Computers and Mathematics with Applications. A nonlinear weighted least-squares finite element method for Stokes equations

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1 Computers Matematics wit Applications 59 () 5 4 Contents lists available at ScienceDirect Computers Matematics wit Applications journal omepage: A nonlinear weigted least-squares finite element metod for Stokes equations Hsue-Cen Lee a, Tsu-Fen Cen b, a General Education Center, Wenzao Ursuline College of Languages, Kaosiung, Taiwan b Department of Matematics, National Cung Ceng University, Mingsiung, Cia-Yi, Taiwan a r t i c l e i n f o a b s t r a c t Article istory: Received July 9 Accepted 5 August 9 Keywords: Least-squares finite element metods Stokes problems Nonlinear weigted function Mass conservation Te paper concerns a nonlinear weigted least-squares finite element metod for te solutions of te incompressible Stokes equations based on te application of te leastsquares minimization principle to an equivalent first order velocity pressure stress system. Model problem considered is te flow in a planar cannel. Te least-squares functional involves te L -norms of te residuals of eac equation multiplied by a nonlinear weigting function mes dependent weigts. Using linear approimations for all variables, by properly adjusting te importance of te mass conservation equation a carefully cosen nonlinear weigting function, te least-squares solutions eibit optimal L -norm error convergence in all unknowns. Numerical solutions of te flow pass troug a 4 to contraction cannel will also be considered. 9 Elsevier Ltd. All rigts reserved.. Introduction In recent years, tere ave been a lot of developments in te application of least-squares metods for te approimation of te flow equations; see, e.g., [ 6]. Te least-squares finite element approac for Stokes problem as been sown to offer several teoretical computational advantages over Galerkin metods for a variety of boundary value problems []. In addition, te algebraic system generated by te discretization is always symmetric positive definite, tere is no compatibility condition between finite element spaces for mied metods, te metod is insensitive to equation type. In [7], Bocev Gunzburger introduce a weigted least-squares functional wit velocity pressure stress formulations involving L -norms of te residuals of eac equation multiplied by a mes dependent weigt. In addition, tey etend te Agmon Douglis Nirenberg a priori estimate to te velocity pressure stress formulation of te Stokes equations. Teir numerical eamples indicate tat te metod is not optimal witout te weigts in te least-squares functional. In addition, if one uses te same linear approimation for all unknowns, te linear weigted least-squares metod is not optimal. Similar results using H -norm least-squares functional for Stokes equations based on te velocity pressure stress formulation are presented in [8]. Regardless of teir advantages, poor mass conservation is reported in least-squares based formulations, see e.g., [9 ]. Note, owever, as indicated in [], results can be improved by sufficiently weigting te mass conservation term. Te purpose of te study is to present a nonlinear weigting function in te least-squares metod based on te velocity pressure stress formulation for Stokes equations. Te analysis of te nonlinear weigted least-squares functional follows te idea introduced in [7]. Te coice of weigts is a focus of te current effort. In tis paper, we implement te nonlinear weigted least-squares formulation using linear basis functions for all variables. Using continuous piecewise linear Corresponding autor. addresses: 873@mail.wtuc.edu.tw (H.-C. Lee), tfcen@mat.ccu.edu.tw (T.-F. Cen). 898-/$ see front matter 9 Elsevier Ltd. All rigts reserved. doi:.6/j.camwa

2 6 H.-C. Lee, T.-F. Cen / Computers Matematics wit Applications 59 () 5 4 finite element spaces for all variables, properly adjusting te importance of te mass conservation wit carefully cosen nonlinear weigting functions, te least-squares solutions eibit optimal L -norm error convergence in all dependent variables. Furter, we etend te implementation to simulate te 4 to contraction problem, []. Here we will point out tat te coice of weigts used to balance te residual contributions is an area tat warrants furter study. Following tis introduction, te stability error estimates of a nonlinear weigted least-squares metod for te Stokes equations are presented in Section. In Section 3, results of various least-squares finite element are provided for te flow in a planar cannel te 4 to contraction problem. Conclusions are presented in Section 4.. Nonlinear