FOURIER ANALYSIS OF MULTIGRID METHODS ON HEXAGONAL GRIDS

Transcription

1 FOURIER ANALYSIS OF MULTIGRID METHODS ON HEXAGONAL GRIDS GUOHUA ZHOU AND SCOTT R. FULTON Abstract. Tis paper applies local Fourier analysis to multigrid metods on exagonal grids. Using oblique coordinates to express te grids and a dual basis for te Fourier modes, te analysis proceeds essentially te same as for rectangular grids. Te framework for one- and two-grid analysis is given, and ten applied to analyze te performance of multigrid metods for te Poisson problem on a exagonal grid. Numerical results confirm te analysis. Uniform exagonal grids provide an approximation to sperical geodesic grids; numerical results for te latter sow similar performance. Wile te analysis is similar to tat for rectangular grids, te results differ somewat: full weigting is superior to injection for restriction, Jacobi relaxation performs about as well as Gauss- Seidel relaxation, and underrelaxation is not required for good performance. Also, coarse-fine or four-color ordering (bot analogues of red-black ordering on te rectangular grid improves te performance of Jacobi relaxation, wit te latter acieving a smooting factor of approximately.25. An especially simple compact fourt-order discretization works well, and te full multigrid algoritm produces te solution to te level of truncation error in work proportional to te number of unknowns. Key words. multigrid, exagonal grid, geodesic grid, local Fourier analysis AMS subject classifications. 65N55, 65N6, 65N22. Introduction. Discretizations of partial differential equations are often based on structured grids. In two spatial dimensions te most common and natural approac is to use rectangular grids. However, exagonal grids offer some advantages. In particular, exagonal grids are more nearly isotropic: eac grid cell adjacent to a given cell is located across a cell wall and te same distance away (cf. Fig. 2. below, in contrast to te rectangular case were neigboring cells sare eiter a wall or a vertex. Consequently, discrete operators may be simpler and truncation errors smaller and more isotropic [6, 7]. Hexagonal grids also play an important role as approximations to te sperical geodesic grids being used increasingly often for numerical modeling in sperical coordinates, e.g., for global climate modeling [4, 5]. Many applications require solving discretized elliptic problems (e.g., te Poisson and modified Helmoltz problems, wic can be accomplised efficiently by multigrid metods [3]. Tese metods combine discretizations on several grids of different mes size, eac covering te wole computational domain; relaxation reduces error components on te scale of eac grid, wit smoot error components reduced by corrections computed on coarser grids. Introduced in te 97s by Brandt [] and oters, multigrid metods ave become a standard approac on rectangular grids. Heikes and Randall [4] introduced a multigrid metod for te Poisson equation on a sperical geodesic grid; tis application provided one motivation for te present study. Local Fourier analysis (LFA is a key tool for studying multigrid metods, providing estimates of convergence rates wic can elp guide te design of algoritms and verify tat tey are implemented correctly. Tis analysis measures te effect of te multigrid components on discrete Fourier components of te error (or combinations of components wic are coupled. Since LFA ignores boundary conditions and assumes constant-coefficient operators, it does not provide rigorous global bounds on errors. However, in many cases it predicts convergence rates correctly, since reducing non-smoot error components by relaxation is fundamentally a local task. Department of Matematics, Clarkson University, Campus Box 585, Potsdam, NY (zoug@clarkson.edu and fulton@clarkson.edu

2 2 G. ZHOU AND S. R. FULTON Fig. 2.. A uniform exagonal grid G. Te cell centers are grid points [indexed troug 6 as in (2.2], and te unit vectors a and a 2 define te oblique coordinate system. Te goal of tis paper is to use LFA to analyze te performance of multigrid metods on uniform exagonal grids. Wile sperical geodesic grids are not perfectly regular and tus not amenable to LFA, tey can be closely approximated by uniform exagonal grids on a plane. Terefore, te results obtained ere sould be useful in designing multigrid metods for geodesic grids. In particular, Heikes and Randall [4] did not report convergence results for teir metod; our analysis provides estimates of te teoretical performance of teir metod and leads to several improvements. Muc of te analysis presented ere follows closely tat for rectangular grids, once we recognize tat exagonal grids are logically rectangular in oblique coordinates. Terefore, wit minor canges we follow te notation and development in []. Section 2 introduces notation for operators and Fourier modes on exagonal grids. Smooting analysis is presented in 3 for smooting operators for wic te Fourier modes are eigenfunctions. Te corresponding two-grid analysis is given in 4, including te extension to tree-grid analysis. Section 5 considers te case of smooting operators wic couple several Fourier modes, and introduces two relaxation scemes wic attempt to mimic te properties of red-black ordering on a rectangular grid. Numerical results are given in 6 wic verify te analytical results. Included ere are results for te Poisson and modified Helmoltz problems on a exagonal grid (wit various relaxation scemes and grid transfers and using second- and fourt-order discretizations and sample results from a sperical geodesic grid confirming tat te exagonal-grid analysis gives useful guidance. Section 7 summarizes our conclusions. 2. Hexagonal grids and operators. A uniform exagonal grid is a collection of gridpoints G := { x j = (j a + j 2 a 2 : j = (j, j 2 Z 2} wic are te centers of exagonal grid cells as sown in Fig. 2., were a = ( ( /2, a 2 = 3/2 are unit vectors defining te oblique coordinate system and is te mes size (distance between cell centers. Tis indexing is equivalent to tat used in [6]. For LFA te grid G is considered to be infinite. We assume standard coarsening: te next coarser grid is G 2 wit gridpoints x 2 j = x 2j for j = (j, j 2 Z 2 as sown in Fig. 2.2, and still coarser grids G 4, G 8,... are defined similarly.

