Multigrid Methods for A Mixed Finite Element Method of The Darcy-Forchheimer Model

Noname manuscript No. (will be inserted by the editor) Multigrid Methods for A Mixed Finite Element Method of he Darcy-Forchheimer Model Jian Huang Long Chen Hongxing Rui Received: date / Accepted: date Abstract An e cient multigrid method for a mixed finite element method of the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. he nonlinear equation can be solved element wise with a closed formulae. he linear saddle point system for the constraint is further reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is found. By comparing the number of iterations and CPU time of di erent solvers in several numerical experiments, our method is shown to convergent with a rate independent of the mesh size and the nonlinearity parameter and the computational cost is nearly linear. Keywords Darcy-Forchheimer model Multigrid method Peaceman-Rachford iteration Introduction Darcy s law u = K rp, he wor of the authors Jian Huang and Hongxing Rui was supported by the National Natural Science Foundation of China Grant No. 933006, 790. Long Chen was supported by NSF Grant DMS-48934 and in part by NIH Grant P50GM7656. he wor of the author Jian Huang was supported by 04 China Scholarship Council (CSC). J. Huang H. Rui School of Mathematic, Shandong University, Jinan, Shandong 5000, China E-mail: yghuangjian@sina.com L.Chen Department of Mathematics, University of California at Irvine, Irvine, CA 9697, USA E-mail: chenlong@math.uci.edu H. Rui E-mail: hxrui@sdu.edu.cn

Jian Huang et al. describes the linear relationship between the velocity of the creep flow and the gradient of the pressure, which is valid when the Darcy velocity is extremely small []. Forchheimer in [8] carried out flow experiments and pointed out that when the velocity of the flow is relatively high, Darcy s law should be replaced by the so-called Darcy-Forchheimer (DF) equation by adding a quadratic nonlinear term to the velocity, shown as follows: K u + u u + rp = 0. A theoretical derivation of the Darcy-Forchheimer equation can be found in [7]. In recent years, many numerical methods of the Darcy-Forchheimer equation have been developed. A mixed finite element method with a semi-discrete scheme for the time dependent problem was considered by Par in []. Pan and Rui in [] gave a mixed element method for the Darcy-Forchheimer problem based on the Raviart-homas (R) element or the Brezzi-Douglas-Marini (BDM) element. Rui and Pan in [5] proposed the bloc-centered finite di erence method for the Darcy-Forchheimer model, which was thought of as the lowest-order Raviart- homas mixed element with proper quadrature formula. Rui, hao and Pan in [6] presented a bloc-centered finite di erence method for Darcy-Forchheimer model with variable Forchheimer number. Wang and Rui in [9] constructed a stabilized Crouzeix-Raviart (CR) element for the Darcy-Forchheimer equation. Rui and Liu in [4] introduced a two-grid bloc-centered finite di erence method for the Darcy- Forchheimer model. Girault and Wheeler in [9] proved the existence and uniqueness of the solution of the Darcy-Forchheimer model. hey approximated the velocity and the pressure by piecewise constant and nonconforming CR elements, respectively. hey also suggested an alternating directions iterative method to solve the nonlinear system. he convergences of both the iterative algorithm and the mixed finite element scheme are also proved. López, Molina and Salas in [0] carried out numerical tests of the methods studied in [9], and compared with the Newton s method for solving this problem. Since it is a nonlinear system, an iterative method should be used, which could be very computationally expensive if the convergence of nonlinear iteration is slow. Multigrid method is one of the most e cient methods on solving the nonlinear elliptic systems. It should be clarified that we no longer have a simple linear residual equation, which is the most significant di erence between linear and nonlinear systems. he multigrid scheme we used here is the most commonly used nonlinear version of multigrid. It is called the full approximation scheme (FAS) [3] because the problem in the coarse grid is solved for the full approximation rather than the correction; see Section 5 for details. We use piecewise constant and piecewise linear polynomial to discretize the velocity and the pressure, respectively. We shall apply FAS to construct an efficient V-cycle multigrid method for the nonlinear Darcy-Forchheimer equation and demonstrate the e ciency of our multigrid method. We use the Peaceman- Rachford (PR) type iteration developed in [9] as a smoother. he most relevant wor is [0] and our improvement are:. We choose a smaller value of the stopping criterion by achieving a better approximation of the pressure accuracy.. We report a better choice of the parameter for decoupling the nonliearity from the constraint rather than the suggested value = for di erent values of,and