weigted least-squares metods for Stokes equations Te section studies a nonlinear weigted least-squares metod for te incompressible Stokes equations based on te velocity pressure stress formulation. Consider te following generalized stationary Stokes problem in an open, boundary two-dimensional wit boundary Γ : η u + p = f in, () u = in, u = U on Γ, were u p denote te velocity pressure fields, η is a constant f U are given functions. We assume tat te pressure p satisfies a zero mean constraint: pd =. ( Let D(u) = ) u + u T denote te symmetric part of te velocity gradient. i.e., te deformation tensor. Defining te stress tensor τ := ηd(u) scaled by η/, we ave te following generalized velocity pressure stress system: τ ηd(u) = F in, u = f in, η τ + p = f 3 in, u = u on Γ, were te function f satisfies te following solvability constraint: f d = u nds. Note tat in two dimensions, te system () as si equations si unknowns. If te tensor F te function f are identically zero, te Stokes equations () is equivalent to te generalized system (). For simplicity, witout loss of generality, we assume tat u =. Let H s (), s be te Sobolev spaces wit te stard inner products (, ) s teir respective norms s. For s =, H s () coincides wit L (). H s () denotes te closure of D (), te linear space of infinitely differentiable functions wit compact supports on, wit respect to te norm s. Denote by L () te subspace of square integrable functions wit zero mean: } L () := {p L () : pd =. For positive values of s, te space H s () is te dual space of H s () wit te norm (φ, v) φ s := sup, v H s () v s were (.,.) is te duality pairing between H s () H s (). Define te product spaces Hs ()d = d i= Hs () H s ()d = d i= H s (). Let H (div; ) = { υ L () d : υ L () } wit te norm ( ) υ H(div;) := υ + υ. In [7], Bocev Gunzburger applied te Agmon Douglis Nirenberg (ADN) teory to establis te following a priori estimate of te first order system (): τ τ + p + u C ηd (u) + C u + q+ η C τ + p, (3) for all q R. q+ q ()

3 H.-C. Lee, T.-F. Cen / Computers Matematics wit Applications 59 () In te following, we describe a nonlinear weigted least-squares metod associated wit te system (). Let Φ := V Q Σ = H () L () L () 3, were L () d is d d system matri functions wose elements are square integrable. For te finite element approimation, we assume tat te domain is a polygon for d = or a polyedron for d = 3 tat Γ is a partition of into finite elements = T Γ T wit = ma{diam(t) : T Γ }. Assume tat te triangulation Γ is regular satisfies te inverse assumption (see [8]). Let Φ := V Q Σ be a finite element subspace of Φ wit te following approimation prosperities: tere eists a positive integer r suc tat te spaces S j approimate optimally wit respect to te space H r+j (), j =,. More precisely, we assume tat for all u H r+j () tere eists an elements u I S j suc tat m, u u Im Cr+j m u r+j. (4) Te least-squares functional for () is defined as follows: ( J(v, q, σ; f) = ws σ ) ηd(v) F + K v f + η σ + q f 3, (5) were K is a positive constant is set to one ere for convenience of te analysis. In te case wen w s =, Deang Gunzburger [3] consider te positive weigt K outside of te residual norm for conservation of mass in (5). Tey indicate tat te rate of convergence can be improved by taking K. In [4], a nonlinear weigted least-squares metod is considered to solve nonlinear yperbolic equations. Following [4], a nonlinear weigting function is employed in [6,5] to solve generalized Newtonian (Carreau model) flow problems. Based on te success of te nonlinear weigted least-squares approac to te generalized Newtonian flow problem, te following weigting function w s is considered in our work. In eac element, te weigt w s is taken as w s = + ( γ ), were te sear rate γ γ w, wit γ = (D(u) : D(u)) double-dot product between two second-order tensors τ σ defined as τ : σ = τ ij σ ji. i,j Te wall sear rate γ w, te maimum value of te sear rate γ, is defined as γ w = τ w η, were η is te viscosity te wall sear stress τ w can be directly obtained from te pressure drop te geometric constants [6]. Te least-squares problem for te first order system () is to minimize te quadratic functional J(u, p, τ; f) over Φ, tat is, we seek (u, p, τ) Φ suc tat J(u, p, τ; f) = inf J(v, q, σ; f). (v,q,σ) Φ Based on te presentation in [7], we establis te ellipticity of te functional J(u, p, τ; ) in Teorem. Teorem. For (u, p, τ) Φ, tere eist constants c C, independent of, suc tat (6) c ( τ + p + u ) J(u, p, τ; ) J(u, p, τ; ) C ( τ + p + u ) (7) (8) for any <. Proof. Let (u, p, τ) Φ. Using te inverse assumption, i.e., u C u, te estimate (3) wit q =, we ave ( τ τ + p + u C ηd (u) + u + η τ + p ).