3 MULTIGRID ON HEXAGONAL GRIDS 3 Fig Relation between fine grid (dots and tin lines and coarse grid (circles and tick lines. Local Fourier analysis (LFA considers te effects of multigrid processing on te functions ϕ (θ, x := exp(iθ x/, were θ R 2 is te Fourier wavenumber relative to G. We will use te term mode to refer to eiter te function ϕ (θ, x or its associated wavenumber θ. On te exagonal grid it is convenient to write θ = θ b + θ 2 b 2, were {b, b 2 } is te dual basis corresponding to {a, a 2 }, i.e., ( ( b = /, b 3 2 = 2/ 3 satisfying a i b j = δ ij. Ten on grid G we ave ϕ (θ, x j = exp[i(j θ + j 2 θ 2 ] for j Z 2. Since ϕ (θ, x = ϕ (θ, x for all x G if and only if θ = θ mod 2π and θ 2 = θ 2 mod 2π, we see tat te Fourier modes ϕ (θ, x are distinguisable on te grid G only for θ Θ := [ π, π 2, wic we refer to as te modes visible on te grid. Note tat for points on grid G 2, ϕ (θ, x 2j = ϕ 2(2θ, x 2 j, i.e., a mode wit wavenumber θ relative to G as wavenumber 2θ relative to G 2. Tus, we can partition te set Θ of modes visible on a grid G into te set Θ low := [ π/2, π/2 2 of low modes, wic are visible on te next coarser grid G 2, and te set Θ ig := Θ\Θ low of ig modes, wic are not. Te discrete operators considered ere are assumed to be linear wit constant coefficients. Suc an operator L corresponds to a stencil [s k ] indexed by k = (k, k 2 Z 2, wic means tat for any grid function u defined on G, L u (x j = k K s k u (x j+k, x j G, (2. were te coefficients s k may depend on (but not on j and K Z 2 is a finite index set. For suc operators te following olds:

4 4 G. ZHOU AND S. R. FULTON Lemma 2.. For any discrete operator L on G described by a difference stencil of te form (2., all Fourier modes ϕ (θ, x are eigenfunctions, wit L ϕ (θ, x = L (θϕ (θ, x, x G were te eigenvalue L (θ := k K s ke i(kθ+k2θ2 is called te symbol of L. For example, te (negative Laplacian operator 2 can be discretized using seven points as ( L u = u u i, (2.2 were for convenience we ave used different indexing: te subscripts and 6 denote te central point and its six nearest neigbors, respectively, as sown in Fig. 2.. Expressing tese points in te oblique coordinate gives te stencil wit corresponding symbol i= 6 L (θ = [3 cos(θ cos(θ 2 cos(θ 2 θ ]. (2.3 It can be sown tat tis discretization is second-order accurate. 3. Smooting analysis. Te performance of a multigrid metod depends in large part on te effectiveness of te relaxation sceme wic smootes te error on eac grid. In tis section we analyze relaxation scemes for wic te Fourier modes are eigenfunctions of te corresponding smooting operators. If we express te exagonal grid using oblique coordinates and Fourier modes using te dual basis, te analysis proceeds essentially te same as for rectangular grids (cf. [, 4.3]. Consider a discretized partial differential equation L u = f, and assume tat a relaxation sceme can be written via an operator splitting L = L + + L (were is invertible as L + L + ū + L ũ = f, (3. were ũ and ū are te old and new approximations to te true discrete solution u, respectively. Subtracting (3. from te discrete equation L u = f sows tat te old and new errors ṽ = u ũ and v = u ū satisfy v = S ṽ, were S := ( L + L is te resulting smooting operator. Applying L+, L, and S to te Fourier modes ϕ (θ, x, we ave te following: Lemma 3.. For any relaxation sceme of te form (3., all Fourier modes ϕ (θ, x for wic L + (θ are eigenfunctions of te smooting operator S, wit S ϕ (θ, x = S (θϕ (θ, x, x G, were te symbol of te smooting operator, given by S (θ := L (θ/ L + (θ, is called te amplification factor.

