Multigrid Methods for A Mixed Finite Element Method of The Darcy-Forchheimer Model
|
|
- Jodie Strickland
- 5 years ago
- Views:
Transcription
1 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 , 790. Long Chen was supported by NSF Grant DMS 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 yghuangjian@sina.com L.Chen Department of Mathematics, University of California at Irvine, Irvine, CA 9697, USA chenlong@math.uci.edu H. Rui hxrui@sdu.edu.cn
2 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
3 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 with the divergence constraint and Neumann boundary conditions, K u + u u + rp = f in, (.) div u = g in, (.) u n = g N (.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 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 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.
5 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. 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. 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 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. 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
7 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 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.
9 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,
10 0 Jian Huang et al. Rate of convergence is CN Rate of convergence is CN u u h L 0 C N p p h L 0 C N H N u u h L C N (a) Convergence Rate of Problem with tol = L C N H N (b) Convergence Rate of Problem with tol = 0 6 Rate of convergence is CN Rate of convergence is CN u u h L C N u u h L C N L C N L C N H N (c) Convergence Rate of Problem with tol = H N (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
11 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 CPU time 4 s 4 s 6 s 6 s 38 s 7 s iter 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 CPU time 53 s 7 s 66 s 7 s 79 s 8 s iter 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.
12 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, e-3.8e-.860e- 6.76e-3.8e-.860e , e-3.960e e- 3.3e-3.960e e ,77.596e e e-.596e e e- 8 38, e-4.85e-4.3e e-4.85e-4.3e- 56,3, e e-5.05e e e-5.05e- 5 5,44,99.986e-4.57e e-3.986e-4.57e e-3 Rate 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, s 0.34 s 0.7 s 6 3 0, s 0.88 s 0.49 s 64 83, s 4.3 s.6 s 8 38, s 6. s 7.4 s 56,3, s 65. s 3.8 s 5 5,44, s s 60.5 s 0 Rate of convergence is CN s C N.4667 ime complexcity s C N u u h L C N ime (s) 0 multigrid N.0566 L C N.0077 p p h H N (a) Convergence rate by using multigrid solver (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.
13 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, s 0.43 s 0.34 s 6 3 0, s. s 0.6 s 64 83, s 5. s.0 s 8 38, s 3.5 s 9.5 s 56,3, s 09.5 s 5.0 s 5 5,44, s 380. s 9. s Rate of convergence is CN u u h L C N p p h L C N.0077 p p h H N ime (s) s C N.4848 s C N.304 multigrid N.087 ime complexcity (a) Convergence rate by using multigrid solver (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, s 0.43 s 0.34 s 6 3 0, s. s 0.65 s 64 83, s 6.6 s.3 s 8 38, s 48.8 s. s 56,3, s s 56.5 s 5 5,44, s 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, s 0.43 s 0.35 s 6 3 0, s. s 0.70 s 64 83, s 7.7 s.4 s 8 38, s 59.5 s 3. s 56,3, s 39.6 s 63.9 s 5 5,44, s 89.4 s 70.4 s
14 4 Jian Huang et al. Rate of convergence is CN ime complexcity u u h L C N p p h L C N.0077 H C N ime (s) s C N.4905 s C N.3406 multigrid N (a) Convergence rate by using multigrid solver (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 ime complexcity 0 0 u u h L C N L C N.0077 H N ime (s) s C N.5 s C N.3468 multigrid N (a) Convergence rate by using multigrid solver (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, s 0.45 s 0.35 s 6 3 0, s. s 0.7 s 64 83, s 7.9 s.6 s 8 38, s 6.9 s 3.5 s 56,3, s 37. s 64.8 s 5 5,44, s s s
15 Multigrid Method of he Darcy-Forchheimer Model 5 Rate of convergence is CN ime complexcity 0 0 u u h L C N p p h L C N.0077 H C N ime (s) s C N.57 s C N.368 multigrid N (a) Convergence rate by using multigrid solver (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 = 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, e e-3.77e- 3.44e e-3.77e , e-.680e e-.740e-.680e e , e e-4 4.4e- 9.04e e-4 4.4e- 8 38, e-3.58e-4.0e- 4.60e-3.50e-4.0e- 56,3,745.33e e-5.05e-.34e e-5.05e- 5 5,44,99.65e e e-3.59e e e-3 Rate 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, s 0.5 s 0.9 s 6 3 0, s.4 s 0.95 s 64 83, s 0.7 s.4 s 8 38, s 9.8 s 8.7 s 56,3, s s 43. s 5 5,44, > 3 hours s 03.3 s
16 6 Jian Huang et al. Rate of convergence is CN ime complexcity u u h L C N ime (s) s C N.689 s C N.569 multigrid N p p h L C N p p h H N (a) Convergence rate by using multigrid solver (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, s 0.5 s 0.36 s 6 3 0, s.5 s 0.96 s 64 83, s.3 s 3.0 s 8 38, s 93.8 s 5.3 s 56,3, s s 65.5 s 5 5,44, > 3 hours s 68.4 s Rate of convergence is CN ime complexcity u u h L C N ime (s) s C N.6954 s C N.576 multigrid N p p h L C N p p h H C N (a) Convergence rate by using multigrid solver (b) ime complexity Fig. 8 Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =0.