4 8 H.-C. Lee, T.-F. Cen / Computers Matematics wit Applications 59 () 5 4 Since L () H (), it follows tat ( τ τ + p + u ) C ηd (u) + u + η τ + p ( ) τ C ηd (u) + u + η τ + p C C τ ηd (u) + + ( γ ) u + η τ + p. Here te constant C = ma{, ( γ w ) }, were γ w is te wall sear rate. Hence, (7) is establised. For te upper bound (8), note tat τ ηd (u) + + ( γ ) u + η τ + p ( ) τ + ηd (u) + u + η τ + p τ + C 3 τ + C 4 u + p. Using te inverse inequalities we ave τ C τ p C p, J(u, p, τ; ) τ + C 5 τ + C 4 u + C 6 p ( ( + C 5 ) τ + C 4 u + ) C 6 p were C = ma {( + C 5 ), C 4, C 6 }. C ( τ + u + p ), Based on [7,8], we establis error estimates of a discrete nonlinear weigted least-squares finite element approimations in te following. In order to apply te results in [7,8], (5) is replaced wit a linearized form. Let (v, q, σ) = ( ṽ, q, σ ) + (v, q, σ ), were (v, q, σ ) ( is te initial guess ṽ, q, σ ) is te correction. Te nonlinear term in te least-squares functional is ten approimated by w s (σ ) (( ηd(v) w σ ) ( ηd(ṽ) + σ )) ηd(v ), were w is computed based on te initial guess (v, q, σ ). Let te initial guess (u, p, τ ) Φ. Ten te finite element approimation to (6) is equivalent to seek for ( u, p, τ ) Φ suc tat B ((u, p, τ ), (v, q, σ)) = F (v, q, σ), for all (v, q, σ) Φ, were B ((u, p, τ ), (v, q, σ)) = F (v, q, σ) = w + K + w ( τ ) ( ηd(u ) : σ ) ηd(v) d + K ( η τ + p ) ( η σ + q ( F τ + ) ( ηd(u ) : σ ) ηd(v) d (f u ) ( v) d + ) d ( u ) ( v) d (f 3 + ) ( η τ p ) η σ + q d. Note tat (9) can also be obtained by minimizing te functional (5) wit w s = w (called J ). Terefore, using arguments similar to tose in [7,8] Teorem, we can establis te unique minimizer of te discrete functional J in te following. (9)

5 H.-C. Lee, T.-F. Cen / Computers Matematics wit Applications 59 () Lemma. For any (u, p, τ) H ()d ( L () H () ) ( L () H (div; ) d), tere eist constants c C, independent of, suc tat c ( τ + p + u ) J (u, p, τ; ), J (u, p, τ; ) C ( τ + τ + u + p ). () If, in addition, (u, p, τ) Φ = V Q Σ te spaces Σ Q satisfy inverse inequalities τ C τ p C p, ten () can be replaced by J (u, p, τ; ) C ( τ + p + u ), for any <. Using Lemma te La Milgram Lemma, te following teorem is proved. Teorem. Te least-squares functional (5) as te unique minimizer out of te space Φ for any <. Te following Lemma, wic is a special case of te results proved in [7], can be derived based on te solvability of (). Lemma. Let U = (u, p, τ) H () d ( L () H () ) ( L () H (div; ) d) U = (u, p, τ ) Φ be as in Lemma. Ten tere eists a constant C suc tat τ τ + p p + u u CB (U U, U U ). Using similar arguments in [7] te approimation properties (4), te following error estimate is establised. Teorem 3. Let U = (u, p, τ) Φ ( H r+ () d H r+ () H r+ () d d) be te solution of te problem () U = (u, p, τ ) Φ denote te solution of te variational problem (9). Ten tere eists a constant C suc tat τ τ + p p + u u ( ) Cr+ τ r+ + p r+ + u r+. Note tat use of continuous piecewise linear polynomials for all unknowns yields te error estimate τ τ + p p + u u C ( τ + p + u ). It is optimal for te velocity in H norm but suboptimal for pressure p stress τ. Altoug te error bounds for te pressure stress are only O() in L, optimal rates of convergence are observed in te numerical results presented in Section 3. It appears tat te nonlinear weigt w s is essential for optimal convergence wen continuous piecewise linear polynomials are used for all unknowns. Te importance of w s in te least-squares approimations is currently under investigation. 