5 MULTIGRID ON HEXAGONAL GRIDS 5 Te effectiveness of relaxation as a smooter is ten measured by te multigrid smooting factor { } (S := sup S (θ : θ Θ ig, (3.2 wic quantifies ow well relaxation reduces te ig-mode error components. Assuming tat te low-mode error components are eliminated by corrections computed on te coarse grids, te smooting factor gives an estimate of te overall efficiency of te multigrid metod. For example, consider te Poisson equation 2 u = f wit te discretization (2.2. Weigted Jacobi (WJ relaxation (as used in [4] can be formulated as ( ū = ( ũ f + ũ i were is a relaxation parameter to allow for under- or over-relaxation. Te corresponding smooting operator is i= (S J (v = ( v i= v i wit symbol S J (θ, = 2 3 [ ( ( ( ] sin 2 θ + sin 2 θ2 + sin 2 θ θ 2. ( For smooting must be positive, and it can be sown tat te smooting factor is { ( } 2 (S J ( = max, 3 3 2, wit te first and second values corresponding to extrema on te inner boundary and interior of te ig-mode region Θ ig, respectively. Te optimum smooting is obtained wit = 2/( , for wic te smooting factor is approximately Also, te simpler unweigted Jacobi sceme (i.e., wit = gives te smooting factor.5, wic is nearly as good; tis differs from te situation on a rectangular grid, in wic underrelaxation ( < is required. Similarly, for (weigted Gauss-Seidel relaxation wit lexicograpic ordering in te oblique coordinate we obtain te symbol S G (θ, = 6( + [ e iθ + e iθ2 + e i(θ2 θ] 6 [ e iθ + e iθ2 + e i(θ2 θ], (3.4 were is again te relaxation parameter. For < < 2 te maximum in (3.2 is obtained on te inner boundary of te ig-mode region Θ ig. Te resulting smooting factor for unweigted Gauss-Seidel ( = is approximately.5399, only marginally iger tan te optimum value.5396 obtained at =.2. Tis result is sligtly worse tan for Jacobi relaxation; tis is te opposite of te situation for te standard five-point discretization on te rectangular grid, but similar to tat for some nine-point discretizations [3].

6 6 G. ZHOU AND S. R. FULTON θ (2 θ (3 /2 θ 2 /π θ ( θ ( /2 /2 /2 θ /π Fig. 4.. Four modes defining te space E θ of armonics. 4. Two-grid analysis. Two-grid analysis improves upon smooting analysis by including te effects of grid transfers in addition to te smooting by relaxation. In a two-grid cycle, te coarse grid correction is computed by calculating te residual on te fine grid G, transferring it to te coarse grid G 2, solving te residual equation tere, and transferring te resulting correction back to te fine grid. Te effect on te fine-grid error is represented by te coarse-grid correction operator K 2 := I I 2(L 2 I 2 L, (4. were I 2 is te restriction operator, I 2 is te prolongation (interpolation operator, I is te identity operator on grid G, and we ave assumed tat te coarse-grid operator L 2 is invertible. A two-grid cycle consisting of ν and ν 2 relaxation sweeps on te fine grid before and after te coarse-grid correction, respectively, can be represented by te two-grid operator M 2 := S ν2 K2 S ν. (4.2 Tis two-grid cycle is an approximation to a multigrid V(ν, ν 2 cycle, wic is obtained by recursively solving te coarse-grid problem by te same approac (i.e., ν and ν 2 relaxation sweeps before and after a coarse-grid correction. Two-grid analysis measures te effect of M 2 on te fine-grid errors; it is more complicated tan smooting analysis since te grid transfers couple te ig and low modes. Again, if we express te exagonal grid using te oblique coordinate and Fourier modes using te dual basis, te analysis proceeds essentially te same as for rectangular grids (cf. [, 4.4]. Te key to te analysis is te use of spaces of armonics wic are invariant under te operators. For any θ = (θ, θ 2 Θ let θ ( := (θ, θ 2, θ ( := (θ, θ 2, θ (2 := (θ, θ 2, θ (3 := (θ, θ 2, were θ i := { θ i + π if θ i <, θ i π if θ i.

7 MULTIGRID ON HEXAGONAL GRIDS 7 Precisely one of tese four modes is a low mode; wile te following development olds for any θ Θ, we can for concreteness assume tat θ ( = θ Θ low as in Fig. 4.. Tese four modes coincide on te coarse-grid points x 2 j = x 2j, i.e., ϕ (θ (l, x 2j = ϕ (θ, x 2j = ϕ 2 (2θ, x 2 j, l =,, 2, 3. Since tey are linearly independent tey form a basis for te space of armonics E θ := span { ϕ (θ (l, : l =,, 2, 3 }. Any grid function ψ E θ can be represented in te form ψ (x = 3 A (l ϕ (θ (l, x, x G. (4.3 l= Since e iθ l = e iθ l, we can also write ψ (x = Ψ (xϕ (θ, x, x G, (4.4 were Ψ as te four-color pattern Ψ (x j = Ψ ( Ψ ( Ψ (2 Ψ (3 if j even and j 2 even, if j odd and j 2 even, if j even and j 2 odd, if j odd and j 2 odd. (4.5 wit values Ψ (l given in terms of coefficients A (l by Ψ ( A ( Ψ ( Ψ (2 = A ( A (2. (4.6 Ψ (3 A (3 Since te matrix in (4.6 is invertible, te two representations (4.3 and (4.4 of ψ E θ are complementary. Under appropriate assumptions, te space E θ is invariant under te operators K 2 and M 2 2, wic terefore may be represented by 4 4 matrices K (θ and M 2(θ, respectively. Tis means tat if ψ E θ 2 ten M ψ E θ as well, wit 2 coefficients in te expansion form (4.3 given by te matrix M (θ times te vector 2 2 of coefficients of ψ. Tis result is establised and te details of K (θ and M (θ are obtained by examining te effects of te discrete operators, smooting, and grid transfers as follows (te proofs are essentially te same as for rectangular grids and ence are omitted. Discrete operators. Given tat te fine-grid operator L is represented by a stencil [s k (], it is a linear operator on E θ. From Lemma 2. te matrix representation is L (θ = diag ( L (θ (, L (θ (, L (θ (2, L (θ (3. Likewise, te coarse-grid operator L 2, given by stencil [s k (2] 2, acting on ϕ 2 (2θ, x produces L 2 ϕ 2 (2θ, x = L 2 (2θϕ 2 (2θ, x, x G 2. (4.7