17 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, s 0.54 s 0.39 s 6 3 0, s.6 s.0 s 64 83, s.8 s 3.8 s 8 38, s 98.3 s 8. s 56,3, s 79.6 s 83.6 s 5 5,44, > 4 hours s 357. s Rate of convergence is CN ime complexcity u u h L C N ime (s) s C N.6846 s C N.5086 multigrid N.0765 L 0 4 C N H N (a) Convergence rate by using multigrid solver (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, s 0.63 s 0.39 s 6 3 0, s.6 s. s 64 83, s. s 4. s 8 38, s 98.8 s.6 s 56,3, s 80.5 s 96. s 5 5,44, > 5 hours 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
18 8 Jian Huang et al. Rate of convergence is CN ime complexcity u u h L C N ime (s) s C N.6896 s C N.57 multigrid N.0837 L 0 4 C N p p h H 0 N (a) Convergence rate by using multigrid solver (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, s 0.6 s 0.4 s 6 3 0, s.8 s. s 64 83, s.8 s 5. s 8 38, s 08.7 s 4.4 s 56,3, s 850. s 08.5 s 5 5,44, > 5 hours s s Rate of convergence is CN 0.5 ime complexcity u u h L C N ime (s) s C N.687 s C N.5007 multigrid N.084 L C N H N (a) Convergence rate by using multigrid solver (b) ime complexity Fig. Convergence rate by using multigrid solver and time complexity by using di erent solvers for Problem with =50.
19 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 = 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(), (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, (973) 8. Forchheimer, P., Wasserbewegung durch Boden,. Ver. Deutsch. Ing., 45, (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, (009). Pan H., Rui, H., Mixed element method for two-dimensional Darcy-Forchheimer model, J.Sci. Comput., 5, (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), (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, (05)
20 0 Jian Huang et al. 7. Ruth, D., Ma, H., On the derivation of the Forchheimer equation by means of the averaging theorem, ransport in Porous Media, 7, 5564 (99) 8. Salas, José J., López, H., Molina, B., An analysis of a mixed finite element method for a DarcyForchheimer model, Mathematical and Computer Modelling 57, (03) 9. Wang, Y., Rui, H., Stabilized CrouzeixRaviart element for DarcyForchheimer model, Numer. Methods Partial Di erential Equations,3, (05)
Multigrid Methods for Saddle Point Problems
Multigrid Methods for Saddle Point Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University Advances in Mathematics of Finite Elements (In
More informationGeometric Multigrid Methods
Geometric Multigrid Methods Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University IMA Tutorial: Fast Solution Techniques November 28, 2010 Ideas
More informationINTERGRID OPERATORS FOR THE CELL CENTERED FINITE DIFFERENCE MULTIGRID ALGORITHM ON RECTANGULAR GRIDS. 1. Introduction
Trends in Mathematics Information Center for Mathematical Sciences Volume 9 Number 2 December 2006 Pages 0 INTERGRID OPERATORS FOR THE CELL CENTERED FINITE DIFFERENCE MULTIGRID ALGORITHM ON RECTANGULAR
More informationAn Algebraic Multigrid Method for Eigenvalue Problems
An Algebraic Multigrid Method for Eigenvalue Problems arxiv:1503.08462v1 [math.na] 29 Mar 2015 Xiaole Han, Yunhui He, Hehu Xie and Chunguang You Abstract An algebraic multigrid method is proposed to solve
More informationEfficient smoothers for all-at-once multigrid methods for Poisson and Stokes control problems
Efficient smoothers for all-at-once multigrid methods for Poisson and Stoes control problems Stefan Taacs stefan.taacs@numa.uni-linz.ac.at, WWW home page: http://www.numa.uni-linz.ac.at/~stefant/j3362/
More informationA MULTIGRID ALGORITHM FOR. Richard E. Ewing and Jian Shen. Institute for Scientic Computation. Texas A&M University. College Station, Texas SUMMARY
A MULTIGRID ALGORITHM FOR THE CELL-CENTERED FINITE DIFFERENCE SCHEME Richard E. Ewing and Jian Shen Institute for Scientic Computation Texas A&M University College Station, Texas SUMMARY In this article,
More informationMultilevel Preconditioning of Graph-Laplacians: Polynomial Approximation of the Pivot Blocks Inverses
Multilevel Preconditioning of Graph-Laplacians: Polynomial Approximation of the Pivot Blocks Inverses P. Boyanova 1, I. Georgiev 34, S. Margenov, L. Zikatanov 5 1 Uppsala University, Box 337, 751 05 Uppsala,
More information1. Introduction. The Stokes problem seeks unknown functions u and p satisfying
A DISCRETE DIVERGENCE FREE WEAK GALERKIN FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU, JUNPING WANG, AND XIU YE Abstract. A discrete divergence free weak Galerkin finite element method is developed
