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 GRIDS JUN S. LEE Abstract. We introduce intergrid operator recently developed for the cell centered finite difference multigrid on rectangular grids. The main idea of operator construction is based on flux continuity and certain ind of interpolation. This operator wors well for solving diffusion equations both with discontinuous coefficient and with smooth coefficient. We disscuss on the construction of prolongation operators and compare this operator with the weight prolongation and trivial injection both theoretically and numerically. We present various numerical results to show that multigrid algorithm with our prolongation wors well for various interface problems.. Introduction The cell-centered finite difference(ccfd) is a finite volume type of method and has been used by many engineers for solving elliptic problems with discontinuous diffusion due to its simplicity and local conservation. Of many such problems a few examples are flows through porous media with different porosity electric currents through material of different conductivities and heat flows through heterogeneous materialsetc.[7 8]. The numerical methods treating such problems are important areas of research. It also arises from the saddle point formulation of the Raviart- Thomas mixed method by taing certain quadrature. On the other hand the multigrid(mg) algorithms has proven to be very effective for a large class of problems[6 5] and has been the subject of extensive research[2 4 ]. The multigrid algorithm for CCFD was considered by a few authors [2 9 2 7] and its W -cycle convergence for Laplace equation was first proved by Bramble et al. in [2]. However their use of natural injection as a prolongation operator is not an optimal choice on triangular and rectangular grids. On rectangular grid the V-cycle multigrid convergence with the natural injection is slow. As is shown by Kwa[] certain weighted prolongation wors much better and guarantees V-cycle convergence. Still it is restricted to problems with smooth coefficients. It does not wor well when the jump of the diffusion coefficient is severe. In early years there have been some efforts to handle discontinuous 2000 Mathematics Subject Classification. primary 65N0 secondary 65F0. Key words and phrases. discontinuous coefficient cell centered finite difference methods finite volume methods multigrid methods. 0 c 2006 Information Center for Mathematical Sciences
04 JUN S. LEE coefficient problems using Galerin coarse grid approximation[2 7]. In the finite difference case Alcouffe et al.[] and Kettler[] suggested a prolongation operator based on the continuity of flux at finer grid points. However in this method stiffness matrices of the coarse grids no longer have 5-point structure and the prolongation is nontrivial. Hence extra cost is needed to generate stiffness matrices and its implementation is difficult. Recently Kwa and Lee developed new prolongation depending on the diffusion and shows W -cycle multigrid convergence[4]. In numerical test the multigrid with this prolongation yields better convergence than with the natural injection. The rest of this paper is as follows. In section 2 we briefly describe CCFD for elliptic problem and multigrid algorithm. In section we consider natural injection and modified bilinear prolongation of multigrid method for CCFD and estimate energy norms to get V -cycle convergence and W -cycle convergence. Finally we give some numerical results for two prolongations in section 4. We report maximum minimum eigenvalues condition numbers and contraction numbers of two multigrid algorithms. 2. Multigrid and cell-centered finite difference method In this section we briefly describe CCFD on rectangular grid and multigrid algorithm to solve the resulting system of linear equations. We first consider the following model problem : (2.) (2.2) p ũ = f in Ω ũ = 0 on Ω. For simplicity we assume that Ω is the unit square and ũ H +α (Ω) for some 0 < α. For = 2 J divide Ω into n n axis-parallel subsquares where n = 2. Such triangulations are denoted by {E }. Each subsquare in E is called a cell and denoted by E ij i j = 2 and has u ij as its value at center. For = 2 J let V denote the space of functions which are piecewise constant on each cell. Integrating by parts (2.) on each cell and replacing the normal derivative ũ/ n on the edges by difference quotient of function u in V we have the finite difference equations as follows: p i /2j (u i j u ij ) p i+/2j (u i+j u ij ) p ij /2 (u ij u ij ) p ij+/2 (u ij+ u ij ) = f ij h 2 where h = h = /2 and p i /2j = p(x i /2 y j ) etc. When p is discontinuous along the interface we tae p i /2j as p i /2j = 2p i jp ij p i j + p ij. Similarly p i+/2j p ij /2 and p ij+/2 are defined. When one of the edge coincides with the boundary of Ω we assume a fictitious value by reflection. For example u 0j is taen as u j and thus ũ n at x = 0 is approximated by 2u j/h. Similar
INTERGRID OPERATORS FOR CCFD MG 05 rules apply to the other parts of the boundary of Ω. After dividing the resulting equation by h 2 we obtain a system of linear equation of the form (2.) A u = f where A is the typical sparse n 2 n 2 symmetric positive definite matrix similar to those arising in the vertex finite difference method and u is the vector whose entries are u ij and f is the vector whose entries are f(x i y j ). We identify the vector u v in V with their matrix representation in R 2. For analysis we define a quadratic form A ( ) on V V in an obvious manner by (2.4) A (u v) = h 2 (A u) ij v ij u v V. ij Then (2.) is equivalent to the following problem: Find u V satisfying (2.5) A (u φ) = (f φ) φ V where ( ) is the L 2 inner product. The error analysis of the cell centered finite difference method is well nown (cf. [5][8][0]). Let Q : L 2 (Ω) V denote the usual L 2 (Ω) projection. If u is the solution of (2.5) then A (u Q ũ u Q ũ) Ch 2 f 2 where is the usual L 2 norm. Given a certain prolongation operator I : V V we define the restriction operator I : V V as its adjoint with respect to ( ) : (I u v) = (u I v) u V v V. Since the space V can be viewed as the space of vectors having entries u ij we also use I and I to denote their matrix representations. Now multigrid algorithm for solving (2.) is defined as follows : Multigrid Algorithm Set B 0 = A 0. For < J assume that B has been defined and define B f for f V as follows: () Set x 0 = 0 and q 0 = 0. (2) Define x l for l =... m() by x l = x l + R (l+m()) (f A x l ). () Define y m() = x m() + I qs where q i for i =... s is defined by [ q i = q i + B P (f 0 A x m() ) A q i ] where P 0 : V V is the L 2 -projection. (4) Define y l for l = m() +... 2m() by (5) Set B f = y 2m(). y l = y l + R (l+m()) (f A y l ).