Multigrid Method of he Darcy-Forchheimer Model 3 show the advantage of our choice by comparing the number of iterations and CPU time. 3. We carry out some experiments to show the e ciency of our multigrid solver by comparing with the PR iterative solvers. 4. We reduce the saddle point system into a Poisson system and demonstrate the e ciency of our approach. he remainder of this article is organized as follows: he model problem is demonstrated in Section. he mixed wea formulation and the discrete wea formulation are presented in Section 3. he PR iteration and an e cient solver for the linear saddle point systems are posted in Section 4. We construct a V-cycle multigrid scheme by applying FAS for the nonlinear problem in Section 5. Some numerical experiments using our multigrid method are carried out in Section 6 to verify that the e ciency of our method in comparison with solving this nonliear problem using the other iterative methods. Finally, conclusions, and further ideas are presented in Section 7. he Problem and Some Notations We consider the steady Darcy-Forchheimer flow of a single phase fluid in a porous medium in a two-dimensional bounded domain,withlipschitzcontinuousboundary @: with the divergence constraint and Neumann boundary conditions, K u + u u + rp = f in, (.) div u = g in, (.) u n = g N on @, (.3) where u and p are the velocity vector and the pressure, respectively;, and are given positive constants that represent the viscosity of the fluid, its density and its dynamic viscosity, respectively; denotes the Euclidean vector norm u = u u, n is the unit exterior normal vector to the boundary of the given domain ; K is the permeability tensor, assumed to be uniformly positive definite and bounded. According to the divergence theorem, g and g N are given given functions satisfying the compatibility condition g (x)dx = @ g N ( )d. (.4) We use the standard notations of the Sobolev spaces and the associated norms, see e.g.[]. We also use the space L 0 () = v L () : v (x) dx =0.

4 Jian Huang et al. 3WeaFormulation In this section, we define the function spaces as follows: X = L 3 (), M = W, 3 () \ L 0 (). where the zero mean value condition is added because p is only defined by the (.)- (.3) up to an additive constant. Given f L 3 (),g L 3 (@), the variational formulation is: find a pair (u,p)inx M such that K u ' dx + u (u ') dx (3.) + rp ' dx = f ' dx, 8' X, rq u dx = gq dx + g N qdx, 8q M. (3.) he variational formulation (3.), (3.) and the original problem (.)-(.3) are equivalent by using the Green s formula: v rqdx= q div v dx + hq, v ni @, 8q M,8v H, (3.3) where H = @ nv L 3 () :divv L 6 5 () o. If the given functions g and g N satisfy the compatibility condition (.4), then the problem has a unique solution (u,p)inx M [9]. Let be a polygon in two dimensions which can be completely triangulated by triangles. Let be a triangulation of, and the triangulations ( =, 3,...)be obtained form via regular subdivision, i.e. edge midpoints in are connected by new edges to form. herefore, is a family of conforming triangulations of, = [ for =,, 3,..., he family is regular (also called non-degenerate) in the sense of Ciarlet [6]. We discretize u and p in di erent finite element spaces. he velocity u is approximated in the following space: n o X = v L () : 8, v P 0, (3.4) and the pressure p is approximated in the following space: M = Q \ L 0 (), (3.5) where P m denotes the space of polynomials of degree m, andq is the linear finite element space. o Q = nq C 0 : 8,q P.

Multigrid Method of he Darcy-Forchheimer Model 5 With these spaces, we can have the -th level discrete formulation of the problem (3.),(3.): K u ' dx + + X X In universal practice, rq u dx = u (u ' ) dx rp ' dx = f ' dx, 8' X, (3.6) gq dx + g N q dx, 8q M. (3.7) @ h =h, for =, 3,... Note that are nested meshes, and thus X X,M M. In [8], the authors demonstrated that the discrete problem has a unique solution. Moreover, if is shape regular with mesh size h and the solution u belongs to W,4 () andp belongs to W, 3 (), then the following error estimations are obtained in [8](heorem 4.0): r (p u u h L () apple Ch u W,4 (), (3.8) p h ) 3 L apple Ch p ( ) W, 3 + u () W,4 (). (3.9) 4NonlinearIteration In this section, we present the Peaceman-Rachford (PR) type method developed in [9] to decouple the nonlinearity and the constraint. First, choose an initial guess u 0,p 0 by solving a linear Darcy equation: as K u 0 ' dx + X X rq u 0 dx = rp 0 ' dx = f ' dx, 8' X, (4.) gq dx + g N q dx, 8q M. (4.) @ he linear Darcy system (4.) and(4.) can be rewritten in the matrix form apple apple apple A B u fd B = 0 p w, (4.3) where A is the symmetric and positive definite matrix associated to the term K u ' dx,