More informationarxiv: v1 [math.na] 11 Jul 2011
Multigrid Preconditioner for Nonconforming Discretization of Elliptic Problems with Jump Coefficients arxiv:07.260v [math.na] Jul 20 Blanca Ayuso De Dios, Michael Holst 2, Yunrong Zhu 2, and Ludmil Zikatanov
More informationPSEUDO-COMPRESSIBILITY METHODS FOR THE UNSTEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS
PSEUDO-COMPRESSIBILITY METHODS FOR THE UNSTEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS Jie Shen Department of Mathematics, Penn State University University Par, PA 1680, USA Abstract. We present in this
More informationA Mixed Nonconforming Finite Element for Linear Elasticity
A Mixed Nonconforming Finite Element for Linear Elasticity Zhiqiang Cai, 1 Xiu Ye 2 1 Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395 2 Department of Mathematics and Statistics,
More informationA Two-grid Method for Coupled Free Flow with Porous Media Flow
A Two-grid Method for Coupled Free Flow with Porous Media Flow Prince Chidyagwai a and Béatrice Rivière a, a Department of Computational and Applied Mathematics, Rice University, 600 Main Street, Houston,
More informationResearch Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations
Abstract and Applied Analysis Volume 212, Article ID 391918, 11 pages doi:1.1155/212/391918 Research Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations Chuanjun Chen
More informationITERATIVE METHODS FOR NONLINEAR ELLIPTIC EQUATIONS
ITERATIVE METHODS FOR NONLINEAR ELLIPTIC EQUATIONS LONG CHEN In this chapter we discuss iterative methods for solving the finite element discretization of semi-linear elliptic equations of the form: find
More informationPARTITION OF UNITY FOR THE STOKES PROBLEM ON NONMATCHING GRIDS
PARTITION OF UNITY FOR THE STOES PROBLEM ON NONMATCHING GRIDS CONSTANTIN BACUTA AND JINCHAO XU Abstract. We consider the Stokes Problem on a plane polygonal domain Ω R 2. We propose a finite element method
More informationAn a posteriori error estimate and a Comparison Theorem for the nonconforming P 1 element
Calcolo manuscript No. (will be inserted by the editor) An a posteriori error estimate and a Comparison Theorem for the nonconforming P 1 element Dietrich Braess Faculty of Mathematics, Ruhr-University
More informationEFFICIENT MULTIGRID BASED SOLVERS FOR ISOGEOMETRIC ANALYSIS
6th European Conference on Computational Mechanics (ECCM 6) 7th European Conference on Computational Fluid Dynamics (ECFD 7) 1115 June 2018, Glasgow, UK EFFICIENT MULTIGRID BASED SOLVERS FOR ISOGEOMETRIC
More informationMultigrid and Domain Decomposition Methods for Electrostatics Problems
Multigrid and Domain Decomposition Methods for Electrostatics Problems Michael Holst and Faisal Saied Abstract. We consider multigrid and domain decomposition methods for the numerical solution of electrostatics
More informationOn Pressure Stabilization Method and Projection Method for Unsteady Navier-Stokes Equations 1
On Pressure Stabilization Method and Projection Method for Unsteady Navier-Stokes Equations 1 Jie Shen Department of Mathematics, Penn State University University Park, PA 1682 Abstract. We present some
More informationA SHORT NOTE COMPARING MULTIGRID AND DOMAIN DECOMPOSITION FOR PROTEIN MODELING EQUATIONS
A SHORT NOTE COMPARING MULTIGRID AND DOMAIN DECOMPOSITION FOR PROTEIN MODELING EQUATIONS MICHAEL HOLST AND FAISAL SAIED Abstract. We consider multigrid and domain decomposition methods for the numerical
More informationA Stable Mixed Finite Element Scheme for the Second Order Elliptic Problems
A Stable Mixed Finite Element Scheme for the Second Order Elliptic Problems Miaochan Zhao, Hongbo Guan, Pei Yin Abstract A stable mixed finite element method (MFEM) for the second order elliptic problems,
More informationComparison of V-cycle Multigrid Method for Cell-centered Finite Difference on Triangular Meshes
Comparison of V-cycle Multigrid Method for Cell-centered Finite Difference on Triangular Meshes Do Y. Kwak, 1 JunS.Lee 1 Department of Mathematics, KAIST, Taejon 305-701, Korea Department of Mathematics,
More informationMathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts 1 and Stephan Matthai 2 3rd Febr
HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts and Stephan Matthai Mathematics Research Report No. MRR 003{96, Mathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL
More informationMultigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids
Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids Long Chen 1, Ricardo H. Nochetto 2, and Chen-Song Zhang 3 1 Department of Mathematics, University of California at Irvine. chenlong@math.uci.edu