3. Numerical results In tis section, two test problems are considered: te flow in te planar cannel te 4 to contraction problem. In our computations, linear basis functions are considered for all variables. Te first problem is te flow in a planar cannel on te square domain [, ] [, ] considered in [7]. A uniform directional triangular mes plotted in Fig. is used for all calculations. Te flow domain is sown in Fig.. Due to te symmetry along y =, te computed domain is reduced to alf. Te velocity u = [u, v] T is specified on te inflow, outflow, wall boundaries. Etra stress τ is specified on te inflow boundary. Pressure p is set to zero at te point were te outflow boundary meets te wall. On te symmetry boundary, te y-component of u τ y vanis. Te eact solutions in Cartesian coordinates are given in [7] by [ u eact = y 4 ], p eact =.

6 H.-C. Lee, T.-F. Cen / Computers Matematics wit Applications 59 () 5 4 Y X Fig.. Triangular mes corresponding to = 6. y u = u = u eact u = u eact τ = τ eact τ y =, v = Fig.. Test domain wit boundary conditions. Note tat te eact solutions are obtained by adding te source term F =, [ ] y f 3 =, te function f is identically zero in (). In our computations, te following least-squares functionals for () are considered:. Te least-squares functional (LS): J(v, q, σ; f) = η σ + q f 3 + σ ηd(v) + v.. Te weigted least-squares functional (WDLS): J(v, q, σ; f) = σ η σ + q f 3 + ηd(v) + K v. 3. Te nonlinear weigted least-squares functional (NL-WDLS): J(v, q, σ; f) = η σ + q f 3 + ws (σ ) ηd(v) + K v, were te nonlinear weigts w s is defined in (6). Te coefficient K = related to te mass conservation is considered first in te WDLS NL-WDLS functionals. Te L errors of various least-squares solutions for Stokes equations are sown in Fig. 3. From Fig. 3, observe tat te improvement

7 H.-C. Lee, T.-F. Cen / Computers Matematics wit Applications 59 () 5 4 a - b - L error in u -3 L error in τ c - L error in p Fig. 3. K = for WDLS NL-WDLS. L errors in (a) u, (b) τ (c) p of LS (o), WDLS (*) NL-WDLS (+) solutions. of te rate of convergence of te WDLS over LS. Tis is consistent wit results obtained for te Stokes equations, [7]. Note also tat te rate of convergence of te NL-WDLS improves over te WDLS. In fact, optimal convergence of te NL-WDLS are obtained in u, τ almost optimal convergence are obtained in p. Based on tese results, te rate of convergence can be restored wit careful selections of nonlinear weigting functions w s. In addition, te results suggest tat te w s considered in our computations is optimal. To investigate te influence of te mass conservation, in Fig. 4, we report results of various weigts K =,,, 6 8 in te nonlinear weigted least-squares (NL-WDLS) functionals. Observe tat optimal rates of convergence in L -norm for u, p τ are obtained by increasing K. Tese results indicate tat te solutions can be improved by increasing te importance of te mass conservation equation relative to te remaining ones. In addition, Fig. 4 sows tat results of te cases wen K =, 6, 8 are almost identical. Based on our eperience, convergent rates for te NL-WDLS solutions wit K = agree well wit tose of K >. Terefore, it is sufficient to coose K = for satisfactory results. To illustrate furter te capability of te NL-WDLS