6 Jian Huang et al. B is the matrix corresponding to X rp ' dx, and f d and w represent the right hand side of (4.) and(4.), respectively. hen, nowing u 0,p 0, construct a sequence u n+,p n+ for n 0intwo steps: Let be a positive parameter chosen to enhance the convergence.. A nonlinear step without constraint: nowing (u n,p n ) compute the intermediate velocity u n+ = u n+ u n by solving the following equation: ' dx + K u n ' dx X u n+ u n+ ' dx + u n+ u n+ ' dx = f ' dx rp n ' dx, 8' X. (4.4). A linear step with constraint: compute u n+,p n+ with the nown u n+ K u n+ ' dx + X rp n+ ' dx f ' dx X u n+ rq u n+ dx = u n+ ' dx, 8' X, (4.5) gq dx + g N q dx, 8q M. (4.6) @ A ey observation in [9] is that because the test functions ',thesolution u n+,andrp n are constant in each element,thenonlinearstep(4.4) canbe solved as follows: where F n+ = un K = u n+ = F n+ (4.7) = + K un r p n + f, K (x) dx, s +4 F n+. In the second step, the linear system (4.5) and(4.6) can be rewritten in the following matrix form: apple apple apple A B u fn+ B =, (4.8) 0 p w

Multigrid Method of he Darcy-Forchheimer Model 7 where A is the matrix corresponding to u n+ ' dx + and f n+ is the vector corresponding to f ' dx + u n+ ' dx K u n+ u n+ ' dx, u n+ ' dx. In [9], the authors proved that (4.), (4.) and(4.5), (4.6) haveauniquesolution. he iterative method is convergent for an arbitrary choice of the initial guess u 0,p 0 and an arbitrary positive. Numerically, choice of will a ect the convergence of the nonlinear iteration. We shall report a good choice of this parameter. We can reduce the linear saddle point system into a SPD system. Because of A and A are symmetric positive definite operators, without loss of generality, we tae (4.8) asanexampletoexpoundanideaasfollows: Eliminate u from the first equation of (4.8), i.e. u = A f n+ and then, substituting to the second equation of (4.8), we get Bp, (4.9) Mp = b, (4.0) where M = B A B,b = B A f n+ w. After solving (4.0), we can get u by solving (4.9). Since A is bloc-diagonal, A can be formed easily. Equation (4.0) isthe linear finite element dscretization of an elliptic equation in primary formulation. Solving the SPD system (4.0) ismucheasierthanthesaddlepointsystem(4.8) and many fast solvers are available. he equivalence between (4.9),(4.0) and(4.8) is obvious. 5MultigridAlgorithm In this section, we consider a generic system of nonlinear equations, L (z) =s where z, s R n.supposethatv is an approximation to the exact solution z. Define the error e and the residual r: e = z v, r = s L(v). We can solve this nonlinear system by using some iterative solver, L (z) =s, here means the discrete problem on the -th level mesh.