More informationB005 A NEW FAST FOURIER TRANSFORM ALGORITHM FOR FLUID FLOW SIMULATION
1 B5 A NEW FAST FOURIER TRANSFORM ALGORITHM FOR FLUID FLOW SIMULATION LUDOVIC RICARD, MICAËLE LE RAVALEC-DUPIN, BENOÎT NOETINGER AND YVES GUÉGUEN Institut Français du Pétrole, 1& 4 avenue Bois Préau, 92852
More informationOn an Approximation Result for Piecewise Polynomial Functions. O. Karakashian
BULLETIN OF THE GREE MATHEMATICAL SOCIETY Volume 57, 010 (1 7) On an Approximation Result for Piecewise Polynomial Functions O. arakashian Abstract We provide a new approach for proving approximation results
More informationWeak Galerkin Finite Element Scheme and Its Applications
Weak Galerkin Finite Element Scheme and Its Applications Ran Zhang Department of Mathematics Jilin University, China IMS, Singapore February 6, 2015 Talk Outline Motivation WG FEMs: Weak Operators + Stabilizer
More informationKasetsart University Workshop. Multigrid methods: An introduction
Kasetsart University Workshop Multigrid methods: An introduction Dr. Anand Pardhanani Mathematics Department Earlham College Richmond, Indiana USA pardhan@earlham.edu A copy of these slides is available
More informationLecture Note III: Least-Squares Method
Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,
More informationENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS
ENERGY NORM A POSTERIORI ERROR ESTIMATES FOR MIXED FINITE ELEMENT METHODS CARLO LOVADINA AND ROLF STENBERG Abstract The paper deals with the a-posteriori error analysis of mixed finite element methods
More informationA mixed finite element approximation of the Stokes equations with the boundary condition of type (D+N)
wwwijmercom Vol2, Issue1, Jan-Feb 2012 pp-464-472 ISSN: 2249-6645 A mixed finite element approximation of the Stokes equations with the boundary condition of type (D+N) Jaouad El-Mekkaoui 1, Abdeslam Elakkad
More informationAMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
A SIMPLE FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU AND XIU YE Abstract. The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in
More informationA note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations
A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations S. Hussain, F. Schieweck, S. Turek Abstract In this note, we extend our recent work for
More informationAnalysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods
Advances in Applied athematics and echanics Adv. Appl. ath. ech., Vol. 1, No. 6, pp. 830-844 DOI: 10.408/aamm.09-m09S09 December 009 Analysis of Two-Grid ethods for Nonlinear Parabolic Equations by Expanded
More informationComputational Linear Algebra
Computational Linear Algebra PD Dr. rer. nat. habil. Ralf-Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2018/19 Part 4: Iterative Methods PD
More informationMULTIGRID PRECONDITIONING FOR THE BIHARMONIC DIRICHLET PROBLEM M. R. HANISCH
MULTIGRID PRECONDITIONING FOR THE BIHARMONIC DIRICHLET PROBLEM M. R. HANISCH Abstract. A multigrid preconditioning scheme for solving the Ciarlet-Raviart mixed method equations for the biharmonic Dirichlet
More informationElliptic Problems / Multigrid. PHY 604: Computational Methods for Physics and Astrophysics II
Elliptic Problems / Multigrid Summary of Hyperbolic PDEs We looked at a simple linear and a nonlinear scalar hyperbolic PDE There is a speed associated with the change of the solution Explicit methods
More informationA Least-Squares Finite Element Approximation for the Compressible Stokes Equations
A Least-Squares Finite Element Approximation for the Compressible Stokes Equations Zhiqiang Cai, 1 Xiu Ye 1 Department of Mathematics, Purdue University, 1395 Mathematical Science Building, West Lafayette,
More informationHigh order, finite volume method, flux conservation, finite element method
FLUX-CONSERVING FINITE ELEMENT METHODS SHANGYOU ZHANG, ZHIMIN ZHANG, AND QINGSONG ZOU Abstract. We analyze the flux conservation property of the finite element method. It is shown that the finite element
More informationTHE solution of the absolute value equation (AVE) of
The nonlinear HSS-like iterative method for absolute value equations Mu-Zheng Zhu Member, IAENG, and Ya-E Qi arxiv:1403.7013v4 [math.na] 2 Jan 2018 Abstract Salkuyeh proposed the Picard-HSS iteration method
More informationMULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* Dedicated to Professor Jim Douglas, Jr. on the occasion of his seventieth birthday.