8 8 G. ZHOU AND S. R. FULTON Note tat wile te discrete operator sould be invertible on a finite grid (wit boundary conditions, often it is not on an infinite grid (e.g., te Laplacian of a nonzero constant is zero. Te two-grid analysis assumes te coarse-grid equation is solved exactly, so we omit te null space of L 2, i.e., Θ = { θ Θ low : L } 2 (2θ =. For example, for te discrete Laplacian (2.2 wit symbol given by (2.3, te null space Θ contains only te single mode θ = (,. Smooting. If te space E θ is invariant under te smooting operator S ten S can be represented by a 4 4 matrix Ŝ(θ. In te special case were all Fourier modes are eigenfunctions of S as in Lemma 3., tis matrix takes te simple form ( Ŝ (θ = diag S (θ (, S (θ (, S (θ (2, S (θ (3. For example, wit WJ or GS relaxation te matrix takes tis form using (3.3 or (3.4, respectively. If te Fourier modes are not eigenfunctions of S ten te matrix Ŝ (θ is not diagonal and must be computed directly; examples are given in 5. Restriction. stencil [ˆt k ] 2 We assume tat te restriction operator I 2 is represented by a indexed by k in some finite index set K R Z 2. Here ˆt k is te weigt of te contribution to te coarse-grid value at x 2 j ( I 2 ψ (x 2 j = from te fine-grid value at x 2j+k, i.e., k K R ˆt k ψ (x 2j+k. (4.8 Suc a restriction satisfies te following: Lemma 4.. If ψ E θ wit coefficients A(l in te representation (4.3, ten te restriction (4.8 produces ψ 2 = I 2ψ satisfying ψ 2 (x = A 2 ϕ 2 (2θ, x for x G 2 wit A 2 = (Ĩ2 (θ (, Ĩ2 (θ (, Ĩ2 (θ (2, Ĩ2 (θ (3 A ( A ( A (2 A (3, (4.9 were Ĩ2 (θ := k K R ˆt k e ik θ is te symbol of te restriction operator. Tus, te restriction I 2 is represented by te 4 matrix on te rigt side of (4.9, wic we denote by Î2 (θ. Peraps te most natural restriction is full weigting, wic on te exagonal grid as te stencil 2 8 and corresponding symbol Ĩ2 (θ = 4 [ + cos(θ + cos(θ 2 + cos(θ 2 θ ]. We will also consider injection, wit stencil [] 2 and corresponding symbol Ĩ2 (θ =. 2

9 MULTIGRID ON HEXAGONAL GRIDS 9 Prolongation. We likewise assume tat te prolongation operator I2 is represented by a stencil ]t k [ 2 indexed by k in some finite index set K P Z 2. Here t k is te weigt of te contribution from coarse-grid value at x 2 j to te fine-grid value at. Tis can be written explicitly in te form x 2j+k ( I 2 ψ 2 (x 2j+k = k t 2k +kψ 2 (x 2 j k, (4. were te sum is over all k Z 2 for wic 2k + k K P. Using te four-color form (4.4 it can be sown tat suc a prolongation satisfies te following: Lemma 4.2. If ψ 2 (x = A 2 ϕ 2 (2θ, x for x G 2, ten te prolongation (4. produces ψ = I2 ψ 2 E θ, and its coefficients A(l in te representation (4.3 satisfy A ( Ĩ A ( 2 (θ( A (2 = Ĩ 2 (θ( Ĩ A (3 2 (θ(2 A 2, (4. Ĩ2 (θ(3 were Ĩ 2 (θ := 4 k K P t k e ik θ is te symbol of te prolongation operator. Tus, te prolongation I2 is represented by te 4 matrix on te rigt side of (4., wic we denote by Î 2 (θ. Te natural prolongation on a exagonal grid is bilinear interpolation (more precisely, linear interpolation between nearest neigbors, wit stencil 2 2 and corresponding symbol Ĩ 2 (θ = 4 [ + cos(θ + cos(θ 2 + cos(θ 2 θ ]. Comparing Lemmas 4. and 4.2 we see tat if ˆt k = t k /4 ten te restriction and prolongation operators are adjoints of eac oter; in particular, tis is true for full weigting and bilinear interpolation. Putting te above pieces togeter establises te following: Teorem 4.3. Suppose tat te discrete operators L and L 2 and grid transfers I 2 and I 2 are represented by stencils. Ten for any θ Θ low\θ te space E θ is invariant under te coarse-grid correction operator K 2, wic is represented by te 4 4 matrix 2 K 2 (θ = Î Î 2(θ ( L2 (2θ Î2 (θ L (θ were Î is te 4 4 identity matrix. Furtermore, if E θ is invariant under te smooting operator S, ten it is also invariant under te two-grid operator M 2, wic is represented by te 4 4 matrix M 2 ν2 2 (θ = Ŝ(θ K (θŝ(θ ν. (4.2 Te effectiveness of te two-grid cycle is ten measured by te asymptotic convergence factor { } ρ 2 (M 2 2 := sup ρ ( M (θ : θ Θ low \Θ, (4.3