8 Jian Huang et al. Because of the iterative nature, multigird ideas should be e ective on the nonlinear problem. he multigrid scheme here we used for this nonlinear problem is the most commonly used nonlinear version of multigrid. It is called the full approximation scheme (FAS) [3] because the problem in the coarse grid is solved for the full approximation z = I v + e rather than the error e.av-cycle multigrid scheme is described as follows: Full Approximation Scheme (FAS). Pre-smoothing: For apple j apple m, relax m times with an initial guess v 0 by v j = R v j. he current approximation v = v m. Restrict the current approximation and its fine grid residual to the coarse grid: r = I (s L (v )) and v = I v. Solve the coarse grid problem: L (z )=L (v )+r. Compute the coarse grid approximation to the error: e = z v. Interpolate the error approximation up to the fine grid and correct the current fine grid approximation: v m+ v + I e. Post-smoothing: For m +apple j apple m +, relax m times by v j = R 0 v j. then we get the approximate solution v m+. Here m denotes the number of presmoothing and post-smoothing steps, R denotes the chosen relaxation method, and I is an intergrid transfer operator from the fine grid to the coarse grid. We apply the PR iteration as the smoother R. We need to switch the ordering of the linear and nonlinear steps in the post-smoothing step in order to eep the symmetry of the V-cycle. It is worth pointing out that although the chosen finite element spaces are nested, the constrained subspaces are non-nested when we interpolated the correction of the velocity, which was obtained in the coarser space, to the finer space. Namely, if we directly interpolated the correction obtained on the coarser grid to the finer grid, the approximation we got do not satisfy the divergence equation in this Darcy-Forchheimer model. We need to construct a L projection to map the correction obtained before into the constrained space in the fine grid. his can be realized by solving a saddle point system: apple apple apple A B 0 B = 0 B, (5.) e u where A is the matrix corresponding to K ' dx + v ( ' ) dx,, represent the error between the restriction of the approximation of velocity and pressure on the finer grid and their approximation obtained on the coarser grid, respectively, and e u is the prolonged correction to the fine space. Again, (5.) can be reduced to a SPD system. We can get = A B through the idea demonstrated in Section 4. hen we obtain a corrected approximation of velocity v = v, which satisfied the divergence equation.

Multigrid Method of he Darcy-Forchheimer Model 9 6NumericalExperiments In this section, we present some numerical results to illustrate the e ciency of our multigrid method for the Darcy-Forchheimer equation (.)-(.3). he following test problems were taen from [0]. All of our experiments are implemented based on the MALAB c software pacage ifem [4]. We choose =, =,K = I. R is a square, Problem : = n o (x, y) R : <x<, <y<. u (x, y) =[x + y, x y], p (x, y) =x 3 + y 3, + p 3 x f (x, y) = 4 +y (x + y)+3x + p 5, x +y (x y)+3y 8 +y, x =, >< y, x =, g N (x, y) = x, y =, >: x, y =. Problem : apple (x +) u (x, y) =, 4 p (x, y) =x 3 + y 3, (x +)(y +), f (x, y) = 6 4 (x+) 4 (x+)(y+) + (x+) 4 + (x+) 4 3 q(x +) +4(y +) +3x 7 q(x +) +4(y +) +3y 5, 8, x =, >< 0, x =, g N (x, y) = x, y =, >: 0, y =. For all above test problems, g = 0. he chosen termination criterion is r = r u + r p apple tol,

0 Jian Huang et al. Rate of convergence is CN 0.508 Rate of convergence is CN 0.5067 u u h L 0 C N 0.5074 p p h L 0 C N 0.77647 0 3 H N 0.508 0 3 u u h L C N 0.5074 0 4 0 5 0 6 (a) Convergence Rate of Problem with tol = 0 5 0 4 0 5 L C N 0.9799 H N 0.5067 0 5 0 6 (b) Convergence Rate of Problem with tol = 0 6 Rate of convergence is CN 0.505 Rate of convergence is CN 0.5068 0 0 0 3 u u h L C N 0.49463 0 3 u u h L C N 0.49463 0 4 L C N 0.85863 0 4 L C N 0.96944 0 5 H N 0.505 0 5 0 6 (c) Convergence Rate of Problem with tol = 0 5 0 5 H N 0.5068 0 5 0 6 (d) Convergence Rate of Problem with tol = 0 6 Fig. Comparison of the convergence rate for Problem and with di erent tols. where 8 < f r u = : f r p = K u n h + u n h u n h + rp n h / f, when f 6=0, K u n h + u n h u n h + rp n h, when f =0. ( g divu n h / g, when g 6=0, g divu n h, when g =0. In the first test, we study that whether the accuracy will change according to di erent choices of tol. Numerical tests were performed for all these problems, and the behaviour of these experiments was similar for all these problems. herefore, here we only posed the results when =, =0. he letter N stands for of p, which is the same as Numbers of vertices, so h = p N, which represents the discretization mesh size in one direction. he results confirmed the convergence order for u u h L and p p h H are O (h). However, the accuracy of the pressure approximations in L -norm depends on tol. From Figure, itisclearthatthel -norm of the pressure approximations when tol =0 6 is better than the one when tol =0 5 for Problem and Problem. Meanwhile, in consideration of the computation