MULTIGRID PRECONDITIONING IN H(div) ON NON-CONVEX POLYGONS* DOUGLAS N ARNOLD, RICHARD S FALK, and RAGNAR WINTHER Dedicated to Professor Jim Douglas, Jr on the occasion of his seventieth birthday Abstract
More informationA High Order Method for Three Phase Flow in Homogeneous Porous Media
A High Order Method for Three Phase Flow in Homogeneous Porous Media Justin Dong Advisor: B eatrice Rivi`ere February 5, 2014 Abstract The modeling of three-phase fluid flow has many important applications
More informationA P4 BUBBLE ENRICHED P3 DIVERGENCE-FREE FINITE ELEMENT ON TRIANGULAR GRIDS
A P4 BUBBLE ENRICHED P3 DIVERGENCE-FREE FINITE ELEMENT ON TRIANGULAR GRIDS SHANGYOU ZHANG DEDICATED TO PROFESSOR PETER MONK ON THE OCCASION OF HIS 6TH BIRTHDAY Abstract. On triangular grids, the continuous
More informationSuperconvergence and H(div) Projection for Discontinuous Galerkin Methods
INTRNATIONAL JOURNAL FOR NUMRICAL MTHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2000; 00:1 6 [Version: 2000/07/27 v1.0] Superconvergence and H(div) Projection for Discontinuous Galerkin Methods Peter Bastian
More information2 IVAN YOTOV decomposition solvers and preconditioners are described in Section 3. Computational results are given in Section Multibloc formulat
INTERFACE SOLVERS AND PRECONDITIONERS OF DOMAIN DECOMPOSITION TYPE FOR MULTIPHASE FLOW IN MULTIBLOCK POROUS MEDIA IVAN YOTOV Abstract. A multibloc mortar approach to modeling multiphase ow in porous media
More informationA Multigrid Method for Two Dimensional Maxwell Interface Problems
A Multigrid Method for Two Dimensional Maxwell Interface Problems Susanne C. Brenner Department of Mathematics and Center for Computation & Technology Louisiana State University USA JSA 2013 Outline A
More information1. Fast Iterative Solvers of SLE
1. Fast Iterative Solvers of crucial drawback of solvers discussed so far: they become slower if we discretize more accurate! now: look for possible remedies relaxation: explicit application of the multigrid
More informationLocal flux mimetic finite difference methods
Local flux mimetic finite difference methods Konstantin Lipnikov Mikhail Shashkov Ivan Yotov November 4, 2005 Abstract We develop a local flux mimetic finite difference method for second order elliptic
More informationb i (x) u + c(x)u = f in Ω,
SIAM J. NUMER. ANAL. Vol. 39, No. 6, pp. 1938 1953 c 2002 Society for Industrial and Applied Mathematics SUBOPTIMAL AND OPTIMAL CONVERGENCE IN MIXED FINITE ELEMENT METHODS ALAN DEMLOW Abstract. An elliptic
More informationDivergence-conforming multigrid methods for incompressible flow problems
Divergence-conforming multigrid methods for incompressible flow problems Guido Kanschat IWR, Universität Heidelberg Prague-Heidelberg-Workshop April 28th, 2015 G. Kanschat (IWR, Uni HD) Hdiv-DG Práha,
More informationPDEs, part 1: Introduction and elliptic PDEs
PDEs, part 1: Introduction and elliptic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Partial di erential equations The solution depends on several variables,
More informationSome New Elements for the Reissner Mindlin Plate Model
Boundary Value Problems for Partial Differential Equations and Applications, J.-L. Lions and C. Baiocchi, eds., Masson, 1993, pp. 287 292. Some New Elements for the Reissner Mindlin Plate Model Douglas
More informationEXISTENCE AND REGULARITY OF SOLUTIONS FOR STOKES SYSTEMS WITH NON-SMOOTH BOUNDARY DATA IN A POLYHEDRON
Electronic Journal of Differential Equations, Vol. 2017 (2017), No. 147, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND REGULARITY OF SOLUTIONS FOR
More informationPartial Differential Equations
Partial Differential Equations Introduction Deng Li Discretization Methods Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006 What do You Stand For? A PDE is a Partial Differential Equation This
More informationMultigrid Method ZHONG-CI SHI. Institute of Computational Mathematics Chinese Academy of Sciences, Beijing, China. Joint work: Xuejun Xu
Multigrid Method ZHONG-CI SHI Institute of Computational Mathematics Chinese Academy of Sciences, Beijing, China Joint work: Xuejun Xu Outline Introduction Standard Cascadic Multigrid method(cmg) Economical
More informationA high order adaptive finite element method for solving nonlinear hyperbolic conservation laws
A high order adaptive finite element method for solving nonlinear hyperbolic conservation laws Zhengfu Xu, Jinchao Xu and Chi-Wang Shu 0th April 010 Abstract In this note, we apply the h-adaptive streamline
More informationVARIATIONAL AND NON-VARIATIONAL MULTIGRID ALGORITHMS FOR THE LAPLACE-BELTRAMI OPERATOR.