10 G. ZHOU AND S. R. FULTON were ρ( denotes te spectral radius. For comparison wit te (one-grid smooting analysis it is convenient to calculate te corresponding two-grid smooting factor 2 := [ ρ 2 (M 2 ] /(ν +ν 2, (4.4 wic estimates te convergence per fine-grid relaxation sweep. Values of 2 will be compared wit te numerical results in 6. Finally, two-grid analysis can be extended recursively to tree (or more grids as described in [, 2]. In some cases tis analysis may be needed, e.g., wen using different discretizations on different grids or different numbers of presmooting and postsmooting sweeps. Altoug we do not consider suc cases ere, te treegrid analysis is a simple extension of te two-grid analysis so we mention it ere for completeness. A tree-grid cycle wit γ coarse-grid corrections per level (e.g., γ = for a V-cycle, γ = 2 for a W-cycle, etc. is represented by te tree-grid operator were M 4 2 M 4 = S ν2 [ I I 2( I2 (M 4 2 γ (L 2 I 2 L ] S ν, is defined by (4.2 wit replaced by 2, i.e., M 4 2 := S ν2 2 Tus, te tree-grid operator M 4 [ I2 I 2 4 (L 4 I 4 2 L 2 ] S ν 2. is obtained by replacing (L 2 in (4. (4.2 by (L 4 2 = (I 2 (M 4 2 γ (L 2. Since eac of te four armonics for te grid G 2 are coupled to four armonics on grid G, te operator M 4 couples armonics in a 6-dimensional space, and tus is 4 represented by a 6 6 matrix M. Te details are te same as for rectangular grids (see [2] and tus will not be presented ere. Te effectiveness of te tree-grid cycle is ten measured by te asymptotic convergence factor { } ρ 3 (M 4 4 := sup ρ ( M (θ : θ Θ low \Θ, (4.5 were in tis context Θ low = [ π/4, π/4 2 and Θ is te null space of L 4. Te corresponding tree-grid smooting factor estimates te convergence per fine-grid relaxation sweep. 3 := [ ρ 3 (M 4 ] /(ν +ν 2, ( Smooting analysis revisited. Te smooting analysis in 3 assumes tat all Fourier modes ϕ (θ, are eigenfunctions of te smooting operator S. However, some smooters do not satisfy tis assumption, but instead couple te ig and low modes. For suc smooters te notation of te previous section allows us to extend te (one-grid smooting analysis, exactly as in te case of rectangular grids (cf. [, 4.5]. Specifically, if te spaces E θ are invariant under te smooting operator S, we can generalize te definition (3.2 of te smooting factor to read { [ ( (S, ν := sup ρ Q2 Ŝ (θ ν] } /ν : θ Θlow, Q 2 were te diagonal matrix represents te perfect coarse-grid correction: wit θ = θ ( 2 Θ low, Q = diag(,,,. An important example of suc a smooter on a rectangular grid is Gauss-Seidel relaxation wit red-black ordering (RB, wic acieves superior performance. Wile tis as no direct analogue on te exagonal grid, two orderings for weigted Jacobi relaxation retain some features of RB relaxation and tus merit investigation.

11 MULTIGRID ON HEXAGONAL GRIDS 5.. Coarse-fine ordering. On a rectangular grid, RB relaxation as te property tat only one of te two partial sweeps canges values at points corresponding to te coarse grid. Tis can be acieved on te exagonal grid by weigted Jacobi relaxation wit coarse-fine (CF ordering, wic is often used in algebraic multigrid metods [2, 8]. One CF sweep consists of two partial sweeps of (weigted Jacobi relaxation at points x j on grid G : coarse sweep: j even and j 2 even, fine sweep: remaining points x j. Te smooting operator for a complete CF sweep is tus S CF = S F SC, were SC and S F are te operators representing te coarse and fine sweeps, respectively. Wen applied to a Fourier mode ϕ (θ,, eac of tese partial sweeps produces a result wit te four-color pattern (4.4 (4.5, so E θ is invariant under SCF. Specifically, for eac Fourier mode te coarse sweep produces S Cϕ (θ, x = Ψ C (xϕ (θ, x for x G, were { SJ Ψ C (x j = (θ, if j even and j 2 even, oterwise, wit S J te symbol of te smooting operator for weigted Jacobi relaxation for te Laplacian, given by (3.3. Using (4.6 to express tis result in terms of te basis of E θ and applying tis result to te four basis modes θ (l, l =,, 2, 3 leads to te matrix representation a b b 2 b 3 Ŝ C (θ = b a b 2 b 3 b b a 2 b 3 (5. b b b 2 a 3 ( ( were a l := 4 SJ (θ (l, + 3 and b l := 4 SJ (θ (l,. Similarly, for te fine sweep we obtain c d d 2 d 3 Ŝ F (θ = d c d 2 d 3 d d c 2 d 3 d d d 2 c 3 ( were c l := S ( J(θ(l, and d l := 4 S J(θ(l,. Note tat S J depends on te relaxation parameter, wic may be cosen differently for te coarse and fine sweeps Four-color ordering. For te Laplacian operator on a rectangular grid, RB relaxation also as te property tat eac partial sweep operates on points wic are decoupled. Tis can be acieved on te exagonal grid by weigted Jacobi relaxation wit four-color (4C ordering (suggested by Ross Heikes. One 4C sweep consists of four partial sweeps of (weigted Jacobi relaxation at points x j on grid G : sweep : points x j wit j even and j 2 even, sweep 2: points x j wit j odd and j 2 even, sweep 3: points x j wit j even and j 2 odd, sweep 4: points x j wit j odd and j 2 odd.

