01 222 11 001 11 2 222 221 11 2 02 11 22 2 1 NNN N hhh h NN nn nn hh hh nn NN n nn tt t nn tt eee e teeCt ee CttuCthh 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 22 2 021 0 01 01 h Further, (37) may be rewritten as 22 2 01 2 01 0 2 2 21 00 00 2 22 121 22 0 22 1 N Nn hh n NN n n htt nn Nnn t n thete CteCtt u Cth h uu (38) Choosing in such way that th 2 00 1 20th , further, (38) can be rewritten as n t t 22 01 0 2 22 1 00 00 2 22 12 1 22 0 1 N Nn hh n NN n h nn Nnn t n ete CteCttu Cth h uu (42) where 0 12 . Using discrete Gronwall’s lemma, we know that 1 22 01 0 2 222 22 1 11 2 02 0 N Nn hh n Nn nn tt t n ete Cttuh 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 1 0 2 2 0 hh TH TH LL uu uu Oh t (44) where, 2 00 12 0 sup sup . n TX TX LL X ntT Nn X NtTn vvv vt 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 h 0 tO 0 h u 5. Conclusions In this paper, we continued our work [17] 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 2 L 1 -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. 2 L 6. 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 [1] R. H. Li, Z. Y. Chen and W. Wu, “Generalized Differ- ence Methods for Differential Equations: Numerical Ana- lysis of Finite Volume Methods,” Marcel Dekker, New York, 2000. [2] Z. Q, Cai and S. F. McCormick, “On the Accuracy of the Finite Volume Element Method for Diffusion Equations on Composite Grids,” SIAM Journal on Numerical Ana- lysis, Vol. 27, 1990, pp. 635-655, 1990. [3] Z. Q. Cai, J. Mandel and S. F. McCormick, “The Finite Volume Element Method for Diffusion Equations on Ge- neral Triangulations,” SIAM Journal on Numerical Ana- lysis, Vol. 28, No. 2, 1991, pp. 392-402. doi:10.1137/0728022 [4] R. E. Bank and D. J. Rose, “Some Error Estimates for the Box Method,” SIAM Journal on Numerical Analysis, Vol. 24, No. 4, 1987, pp. 777-787. doi:10.1137/0724050 [5] V. Patankar, “Numerical Heat Transfer and Fluid Flow,” McGraw-Hill, New York, 1980. [6] J. Douglas Jr. and T. F. Russell, “Numerical Methods for Convection-Dominated Diffusion Problems Based on Com- bining the Method of Characteristics with Finite Element or Finite Difference Procedures,” SIAM Journal on Nu- merical Analysis, Vol. 19, No. 5, 1982, pp. 871-885. doi:10.1137/0719063 [7] D. B. Spalding, “A Novel Finite Difference Formulation for Differential Equations Involving Both First and Sec- ond Derivatives,” International Journal for Numerical Me- thods in Engineering, Vol. 4, No. 4, 1973, pp. 551-559. doi:10.1002/nme.1620040409 [8] K. Baba and M. Tabata, “On a Conservative Upwind Fi- nite Element Scheme for Convective Diffusion Equa- tions,” RAIRO Analyse Numériqe, Vol. 15, No. 1, 1981, pp. 3-25. [9] M. Tabata, “Uniform Convergence of the Upwind Finite Element Approximation for Semi-Linear Parabolic Prob- lems,” Journal of Mathematics of Kyoto University, Vol. Copyright © 2011 SciRes. AJCM  F. Z. GAO ET AL. Copyright © 2011 SciRes. AJCM 270 18, No. 2, 1978, pp. 307-351. [10] M. Tabata, “A Finite Element Approximation Correspon- ding to the Upwind Finite Differencing,” Memoirs of Numerical Mathematics, Vol. 4, 1977, pp. 47-63. [11] M. Tabata, “Conservative Upwind Finite Element Appro- ximation and Its Applications, Analytical and Numerical Approaches to Asymptopic Problem in Analysis,” North- Holland, Amsterdam, 1981, pp. 369-387. [12] Y. R. Yuan, “The Upwind Finite Difference Fractional Steps Methods for Two-phase Compressible Flow in Po- rous Media,” Numerical Methods for Partial Differential Equations, Vol. 19, No. 1, 2003, pp. 67-88. doi:10.1002/num.10036 [13] D. Liang, “A Kind of Upwind Schemes for Convection Diffusion Equations,” Mathematical Numerical Sinica, Vol. 2, 1991, pp. 133-141. [14] Y. H. Li and R. H. Li, “Generalized Difference Methods on Arbitrary Quadrilateral Netwoks,” Journal of Compu- tational Mathematics, Vol. 17, No. 6, 1999, pp. 653-672. [15] M. Feistauer, J. Felcman and M. Lukáčová-Medvid'ová, “On the Convergence of a Combined Finite Volume-Finite Element Method for Nonlinear Convection-Diffusion Pro- blems,” Numerical Methods for Partial Differential Equa- tions, Vol. 13, No. 2, 1997, pp. 163-190. doi:10.1002/(SICI)1098-2426(199703)13:2<163::AID-N UM3>3.0.CO;2-N [16] M. Feistauer, J. Slavik and P. Stupka, “On the Conver- gence of a Combined Finite Volume-Finite Element Me- thod for Nonlinear Convection-Diffusion Problems,” Nu- merical Methods for Partial Differential Equations, Vol. 15, No. 2, 1999, pp. 215-235. doi:10.1002/(SICI)1098-2426(199903)15:2<215::AID-N UM6>3.0.CO;2-1 [17] F. Z. Gao and Y. R. Yuan, “An upwind Finite Volume Element Method Based on Quadrilateral Meshes for Non- linear Convection-Diffusion Problems,” Numerical Me- thods for Partial Differential Equations, Vol. 25, No. 5, 2009, pp. 1067-1085. doi:10.1002/num.20387 [18] P. G. Ciarlet, “The Finite Element Method for Elliptic Pro- blems,” North-Holland, Amsterdam, 1978. [19] R. A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975. [20] S. C. Brenner and L. R. Scott, “The Mathematics Theory of Finite Element Methods,” Springer-Verlag, New York, 1994.
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