Multigrid Method of he Darcy-Forchheimer Model able Comparison of di erent values of with h = 64 for =0, 0, 30. Problem Problem Problem =0 =0 =30 = =/0 = =/0 = =/30 iter 9 73 457 05 686 0 CPU time 4 s 4 s 6 s 6 s 38 s 7 s iter 30 7 459 83 688 9 CPU time 3 s 0 s 6 s s 38 s s able Comparison of di erent values of with h = 64 for =40, 50, 60. Problem Problem Problem =40 =50 =60 = =/40 = =/50 = =/60 iter 94 6 43 9 37 3 CPU time 53 s 7 s 66 s 7 s 79 s 8 s iter 97 98 46 05 376 3 CPU time 5 s s 65 s s 79 s s cost, the su ciently accurate results were achieved when tol =0 6 for Problem and. herefore, the value of tol =0 6 is used in the remaining numerical experiments. Instead, in [0] the authors use tol =.95h, which only considered the L -norm approximation for velocity. In the second test, we give an empirical choice of parameter = /.As it is mentioned in [9], the PR nonlinear iteration converges for any >0. Its convergence rate, however, is very sensitive to the choice of this parameter. From the convergence proof of the PR algorithm in [9], we inferred that the choices of depends on. Here we tested =/, then compared with the choice = in two aspects: the number of iterations (abbreviated as iter) and CPU time. As shown in able and, this choice of the parameter is much better than the fixed selection for di erent values of. herefore, this choice of will be used in the remaining numerical experiments. In the following experiments, we compare the multigrid method with the PR iterative method for solving this nonlinear systems. Here we choose m = 3 for all the following tests. It means that we apply three PR iterations in the pre-smoothing step and post-smoothing step, respectively. In order to eep the symmetry of the V-cycle, we switch the ordering of the linear and nonlinear steps in the postsmoothing step. We set h =/6 as the initial mesh. In the tables, we use the following symbols: E u,0 = u u h L,E p,0 = p p h L,E p, = p p h H. he PR solver is denoted by s, whereas the multigrid solver is denoted by m. I - number of iterations, and CPU - CPU time. s represents that we solve those linear saddle point systems directly in each step while we use the PR iterative method to solve this nonlinear system, s is that we solve the SPD system mentioned in Section 4 rather than solving the saddle point system directly. Our multigrid solver is constructed based on s. In all examples we achieve optimal order convergence of u u h L and p p h H. Since our focus is on the e - ciency of solvers, we mainly report the comparison of the number of iterations and CPU time by using di erent solvers.

Jian Huang et al. able 3 Comparison of error of Problem by using di erent solvers with =0, =/0. h dof E s u,0 E s p,0 E s p, E m u,0 E m p,0 E m p, 5,85 6.76e-3.8e-.860e- 6.76e-3.8e-.860e- 6 3 0,609 3.3e-3.960e-3 8.979e- 3.3e-3.960e-3 8.979e- 64 83,77.596e-3 7.404e-4 4.440e-.596e-3 7.404e-4 4.440e- 8 38,93 7.95e-4.85e-4.3e- 7.953e-4.85e-4.3e- 56,3,745 3.97e-4 4.68e-5.05e- 3.973e-4 4.68e-5.05e- 5 5,44,99.986e-4.57e-5 5.55e-3.986e-4.57e-5 5.55e-3 Rate -.0.05.0.0.05.0 able 4 Comparison of number of iterations and CPU time of Problem by using di erent solvers with =0. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 4 0.6 s 0.34 s 0.7 s 6 3 0,609 59 4. s 0.88 s 0.49 s 64 83,77 73 4 5.0 s 4.3 s.6 s 8 38,93 84 4 34.9 s 6. s 7.4 s 56,3,745 95 4 95. s 65. s 3.8 s 5 5,44,99 99 3 687.7 s 875.4 s 60.5 s 0 Rate of convergence is CN 0.507 0 4 s C N.4667 ime complexcity 0 0 3 s C N.654 0 3 u u h L C N 0.5099 ime (s) 0 multigrid N.0566 L 0 0 4 C N.0077 p p h H N 0.507 0 5 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 0 0 4 0 5 0 6 (b) ime complexity Fig. Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =0. It can be observed that our multigrid solver required significantly fewer iterations and CPU time than the other two solvers, and the time complexity of our multigrid solver is almost O(N), in contrast, the time complexity of the other two solvers seems to be more computationally expensive than that of our solver, which demonstrated the e ciency of our approach. As shown in able 9 and 6, the number of iterations are compared for di erent beta and it is demonstrated that our multigrid method is robust to both h and.