12 2 G. ZHOU AND S. R. FULTON Te smooting operator for te full sweep is S 4C = S (4 S(3 S(2 S(, were S(l is te operator for sweep l =, 2, 3, 4. Sweep is te same as te coarse sweep of CF relaxation, so it is represented by te matrix Ŝ( = ŜC given by (5.. Te remaining sweeps are represented by te corresponding matrices a b b 2 b 3 a b b 2 b 3 Ŝ (2 (θ = b a b 2 b 3 b b a 2 b 3, Ŝ(3 (θ = b a b 2 b 3 b b a 2 b 3, b b b 2 a 3 b b b 2 a 3 and Ŝ (4 (θ = a b b 2 b 3 b a b 2 b 3 b b a 2 b 3. b b b 2 a 3 As before, a different relaxation parameter could be used for eac of te four sweeps. Bot of te above scemes offer potential advantages: te CF sceme preserves te symmetry of te exagonal grid, and te 4C sceme requires less storage. Values of te smooting factors for tese scemes are given in te following section. 6. Comparison wit numerical results. Te canonical model problem for elliptic equations is te Poisson problem. Here we consider 2 u(x, y = f(x, y on Ω = [, 2] [, 3], (6. wit periodic boundary conditions in x and y. Te true solution is specified as u(x, y = cos(πx sin(2πy/ 3 and te corresponding forcing f is computed analytically using (6.. Unless oterwise specified, tis problem is discretized using te second-order discretization (2.2 on a uniform exagonal grid wit mes size = /32 (i.e., 64 grid points in eac of te oblique coordinates and te restriction operator is full weigting. Coarser grids ave mes sizes successively doubled, wit = on te coarsest grid. We also consider (briefly a fourt-order discretization, te related modified Helmoltz problem, and sperical geodesic grids below. For an algoritm consisting of repeated V(ν, ν 2 cycles, te observed residual reduction per fine-grid relaxation sweep is measured by te factor N := ( /(ν+ν r 2, r were r := f L ũ is te residual before one V-cycle and r is te corresponding [ ] /2. residual after te cycle, and te norm is defined as r := i j (r ij 2 2 Te factor N sould closely approximate te one- and two-grid smooting factors and 2 ; values reported ere are te asymptotic values observed during cycles. Figure 6. sows te performance (analysis and actual of te four relaxation scemes considered above, using multigrid V(, cycles. Te curves plot te smooting factors per sweep (, 2, or N as functions of te relaxation parameter ; for te CF and 4C scemes te same value of was used for eac partial sweep. In eac case te numerical results agree well wit te analytical results (especially te two-grid analysis, and te optimal relaxation parameter is close enoug to =

13 MULTIGRID ON HEXAGONAL GRIDS 3 Weigted Jacobi Gauss Seidel.4.2 numerical one grid two grid Coarse Fine Four Color Fig. 6.. Smooting factors (one-grid, two-grid, and numerical as functions of te relaxation parameter for te Poisson problem using te four relaxation scemes as labeled and V(, cycles. Weigted Jacobi Gauss Seidel.4.2 numerical one grid two grid Coarse Fine Four Color Fig Same as Fig. 6. except for V(, 2 cycles. tat using te simpler unweigted relaxation would perform nearly as well. Very similar results are obtained for V(, 2 cycles (see Fig. 6.2, and te tree-grid analysis provides similar estimates for tis problem as expected (see Fig. 6.3.

14 4 G. ZHOU AND S. R. FULTON V(, V(,2.4 numerical.2 one grid two grid tree grid V(2, V(2, Fig Smooting factors (one-grid, two-grid, tree-grid, and numerical for weigted Jacobi relaxation and V(ν, ν 2 cycles as labeled. Te above results were all produced using full weigting as te restriction operator. Using injection instead degrades te performance, as sown in Fig. 6.4 for te WJ and GS scemes. Also, using injection wit WJ relaxation requires underrelaxation ( < for optimum performance, wic is not required wit full weigting. Te difference between te analytical and numerical results may be related to te fact tat injection (unlike full weigting does not preserve te integral of te residual it transfers, wic means tat te rigt-and side of te coarse-grid problem (in te periodic case considered ere no longer satisfies te compatibility condition. Indeed, numerical results (not sown for te corresponding Diriclet problem wic does not ave a compatibility condition sow better agreement wit te analysis. Since using injection leads to larger smooting factors tan does full weigting (in te cases were it works at all, it will not be considered furter ere. Te coarse-fine and four-color scemes analyzed in 5 permit using a different relaxation parameter for eac partial sweep. For te coarse-fine sceme te dependence on te two relaxation parameters ( and 2 for te coarse and fine sweeps, respectively is sown in Fig. 6.5, wic gives te smooting factors 2 and N for V(, cycles; te agreement between analysis and numerical results is good. However, te numerical values summarized in Table 6. sow tere is not a large difference between te smooting wit optimal relaxation parameters and te simpler unweigted sceme ( = 2 =. In view of tis result a similar optimization was not explored for te four-color sceme (wic already acieves excellent performance witout weigting. Table 6.2 summarizes te smooting factors (analytical and numerical for te four relaxation scemes considered ere (all unweigted, i.e., wit =. In eac case te analysis predicts te actual performance wit reasonable accuracy, wit te Since using injection makes sense only wen te residual is smoot, no results are sown for te CF and 4C scemes wit injection (in fact, it performs poorly for tese scemes.