scemes, te following bencmark problem as been cosen: te Stokes flow pass troug a 4 to contraction cannel. Te computational domain te boundary conditions are described in Fig. 5. Due to te symmetry along y =, te computation domain is reduced to alf. Te ratio of te eigt of downstream upstream cannels is set to 4. Te boundary conditions are taken from tose given in []. On te symmetry boundary, te y-component of u τ y vanis. Pressure p is set to zero on te outlet of te domain. On te wall, all components of u are zero. In our computations, 5 is used wic corresponds to te upstream lengt X u = 8L te downstream lengt X d = L, were L is te downstream cannel alf-widt. In [], L 5L is considered a transient finite element metod is used to obtain ig-resolution solutions for viscoelastic 4 to planar contraction flow problems using a minimum spacing of.56. Altoug our domain 8L L is muc smaller tan te domain used in [], te flow is fully developed in tis region. Terefore, our results can be compared to tose obtained in []. In our computations, as illustrated in Fig. 6, te Union Jack grid wit te uniform mes spacing of.79 on [, ].5 on [, 5] is considered. In [], suc grid is illustrated to ave special properties not necessary possessed by oter configurations. Te computational results are presented based on te velocity component along te ais of symmetry corner vorte beaviors.

8 H.-C. Lee, T.-F. Cen / Computers Matematics wit Applications 59 () 5 4 a - b - L error in u -3 L error in τ c - L error in p Fig. 4. L errors in (a) u, (b) τ (c) p of NL-WDLS solutions using K = (o), K = (.), K = (), K = 6 (+) K = 8 (*). u = X R y u, τ 4L u = L p = X u τ y =, v = X d Fig. 5. Te 4 to contraction domain wit boundary conditions. Te WDLS NL-WDLS functionals are bot considered for tis 4 to contraction problem. Recall tat for te flow in a planar cannel, it is necessary to coose te proper weigt K in tese functionals for optimal results. Using weigts ranging from K = to 8 in te WDLS NL-WDLS functionals, te solutions u along te ais of symmetry are plotted in Figs. 7 8, respectively. Te results indicate tat wen K > 5, u along te ais of symmetry for te WDLS NL-WDLS solutions are similar agree well wit tose of K = 5. It is tus sufficient to coose K = 5 for satisfactory results. In addition, as illustrated from Figs. 7 8, u along y = on te outlet using te WDLS NL-WDLS metods are over.375 approimately.375, respectively. In Fig. 9, te streamline patterns of te WDLS NL-WDLS formulations using K = 5 are plotted. Observe tat te sizes of te corner vorte X R using te WDLS NL-WDLS are approimately.54.37,

9 H.-C. Lee, T.-F. Cen / Computers Matematics wit Applications 59 () Y X Fig. 6. Union Jack grid wit a minimum mes lengt of.79 near te singularity u(,) K= K= 3 K= 4 K= 5 K= 6 K= 8 u(,) K= K= 3 K= 4 K= 5 K= 6 K= Fig. 7. WDLS metod. Plots of u (left) along te ais of symmetry using K = (- -), K = 3 ( ), K = 4 (o), K = 5 (*), K = 6 (.) K = 8 (+) near te outlet (rigt). u(,) K=. K= 3 K= 4.5 K= 5 K= 6. K= u(,) K= K= 3 K= 4 K= 5 K= 6 K= Fig. 8. NL-WDLS metod. Plots of u (left) along te ais of symmetry using K = (- -), K = 3 ( ), K = 4 (o), K = 5 (*), K = 6 (.) K = 8 (+) near te outlet (rigt). respectively. Note tat in [], u along y = on te outlet is approimately.375 X R is approimately.375. Terefore, te NL-WDLS metod outperforms te WDLS metod gives results wic are compatible to tose presented in [].