Multigrid Method of he Darcy-Forchheimer Model 3 able 5 Comparison of number of iterations and CPU time of Problem by using di erent solvers with =0. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 49 0.68 s 0.43 s 0.34 s 6 3 0,609 75 6.7 s. s 0.6 s 64 83,77 05 6 4.6 s 5. s.0 s 8 38,93 6 5 95. s 3.5 s 9.5 s 56,3,745 39 5 3. s 09.5 s 5.0 s 5 5,44,99 53 5 0354. s 380. s 9. s 0 0 0 3 Rate of convergence is CN 0.5587 u u h L C N 0.5589 p p h L C N.0077 p p h H N 0.5587 ime (s) 0 4 0 3 0 s C N.4848 s C N.304 multigrid N.087 ime complexcity 0 0 4 0 0 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 4 0 5 0 6 (b) ime complexity Fig. 3 Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =0. able 6 Comparison of number of iterations and CPU time of Problem by using di erent solvers with =30. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 50 0.70 s 0.43 s 0.34 s 6 3 0,609 8 6 3.0 s. s 0.65 s 64 83,77 0 6 8.6 s 6.6 s.3 s 8 38,93 54 6 4.3 s 48.8 s. s 56,3,745 68 6 554.7 s 308.3 s 56.5 s 5 5,44,99 85 5 857.3 s 667.7 s 54.6 s able 7 Comparison of number of iterations and CPU time of Problem by using di erent solvers with =40. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 50 0.80 s 0.43 s 0.35 s 6 3 0,609 8 7 3. s. s 0.70 s 64 83,77 6 7 30.5 s 7.7 s.4 s 8 38,93 7 6 69.9 s 59.5 s 3. s 56,3,745 9 6 786.3 s 39.6 s 63.9 s 5 5,44,99 07 6 3504. s 89.4 s 70.4 s

4 Jian Huang et al. Rate of convergence is CN 0.54598 ime complexcity 0 0 0 3 u u h L C N 0.5433 p p h L C N.0077 H C N 0.54598 3 ime (s) 0 4 0 3 0 0 s C N.4905 s C N.3406 multigrid N.0979 0 4 0 0 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 4 0 5 0 6 (b) ime complexity Fig. 4 Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =30. Rate of convergence is CN 0.5673 ime complexcity 0 0 u u h L C N 0.56075 L C N.0077 H N 0.5673 ime (s) 0 4 0 3 0 s C N.5 s C N.3468 multigrid N.07 0 3 0 0 4 0 0 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 4 0 5 0 6 (b) ime complexity Fig. 5 Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =40. able 8 Comparison of number of iterations and CPU time of Problem by using di erent solvers with =50. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 50 0.9 s 0.45 s 0.35 s 6 3 0,609 8 7 3. s. s 0.7 s 64 83,77 9 7 3.4 s 7.9 s.6 s 8 38,93 8 7 3.6 s 6.9 s 3.5 s 56,3,745 0 6 955.6 s 37. s 64.8 s 5 5,44,99 6 4733.5 s 009.4 s 306.8 s