VARIATIONAL AND NON-VARIATIONAL MULTIGRID ALGORITHMS FOR THE LAPLACE-BELTRAMI OPERATOR. ANDREA BONITO AND JOSEPH E. PASCIAK Abstract. We design and analyze variational and non-variational multigrid algorithms
More informationAdaptive methods for control problems with finite-dimensional control space
Adaptive methods for control problems with finite-dimensional control space Saheed Akindeinde and Daniel Wachsmuth Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy
More informationA posteriori error estimates for non conforming approximation of eigenvalue problems
A posteriori error estimates for non conforming approximation of eigenvalue problems E. Dari a, R. G. Durán b and C. Padra c, a Centro Atómico Bariloche, Comisión Nacional de Energía Atómica and CONICE,
More informationTo link to this article:
This article was downloaded by: [Wuhan University] On: 11 March 2015, At: 01:08 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer
More informationNewton s Method and Efficient, Robust Variants
Newton s Method and Efficient, Robust Variants Philipp Birken University of Kassel (SFB/TRR 30) Soon: University of Lund October 7th 2013 Efficient solution of large systems of non-linear PDEs in science
More informationA NEW FAMILY OF STABLE MIXED FINITE ELEMENTS FOR THE 3D STOKES EQUATIONS
MATHEMATICS OF COMPUTATION Volume 74, Number 250, Pages 543 554 S 0025-5718(04)01711-9 Article electronically published on August 31, 2004 A NEW FAMILY OF STABLE MIXED FINITE ELEMENTS FOR THE 3D STOKES
More informationA MULTIGRID METHOD FOR THE PSEUDOSTRESS FORMULATION OF STOKES PROBLEMS
SIAM J. SCI. COMPUT. Vol. 29, No. 5, pp. 2078 2095 c 2007 Society for Industrial and Applied Mathematics A MULTIGRID METHOD FOR THE PSEUDOSTRESS FORMULATION OF STOKES PROBLEMS ZHIQIANG CAI AND YANQIU WANG
More informationA gradient recovery method based on an oblique projection and boundary modification
ANZIAM J. 58 (CTAC2016) pp.c34 C45, 2017 C34 A gradient recovery method based on an oblique projection and boundary modification M. Ilyas 1 B. P. Lamichhane 2 M. H. Meylan 3 (Received 24 January 2017;
More informationDomain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions
Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions Bernhard Hientzsch Courant Institute of Mathematical Sciences, New York University, 51 Mercer Street, New
More informationMultigrid Methods and their application in CFD
Multigrid Methods and their application in CFD Michael Wurst TU München 16.06.2009 1 Multigrid Methods Definition Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential
More informationConstrained Minimization and Multigrid
Constrained Minimization and Multigrid C. Gräser (FU Berlin), R. Kornhuber (FU Berlin), and O. Sander (FU Berlin) Workshop on PDE Constrained Optimization Hamburg, March 27-29, 2008 Matheon Outline Successive
More informationSolving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners
Solving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners Eugene Vecharynski 1 Andrew Knyazev 2 1 Department of Computer Science and Engineering University of Minnesota 2 Department
More informationON THE MULTIPOINT MIXED FINITE VOLUME METHODS ON QUADRILATERAL GRIDS
ON THE MULTIPOINT MIXED FINITE VOLUME METHODS ON QUADRILATERAL GRIDS VINCENT FONTAINE 1,2 AND ANIS YOUNES 2 1 Laboratoire de Physique des Bâtiments et des Systèmes, Université de La Réunion, 15 avenue