06 JUN S. LEE If s = we obtain V (m() m())-cycle and if s = 2 we obtain W (m() m())- cycle. If m() varies with we call it variable V -cycle. Here R (l) is a smoother on V which alternates between R and its adjoint R t and m() is the number of smoothings which can vary depending on. The smoother R as usual can be taen as the Jacobi or Gauss-Seidel relaxation.. Prolongation operators on rectangular grids In this section first we briefly introduce trivial injection and bilinear prolongation. Multigrid with bilinear prolongation operator wors well for smooth problems but not for nonsmooth problems. On the other hand multigrid with trivial injection converges for nonsmooth problems but not for smooth problems. Next we consider how to extend bilinear prolongation operator to modified one which wors well both for smooth and nonsmooth problems. Let E ij be a cell at level and v ij be the values at the center of each cells. Let u IJ be the value of upper right subcell of E ij and u IJ u I J and u I J be those of lower right upper left lower left subcells of E ij. (See Figure (b)) Here we use the notation I = I + J = J etc. We first define trivial injection. For v V define u = I t v as follows: (.) u IJ = v ij u I J = v i+j u IJ = v ij+ u I J = v i+j+. Next we define bilinear prolongation operator. For v V define u = I b v as follows: (.2) u IJ = 9v ij + v ij+ + v i+j + v i+j+ u I J = 9v i+j + v ij + v i+j+ + v ij+ u IJ = 9v ij+ + v ij + v i+j+ + v i+j u I J = 9v i+j+ + v i+j + v ij+ + v ij. For trivial injection we have the following energy norm estimation: Proposition.. We assume that p is piecewise constant on each cell. We have (.) A (I t v I t v) 2A (v v) for v V. This result together with the regularity and approximation property which can be shown exactly the same way as in [] we can deduce that W -cycle multigrid algorithm converges when p is piecewise constant. Note that even when p is discontinuous W -cycle multigrid convergence is guaranteed. On the other hand by the technique in [ 4] we have the following energy norm estimation for bilinear prolongation: Proposition.2. We have (.4) A (I b v I b v) C A (v v) for v V. where C = if p is constant and + C(p)h if p is Lipschitz continuous.