10 4 H.-C. Lee, T.-F. Cen / Computers Matematics wit Applications 59 () 5 4 a 3 b Y.5 Y X X Fig. 9. Streamlines in u of (a) WDLS (b)nl-wdls metods. 4. Conclusions We ave presented a nonlinear weigted least-squares finite element approimation to Stokes problems. Comparisons are made wit least-squares formulations wit no weigts wit a simpler mes dependent weigting sceme. Based on te above, tere is a significant difference between te results among te least-squares approaces, confirming tat using linear polynomials in all variables, we are able to obtain optimal convergence in all variables in te NL-WDLS metods. Tis resolves one of te difficulties associated wit low order basis functions used in least-squares metod for Stokes equations, [7,8]. In addition, we ave illustrated te importance of te mass conservation equation in te least-squares functionals. In particular, using K = 5, te solutions can be improved greatly to acieve accurate results in te 4 to contraction problem. Tese results suggest tat te mass conservation constant varies wit te problems. Furtermore, numerical eperiments indicate tat te metod can be etended to more general 4 to contraction problems witout major difficulty. Finally, te error estimates in Section do not reflect te optimal convergence obtained in te NL-WDLS metods wen linear polynomials are used in all variables. Tese issues will be investigated furter in te future. Acknowledgements Te first autor was supported in part by Wenzao Ursuline College of Languages of Taiwan. Te second autor was supported in part by te National Science Council of Taiwan under contract number 96-5-M References [] P.B. Bocev, M.D. Gunzburger, Finite element metods of least-squares type, SIAM, Review 4 (998) [] A. Bose, G.F. Carey, Least-squares p-r finite element metods for incompressible non-newtonian flows, Comput. Metods Appl. Mec. Engrg. 8 (999) [3] Z. Cai, T.A. Manteufeel, S.F. Mccormick, First-order system least-squares for velocity-vorticity-pressure form of te Stokes equations, wit application to linear elastically, Eectronic Transactions on Numerical Analysis 3 (995) [4] G.F. Carey, A.I. Pelivanov, Y. Sen, A. Bose, K.C. Wang, Least-squares finite elements for fluid flow transport, Int. J. Numer. Metods Fluids 7 (998) [5] B.N. Jiang, C.L. Cang, A Least-squares finite elements for te Stokes problem, Comput. Metods Appl. Mec. Engrg. 78 (99) [6] H.C. Lee, Adaptive least-squares finite elements metods for viscoelastic flow problems, P.D. Tesis, Dept. of Matematics, National Cung Ceng University, Taiwan, 8. [7] P.B. Bocev, M.D. Gunzburger, Least-squares for te velocity pressure stress formulation of te Stokes equations, Comput. Metods Appl. Mec. Engrg. 6 (995) [8] S.D. Kim, B.C. Sin, H least-squares metod for te velocity pressure stress formulation of Stokes equation, Appl. Numer. Mat. 4 () [9] C.L. Cang, J. Nelson, Least-squares finite element metod for te Stokes problem wit zero residual of mass conservation, SIAM J. Numer. Anal. 34 (997) [] P. Bolton, R.W. Tatcer, On mass conservation in least-squares metods, J. Comput. Pys. 3 (5) [] V. Prabakar, J.P. Pontaza, J.N. Reddy, A collocation penalty least-squares finite element formulation for incompressible flows, Comput. Metods Appl. Mec. Engrg. 97 (8) [] J.M. Kim, C. Kim, J.H. Kim, C. Cunga, K.H. Ana, S.J. Lee, Hig-resolution finite element simulation of 4: planar contraction flow of viscoelastic fluid, J. Non-Newtonian Fluid Mec. 9 (5) [3] J.M. Deang, M.D. Gunzburger, Issues related to least-squares finite element metods for te Stoke equations, SIAM J. SCI. Comput. (3) (998) [4] T.F. Cen, Weigted least-squares approimations for nonlinear yperbolic equations, Comput. Mat. Appl. 48 (4) [5] T.F. Cen, C.L. Co, H.C. Lee, K.L. Tung, Least-squares finite elements for generalized Newtonian viscoelastic flows, Applied Numerical Matematics, 9 (submitted for publication). [6] Y. Son, Determination of sear viscosity sear rate from pressure drop flow rate relationsip in a rectangular cannel, Polymer 48 (7) [7] A. Fortin, R. Guenette, R. Pierre, On te discrete EVSS metod, Comput. Metods Appl. Mec. Engrg. 89 () 39.