Multigrid Method of he Darcy-Forchheimer Model 5 Rate of convergence is CN 0.5887 ime complexcity 0 0 u u h L C N 0.5793 p p h L C N.0077 H C N 0.5887 3 ime (s) 0 4 0 3 0 s C N.57 s C N.368 multigrid N. 0 3 0 0 4 0 0 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 4 0 5 0 6 (b) ime complexity Fig. 6 Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =50. able 9 Comparison of iteration steps of multigrid solver according to di erent h and Problem with =/. for h =0 =0 =30 =40 =50 4 6 6 7 7 3 64 4 6 6 7 7 8 4 5 6 6 7 56 4 5 6 6 6 5 3 5 5 6 6 able 0 Comparison of error of Problem by using di erent solvers with =0, =/0. h dof E s u,0 E s p,0 E s p, E m u,0 E m p,0 E m p, 5,85 3.44e- 5.863e-3.77e- 3.44e- 5.863e-3.77e- 6 3 0,609.740e-.680e-3 8.846e-.740e-.680e-3 8.846e- 64 83,77 9.048e-3 4.676e-4 4.4e- 9.04e-3 4.676e-4 4.4e- 8 38,93 4.60e-3.58e-4.0e- 4.60e-3.50e-4.0e- 56,3,745.33e-3 3.85e-5.05e-.34e-3 3.50e-5.05e- 5 5,44,99.65e-3 8.448e-6 5.54e-3.59e-3 8.88e-6 5.54e-3 Rate - 0.970.904.009 0.97.90.009 able Comparison of number of iterations and CPU time of Problem by using di erent solvers with =0. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 75 0.87 s 0.5 s 0.9 s 6 3 0,609 5 4.0 s.4 s 0.95 s 64 83,77 7 5 4.6 s 0.7 s.4 s 8 38,93 69 5 437.3 s 9.8 s 8.7 s 56,3,745 48 4 3983. s 733.5 s 43. s 5 5,44,99 686 4 > 3 hours 633.6 s 03.3 s

6 Jian Huang et al. Rate of convergence is CN 0.5043 ime complexcity 0 0 0 3 u u h L C N 0.4859 ime (s) 0 4 0 3 0 s C N.689 s C N.569 multigrid N 0.9878 0 4 p p h L C N 0.95487 0 0 5 p p h H N 0.5043 0 0 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 4 0 5 0 6 (b) ime complexity Fig. 7 Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =0. able Comparison of number of iterations and CPU time of Problem by using di erent solvers with =0. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 84 0.94 s 0.5 s 0.36 s 6 3 0,609 7 4. s.5 s 0.96 s 64 83,77 83 7 45.8 s.3 s 3.0 s 8 38,93 88 7 465.8 s 93.8 s 5.3 s 56,3,745 459 6 4338.6 s 776.5 s 65.5 s 5 5,44,99 73 5 > 3 hours 6350.6 s 68.4 s Rate of convergence is CN 0.5054 ime complexcity 0 0 0 3 u u h L C N 0.4759 ime (s) 0 4 0 3 0 s C N.6954 s C N.576 multigrid N.0404 0 4 p p h L C N 0.948 0 p p h H C N 0.5054 3 0 5 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 0 0 4 0 5 0 6 (b) ime complexity Fig. 8 Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =0.

Multigrid Method of he Darcy-Forchheimer Model 7 able 3 Comparison of number of iterations and CPU time of Problem by using di erent solvers with =30. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 9 0.96 s 0.54 s 0.39 s 6 3 0,609 8 9 4.6 s.6 s.0 s 64 83,77 9 9 46.5 s.8 s 3.8 s 8 38,93 96 9 46.9 s 98.3 s 8. s 56,3,745 468 8 44.9 s 79.6 s 83.6 s 5 5,44,99 746 7 > 4 hours 6440.3 s 357. s Rate of convergence is CN 0.5073 ime complexcity 0 0 0 3 u u h L C N 0.46878 ime (s) 0 4 0 3 0 s C N.6846 s C N.5086 multigrid N.0765 L 0 4 C N 0.94006 H N 0.5073 0 0 0 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 4 0 5 0 6 (b) ime complexity Fig. 9 Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =30. able 4 Comparison of number of iterations and CPU time of Problem by using di erent solvers with =40. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 00.0 s 0.63 s 0.39 s 6 3 0,609 34 4.6 s.6 s. s 64 83,77 98 49.9 s. s 4. s 8 38,93 306 0 500.4 s 98.8 s.6 s 56,3,745 48 9 453.5 s 80.5 s 96. s 5 5,44,99 759 8 > 5 hours 6875. s 405. s 7Conclusions In this paper, we constructed a nonlinear multigrid method for a mixed finite element method of the two-dimensional Darcy-Forchheimer model. We presented a comparative study between the multigrid solver and the PR iterative solver, at the same time compared CPU time of the e cient solver of solving the SPD systems with that obtained by solving the linear saddle point systems directly. We too into account the pressure accuracy when we set the termination criterion, and chose a better value of the stopping criterion tol. In comparison with the authors in [0] alwayschose = for di erent values of, we reported a better choice and