INTERGRID OPERATORS FOR CCFD MG 07 In this case we can deduce that V -cycle multigrid algorithm converges when p is constant and is a good preconditioner when p is smooth. But the V -cycle convergence is not guaranteed when p is nonsmooth. In fact the energy norm of I b goes to infinity as the jump of p grows and numerical result shows that it diverges when p is not smooth. So we want to devise a new prolongation whose energy norm remains bounded even if the jump ratio of the coefficients approaches. For a motivation we consider the following one dimensional diffusion equation on [0 ]. d ( p du(x) ) = f in (0 ) dx dx u(0) = u() = 0. Let x i i = 2 n be the center of equally spaced grid on [0 ] and h be the mesh size. When this grid is refined new center will have locations at x i+/4 x i+/4 etc. Now we consider how to define the values at x i+/4. A natural way is to impose the flux continuity at x i+/4 so that the following equality holds: Solving we get p i v i+/4 v i h 4 v i+ v i+/4 = p i+. h 4 v i+/4 = p iv i + p i+ v i+. p i + p i+ This can be viewed as the linear interpolation of neighboring point with diffusion weights. Motivated by this we define a new prolongation I m for two dimensional problem as follows: u IJ = 9p ijv ij + p ij+ v ij+ + p i+j v i+j + p i+j+ v i+j+ 9p ij + p i+j + p ij+ + p i+j+ u I J = 9p i+jv i+j + p ij v ij + p i+j+ v i+j+ + p ij+ v ij+ 9p i+j + p ij + p i+j+ + p ij+ u IJ u I J = 9p ij+v ij+ + p ij v ij + p i+j+ v i+j+ + p i+j v i+j 9p ij+ + p ij + p i+j+ + p i+j =9p i+j+v i+j+ + p i+j v i+j + p ij+ v ij+ + p ij v ij 9p i+j+ + p i+j + p ij+ + p ij. It is easy to chec that this prolongation operator reduces to the bilinear one when the diffusion p is constant. For modified bilinear prolongation we have the following: Proposition.. Let p = p for x < /2 and p = p 2 for x > /2. Then for any pair of positive constants p and p 2 we have (.5) A (I m v I m v) 8 A (v v) v V. Now we consider regularity and approximation property. Since p is discontinuous it is not trivial to prove this property. Fortunately by adopting a nearby
08 JUN S. LEE I J IJ I J I 2 J (i j) (i + j) I J IJ I J I 2 J (i j) (i + j) I J IJ I J I 2 J I J 2 IJ 2 I J 2 I 2 J 2 i i + x = 2 (a) (b) Figure. Bilinear contributions and indices (I = I J = J + ) smooth problem which has enough regularity Kwa and Lee showed regularity and approximation property of discontinuous problem(see for the detail[4]). Lemma.. Let the operator P be elliptic projection. Then ( ) A u A ((I I P )u u) C α A (u u) α λ u V holds for α = 2. Here λ is the largest eigenvalue of A and I is I t Ib or I m. This together with (.) we obtain the following result for I t. Theorem.. Let E = I B A in multigrid algorithm with s = 2(W -cycle algorithm). Then for any m we have (.6) where δ = A (E u E u) δ A (u u) u V M M+. In particular W -cycle wors with one smoothing. m By the energy norm estimates (.4) and (.5) we have the following results for I m. Theorem.2. First let p be constant and let E = I B A in multigrid algorithm with s = (V -cycle algorithm). Then for any m we have 0 A (E u u) δ A (u u) u V
INTERGRID OPERATORS FOR CCFD MG 09 where δ = CM CM+ m. Second let p has a simple discontinuity and let E = I B A in multigrid algorithm with s = 2(W -cycle algorithm). Then for any m we have (.6). Remar.. In case of the modified prolongation I m more general discontinuity can be handled similarly as long as the discontinuity arise as jumps along some line segments parallel to the axes. 4. Numerical Results We consider the following problem on the unit square(rectangular grid): (4.) p ũ = f in Ω = (0 ) 2 ũ = 0 on Ω. In all problems we use one pre-post Gauss-Seidel relaxationi.e. V ( ). We report the eigenvalues condition numbers and reduction factors of trivial bilinear and modified bilinear operators. Problem. The domain is the unit square and the diffusion coefficient p is. In this case the modified bilinear prolongation is similar to the weighted prolongation and wors better than the trivial injection. Table. V ( )-cycle with natural injection Figure 2 (a). / 0.784.444.842 0.28 /64 0.78.54.960 0.09 /28 0.784.68 2.060 0.40 Table 2. V ( )-cycle with modified bilinear prolongation Figure 2 (a). / 0.69 0.999.444 0.075 /64 0.688 0.999.454 0.08 /28 0.68 0.999.46 0.08 Problem 2. The domain is still the unit square and has simple discontinuity on the line x = /2 and 0 < y < (See Figure 2 (b)). We observed that multigrid with standard bilinear prolongation diverges. Multigrid algorithms with trivial and modified bilinear prolongation converge. But the modified bilinear prolongation converges faster and the contraction number is independent of number of levels.
0 JUN S. LEE Table. V ( )-cycle with natural injection p = p 2 = 0 Figure 2 (b). / 0.769.49.98 0.20 /64 0.770.574 2.045 0.298 /28 0.779.649 2. 0.89 Table 4. V ( )-cycle with bilinear prolongation p = p 2 = 0 Figure 2 (b). / 0.405 24.258 59.82 > /64 0.77 0.457 4.04 > /28 0.876 5.969 4.065 > Table 5. V ( )-cycle with modified bilinear prolongation p = p 2 = 0 Figure 2 (b). / 0.628 0.999.59 0.9 /64 0.605 0.999.652 0.22 /28 0.592 0.999.690 0.24 p = p p 2 (a)smooth (b)nonsmooth Figure 2. test problems References [] R. E. Alcouffe A. Brandt J. E. Dendy and J. W. Painter The multi-grid method for the diffusion equation with strongly discontinuous coefficients SIAM J. Sci. Stat. Comput. 2 40 455(98). [2] J. Bramble R. Ewing J. Pascia and J. Shen The analysis of multigrid algorithms for cell centered finite difference methods Adv. Comput. Math. 5 5 29 (996).
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