15 MULTIGRID ON HEXAGONAL GRIDS 5 Weigted Jacobi V(, Gauss Seidel V(,.4.2 numerical one grid two grid Weigted Jacobi V(,2 Gauss Seidel V(, Fig Same as Figs. 6. and 6.2 except using injection analytical: numerical: N Fig Two-grid smooting factor 2 and numerical smooting factor N for weigted Jacobi relaxation wit coarse-fine ordering as functions of te relaxation parameters and 2 for te coarse and fine sweeps, respectively. two-grid analysis faring sligtly better. All scemes exibit smooting factors wic are nearly independent of te mes size. Numerical smooting factors for V(2, cycles (not sown are close to tose for V(, 2 cycles but typically sligtly larger. Te best performance is acieved by te four-color (4C sceme, and is comparable to te RB sceme on a rectangular grid. A remarkable feature of te exagonal grid is tat it provides an especially simple fourt-order discretization of te Poisson problem. In te Appendix it is sown tat te discretization ( u u i = 3 4 f + 6 f i ( i= i=

16 6 G. ZHOU AND S. R. FULTON optimal parameters smooting factor cycles factor 2 optimal unweigted V(, N V(, N Table 6. Smooting factors (per sweep for te CF sceme wit optimal relaxation parameters and corresponding values witout weigting. analytical numerical relaxation cycles one-grid two-grid = 32 = 64 = 28 WJ V(, V(, GS V(, V(, CF V(, V(, C V(, V(, Table 6.2 Smooting factors (per sweep for te second-order Poisson problem for four (unweigted relaxation scemes wit full weigting. as fourt-order accuracy. Tis is te exagonal-grid analog of te so-called Merstellen Verfaren discretization [9], but it is markedly simpler: it involves only seven points, and te weigts at all surrounding points are te same bot for te solution and te forcing. Furtermore, te operator on te left is identical to tat in te secondorder discretization (2.2, so only te rigt-and side of te second-order discretization needs to be modified to obtain fourt-order accuracy (in te multigrid context tis must be done only on te finest grid. Te analytical smooting factors for te two discretizations are identical; numerical smooting factors for te fourt-order sceme (not sown matc tose sown in Table 6.2 for te second-order sceme quite closely. A sligt generalization of te Poisson equation is te modified Helmoltz equation λu(x, y 2 u(x, y = f(x, y (6.3 were λ is a positive constant. Using te second-order discretization (2.2 for te Laplacian leads to te discretization ( λu u u i = f. i=

17 MULTIGRID ON HEXAGONAL GRIDS 7 Jacobi numerical one grid two grid Four Color λ λ Fig Smooting factors (one-grid, two-grid, and numerical as functions of te parameter λ for te modified Helmoltz problem using te (unweigted Jacobi and four-color scemes wit V(, cycles. Te symbol for weigted Jacobi relaxation is S J (θ, = ( λ [cos(θ + cos(θ 2 + cos(θ θ 2 ] were is te relaxation parameter. Smooting factors (, 2, and N for tis problem (wit = are sown in Fig. 6.6 as functions of λ, wit te numerical results computed for te same domain, grid, and analytical solution as used above. As expected, te convergence improves as λ increases; again, te agreement between analytical and numerical results is substantial. Te metods examined above can also be incorporated into te Full Multigrid (FMG algoritm []. Tis quasi-direct algoritm combines nested iteration wit a fixed number of multigrid cycles to solve te problem on eac successive grid from coarsest to finest. Properly designed, tis sould solve te problem to te level of truncation error wit work proportional to te number of unknowns on te finest grid. Here we consider te following two algoritms for te Poisson problem, bot using te (unweigted four-color sceme and te same grids used above: FMG2: Uses second-order discretization, one V(, cycle per level, and fourt-order initial interpolation. FMG4: Uses fourt-order discretization on currently finest level, two V(, cycles per level, and sixt-order initial interpolation. Tables 6.3 and 6.4 summarize te results from te algoritms FMG2 and FMG4, respectively. In bot cases te truncation error τ and discretization error ε decrease wit at te proper rates, and te problem is solved to or below te level of truncation error, i.e., r τ and v < ε. Finally, wile our analysis is for uniform exagonal grids, it also provides useful estimates for sperical geodesic grids, wic consist of many quasi-uniform exagonal cells togeter wit twelve pentagonal cells. Heikes and Randall [4] introduced a multigrid metod for te Poisson problem on suc grids wic uses weigted Jacobi relaxation and injection. As sown in Fig. 6.7 (top left panel, te exagonal-grid analysis accurately predicts te performance of tis metod, and underrelaxation ( < is required for best performance. As before, better agreement wit analysis (and better performance witout weigting is obtained using full weigting (top rigt panel. On te geodesic grid takes te form [cf. (4.8] n i= A i(ψ i, ( I 2 ψ = A (ψ + 2 A + n 2 i= A i