8 Jian Huang et al. Rate of convergence is CN 0.5094 ime complexcity 0 0 0 3 u u h L C N 0.46385 ime (s) 0 4 0 3 0 s C N.6896 s C N.57 multigrid N.0837 L 0 4 C N 0.9475 p p h H 0 N 0.5094 0 0 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 4 0 5 0 6 (b) ime complexity Fig. 0 Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =40. able 5 Comparison of number of iterations and CPU time of Problem by using di erent solvers with =50. h dof I(s) I(m) CPU(s) CPU(s) CPU(m) 5,85 09. s 0.6 s 0.4 s 6 3 0,609 4 4.8 s.8 s. s 64 83,77 05 5.7 s.8 s 5. s 8 38,93 33 53.3 s 08.7 s 4.4 s 56,3,745 489 0 479.3 s 850. s 08.5 s 5 5,44,99 770 9 > 5 hours 6900.3 s 458.6 s Rate of convergence is CN 0.5 ime complexcity 0 0 0 3 u u h L C N 0.46008 ime (s) 0 4 0 3 0 s C N.687 s C N.5007 multigrid N.084 L C N 0.94685 0 0 4 H N 0.5 0 0 0 4 0 5 0 6 (a) Convergence rate by using multigrid solver 0 4 0 5 0 6 (b) ime complexity Fig. Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =50.

Multigrid Method of he Darcy-Forchheimer Model 9 able 6 Comparison of iteration steps of multigrid solver according to di erent h and Problem with =/. for h =0 =0 =30 =40 =50 5 7 9 3 64 5 7 9 8 5 7 9 0 56 4 6 8 9 0 5 4 5 7 8 9 compared with the previous choice through comparing the number of iterations and CPU time. he results obtained from our tests indicate that the multigrid solver is very e cient for numerically solving this nonlinear elliptic equation. he number of iterations and CPU time for using multigrid solver are shown to be significantly less than that obtained by using the other solvers. In the future wor, we shall extend our results to two directions. One is that we would lie to find a better smoother, which is used in the pre-smoothing and postsmoothing step, to reduce CPU time and mae the multigrid solver more e cient. Another is that we intend to carry out some studies on the three-dimensional Darcy-Forchheimer problem and the real application in a porous medium. References. Adams, R. A., Sobolev Spaces, Academic Press, New Yor (975). Aziz, K., Settari, A., Petroleum Reservoir Simulation, Applied Science Publishers LD, London (979) 3. Briggs, W. L., Henson, V. E., McCormic, S. F., A Multigrid tutorial, ndedition, SIAM, Philadelphia (000) 4. Chen, L., ifem: an integrated finite element methods pacage in MALAB, echnical Report, echnical Report, University of California at Irvine (009) 5. Chen, L., Multigrid methods for saddle point systems using constrained smoothers, Computers and Mathematics with Applications, 70(), 854-866 (05) 6. Ciarlet, P. G., he Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New Yor, Oxford (978) 7. Crouzeix, M., Raviart, P. A., Conforming and non-conforming finite element methods for solving the stationary Stoes problem, RAIRO Anal. Numér, 8, 33-76 (973) 8. Forchheimer, P., Wasserbewegung durch Boden,. Ver. Deutsch. Ing., 45, 78788 (90) 9. Girault, V., Wheeler, M. F., Numerical discretization of a Darcy-Forchheimer model, Numer. Math., 0, 6-98 (008) 0. López, H., Molina, B., Salas, José J., Comparison between di erent numerical discretizations for a DarcyForchheimer model, ENA 34, 87-03 (009). Pan H., Rui, H., Mixed element method for two-dimensional Darcy-Forchheimer model, J.Sci. Comput., 5, 563587 (0). Par, E. J., Mixed finite element method for generalized Forchheimer flow in porous media, Numer. Methods Partial Di erential Equations,, 38 (005) 3. Peaceman, D. W., Rachford, H. H., he numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math., 3, 8-4 (955) 4. Rui, H., Liu, W., A two-grid bloc-centered finite di erence method for Darcy-Forchheimer flow in porous media, SIAM J. Numer. Anal., 53(4), 94-96 (05) 5. Rui, H., Pan H., A bloc-centered finite di erence method for the Darcy-Forchheimer model, SIAM J. Numer. Anal., 50(5), 663 (0) 6. Rui, H., hao, D., Pan H., A bloc-centered finite di erence method for Darcy-Forchheimer model with variable Forchheimer number, Numer. Methods Partial Di erential Equations,3, 603-6 (05)

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