More informationME 608 Numerical Methods in Heat, Mass, and Momentum Transfer
ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer Solution to Final Examination Date: May 4, 2011 1:00 3:00 PM Instructor: J. Murthy Open Book, Open Notes Total: 100 points Use the finite volume
More informationSolving the stochastic steady-state diffusion problem using multigrid
IMA Journal of Numerical Analysis (2007) 27, 675 688 doi:10.1093/imanum/drm006 Advance Access publication on April 9, 2007 Solving the stochastic steady-state diffusion problem using multigrid HOWARD ELMAN
More informationSUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS
SUPERCONVERGENCE PROPERTIES FOR OPTIMAL CONTROL PROBLEMS DISCRETIZED BY PIECEWISE LINEAR AND DISCONTINUOUS FUNCTIONS A. RÖSCH AND R. SIMON Abstract. An optimal control problem for an elliptic equation
More informationMultigrid absolute value preconditioning
Multigrid absolute value preconditioning Eugene Vecharynski 1 Andrew Knyazev 2 (speaker) 1 Department of Computer Science and Engineering University of Minnesota 2 Department of Mathematical and Statistical
More informationNumAn2014 Conference Proceedings
OpenAccess Proceedings of the 6th International Conference on Numerical Analysis, pp 198-03 Contents lists available at AMCL s Digital Library. NumAn014 Conference Proceedings Digital Library Triton :
More informationIterative Methods and Multigrid
Iterative Methods and Multigrid Part 1: Introduction to Multigrid 1 12/02/09 MG02.prz Error Smoothing 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Initial Solution=-Error 0 10 20 30 40 50 60 70 80 90 100 DCT:
More informationETNA Kent State University
g ' Electronic Transactions on umerical Analysis. Volume 17 pp. 112-132 2004. Copyright 2004. SS 10-913. COVERGECE OF V-CYCE AD F-CYCE MTGRD METHODS FOR THE BHARMOC PROBEM SG THE MOREY EEMET JE ZHAO Abstract.
More information6. Multigrid & Krylov Methods. June 1, 2010
June 1, 2010 Scientific Computing II, Tobias Weinzierl page 1 of 27 Outline of This Session A recapitulation of iterative schemes Lots of advertisement Multigrid Ingredients Multigrid Analysis Scientific
More informationMultigrid Methods for Maxwell s Equations
Multigrid Methods for Maxwell s Equations Jintao Cui Institute for Mathematics and Its Applications University of Minnesota Outline Nonconforming Finite Element Methods for a Two Dimensional Curl-Curl
More informationAltered Jacobian Newton Iterative Method for Nonlinear Elliptic Problems
Altered Jacobian Newton Iterative Method for Nonlinear Elliptic Problems Sanjay K Khattri Abstract We present an Altered Jacobian Newton Iterative Method for solving nonlinear elliptic problems Effectiveness
More informationOn Multigrid for Phase Field
On Multigrid for Phase Field Carsten Gräser (FU Berlin), Ralf Kornhuber (FU Berlin), Rolf Krause (Uni Bonn), and Vanessa Styles (University of Sussex) Interphase 04 Rome, September, 13-16, 2004 Synopsis
More informationMULTIGRID METHODS FOR NONLINEAR PROBLEMS: AN OVERVIEW
MULTIGRID METHODS FOR NONLINEAR PROBLEMS: AN OVERVIEW VAN EMDEN HENSON CENTER FOR APPLIED SCIENTIFIC COMPUTING LAWRENCE LIVERMORE NATIONAL LABORATORY Abstract Since their early application to elliptic
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 37, pp. 166-172, 2010. Copyright 2010,. ISSN 1068-9613. ETNA A GRADIENT RECOVERY OPERATOR BASED ON AN OBLIQUE PROJECTION BISHNU P. LAMICHHANE Abstract.
More informationThe Conjugate Gradient Method
The Conjugate Gradient Method Classical Iterations We have a problem, We assume that the matrix comes from a discretization of a PDE. The best and most popular model problem is, The matrix will be as large
More information/00 $ $.25 per page
Contemporary Mathematics Volume 00, 0000 Domain Decomposition For Linear And Nonlinear Elliptic Problems Via Function Or Space Decomposition UE-CHENG TAI Abstract. In this article, we use a function decomposition
More informationVolterra integral equations solved in Fredholm form using Walsh functions
ANZIAM J. 45 (E) ppc269 C282, 24 C269 Volterra integral equations solved in Fredholm form using Walsh functions W. F. Blyth R. L. May P. Widyaningsih (Received 8 August 23; revised 6 Jan 24) Abstract Recently
More informationNon-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions
Non-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions Wayne McGee and Padmanabhan Seshaiyer Texas Tech University, Mathematics and Statistics (padhu@math.ttu.edu) Summary. In
More informationAn Iterative Substructuring Method for Mortar Nonconforming Discretization of a Fourth-Order Elliptic Problem in two dimensions
An Iterative Substructuring Method for Mortar Nonconforming Discretization of a Fourth-Order Elliptic Problem in two dimensions Leszek Marcinkowski Department of Mathematics, Warsaw University, Banacha
More informationA POSTERIORI ERROR ESTIMATES OF TRIANGULAR MIXED FINITE ELEMENT METHODS FOR QUADRATIC CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEMS
A POSTERIORI ERROR ESTIMATES OF TRIANGULAR MIXED FINITE ELEMENT METHODS FOR QUADRATIC CONVECTION DIFFUSION OPTIMAL CONTROL PROBLEMS Z. LU Communicated by Gabriela Marinoschi In this paper, we discuss a
More informationOptimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 1/36
Optimal multilevel preconditioning of strongly anisotropic problems. Part II: non-conforming FEM. Svetozar Margenov margenov@parallel.bas.bg Institute for Parallel Processing, Bulgarian Academy of Sciences,
More informationNONCONFORMING MIXED ELEMENTS FOR ELASTICITY
Mathematical Models and Methods in Applied Sciences Vol. 13, No. 3 (2003) 295 307 c World Scientific Publishing Company NONCONFORMING MIXED ELEMENTS FOR ELASTICITY DOUGLAS N. ARNOLD Institute for Mathematics
More informationAspects of Multigrid
Aspects of Multigrid Kees Oosterlee 1,2 1 Delft University of Technology, Delft. 2 CWI, Center for Mathematics and Computer Science, Amsterdam, SIAM Chapter Workshop Day, May 30th 2018 C.W.Oosterlee (CWI)
More informationA Robust Preconditioner for the Hessian System in Elliptic Optimal Control Problems
A Robust Preconditioner for the Hessian System in Elliptic Optimal Control Problems Etereldes Gonçalves 1, Tarek P. Mathew 1, Markus Sarkis 1,2, and Christian E. Schaerer 1 1 Instituto de Matemática Pura
More informationEfficient numerical solution of the Biot poroelasticity system in multilayered domains
Efficient numerical solution of the Biot poroelasticity system Anna Naumovich Oleg Iliev Francisco Gaspar Fraunhofer Institute for Industrial Mathematics Kaiserslautern, Germany, Spain Workshop on Model
More informationPreconditioning in H(div) and Applications
1 Preconditioning in H(div) and Applications Douglas N. Arnold 1, Ricard S. Falk 2 and Ragnar Winter 3 4 Abstract. Summarizing te work of [AFW97], we sow ow to construct preconditioners using domain decomposition
More informationDiscrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction
Discrete Projection Methods for Incompressible Fluid Flow Problems and Application to a Fluid-Structure Interaction Problem Jörg-M. Sautter Mathematisches Institut, Universität Düsseldorf, Germany, sautter@am.uni-duesseldorf.de
More informationAdaptive C1 Macroelements for Fourth Order and Divergence-Free Problems
Adaptive C1 Macroelements for Fourth Order and Divergence-Free Problems Roy Stogner Computational Fluid Dynamics Lab Institute for Computational Engineering and Sciences University of Texas at Austin March
More informationA local-structure-preserving local discontinuous Galerkin method for the Laplace equation
A local-structure-preserving local discontinuous Galerkin method for the Laplace equation Fengyan Li 1 and Chi-Wang Shu 2 Abstract In this paper, we present a local-structure-preserving local discontinuous
More information(1:1) 1. The gauge formulation of the Navier-Stokes equation We start with the incompressible Navier-Stokes equation 8 >< >: u t +(u r)u + rp = 1 Re 4
Gauge Finite Element Method for Incompressible Flows Weinan E 1 Courant Institute of Mathematical Sciences New York, NY 10012 Jian-Guo Liu 2 Temple University Philadelphia, PA 19122 Abstract: We present
More informationINNOVATIVE FINITE ELEMENT METHODS FOR PLATES* DOUGLAS N. ARNOLD
INNOVATIVE FINITE ELEMENT METHODS FOR PLATES* DOUGLAS N. ARNOLD Abstract. Finite element methods for the Reissner Mindlin plate theory are discussed. Methods in which both the tranverse displacement and
More informationA NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION
A NOTE ON THE LADYŽENSKAJA-BABUŠKA-BREZZI CONDITION JOHNNY GUZMÁN, ABNER J. SALGADO, AND FRANCISCO-JAVIER SAYAS Abstract. The analysis of finite-element-like Galerkin discretization techniques for the
More information