18 8 G. ZHOU AND S. R. FULTON τ τ 2 / τ r / τ ε ε 2 / ε v / ε e e e e e e e e e e e e Table 6.3 Results from algoritm FMG2. Here τ = f L u is te truncation error, r = f L ũ is te residual, ε = u u is te discretization error, and v = u ũ is te algebraic error, were u, u, and ũ denote te continuous, discrete, and approximate discrete solutions, respectively. τ τ 2 / τ r / τ ε ε 2 / ε v / ε 8 3.8e e e e e e e e e e e e Table 6.4 Results from algoritm FMG4 (notation as in Table 6.3. were te subscripts and i index te coarse-grid point and te n surrounding finegrid points (n = 5 or 6 and A and A i denote te areas of te corresponding fine-grid cells. As suggested by te exagonal-grid analysis, te CF and 4C scemes perform well on te geodesic grid (bottom two panels, wit te (unweigted 4C sceme acieving a smooting factor of about.3, again comparable to te RB sceme on a rectangular grid. 7. Summary and Conclusions. Local Fourier analysis of multigrid metods on exagonal grids closely parallels tat on rectangular grids, provided we express exagonal grids using oblique coordinates and Fourier modes using a dual basis. Applying tis framework to te Poisson problem we ave obtained analytical smooting factors (using one-grid and two-grid analyses for several relaxation scemes and grid transfers and verified tese wit numerical calculations. In particular, we find tat: (i Using full weigting, eac relaxation sceme tested does not require underor over-relaxation for nearly optimal performance; (ii Jacobi relaxation (wic is parallelizable and preserves te symmetry of te grid performs nearly as well as Gauss-Seidel relaxation (wic is not parallelizable and does not preserve symmetry; (iii Jacobi relaxation wit four-color ordering as a smooting factor of approximately.25 for V(, cycles, making it comparable to Gauss-Seidel relaxation wit red-black ordering on te rectangular grid. (iv An especially simple compact fourt-order discretization exists for te Pois-

19 MULTIGRID ON HEXAGONAL GRIDS 9 Weigted Jacobi (injection Weigted Jacobi (full weigting.4.2 numerical one grid two grid Coarse Fine Four Color Fig Numerical smooting factors for te Poisson problem on a sperical geodesic grid wit,242 cells (five grid levels, compared to te analytical one- and two-grid smooting factors for a uniform exagonal grid. Te Jacobi results use V(, 2 cycles; te oters use V(, cycles and full weigting. son equation on a uniform exagonal grid; (v Second- and fourt-order FMG algoritms on te exagonal grid solve te Poisson problem to te level of truncation error in work proportional to te number of unknowns. (vi Te exagonal-grid analysis gives quantitatively correct guidance for developing multigrid solvers for sperical geodesic grids. Acknowledgments. Te autors tank Ross Heikes for elpful comments and copies of is geodesic grid codes, and te anonymous referees for teir suggestions. Appendix. Fourt-Order Compact Discretization. In [6] te existence of a compact fourt-order discretization of te Poisson equation 2 u = f on a uniform exagonal grid G is mentioned, but te formula is not given. To derive it, we start wit te compact symmetric discretization b b d d L u := 2 b a b u = d c d f =: I f, b b d d were f = f G and te constants a, b, c, and d are to be cosen. If te true solution u is in C 6 ten Taylor expansions yield ( a 6b L u = 2 u 3b 2 2 u + 3b ( 2 u + O( 4

20 2 G. ZHOU AND S. R. FULTON for te left-and side and I f = (c + 6df + 3d f + O( 4 for te rigt-and side. Using f = 2 u te truncation error is tus τ :=L u I f ( a 6b = 2 u + [ 3b 2 ] ( 3b (c + 6d 2 u d 2 2 ( 2 u + O( 4. 2 Te discretization will be consistent (τ as if a = 6b and 3b/2 = c + 6d, and fourt-order accurate if 3b/32 = 3d/2. Solving tese equations yields a = 6b, c = 9b/8, and d = b/6. Coosing te normalization b = 2/3 (so tat L u 2 u and I f f as ten gives te discretization u = 8 24 f wic is (6.2. Since tis discretization relies on te fact tat f = 2 u, it cannot be extended to te modified Helmoltz equation. REFERENCES [] A. Brandt, Multi-level adaptive solutions to boundary-value problems, Mat. Comp., 3 (977, pp [2] M. Brezina, A. J. Cleary, R. D. Falgout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, and J. W. Ruge, Algebraic multigrid based on element interpolation (AMGe, SIAM J. Sci. Comput., 22 (2, pp [3] W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial (second edition, Society for Industrial and Applied Matematics, 2. [4] R. Heikes and D. A. Randall, Numerical integration of te sallow-water equations on a twisted icosaedral grid. Part I: Basic design and results of tests, Mon. Wea. Rev., 23 (995, pp [5], Numerical integration of te sallow-water equations on a twisted icosaedral grid. Part II: A detailed description of te grid and an analysis of numerical accuracy, Mon. Wea. Rev., 23 (995, pp [6] V. L. Makarov, S. V. Makarov, and M. N. Moskal kov, Spectral properties of te difference Laplacian on a exagonal mes and teir application, Differ. Equ., 29 (993, pp [7] S. Ni cković, M. B. Gavrilov, and I. A. To sić, Geostropic adjustment on exagonal grids, Mon. Wea. Rev., 3 (22, pp [8] J. W. Ruge and K. Stuüben, Algebraic multigrid, in Multigrid Metods, S. F. McCormick, ed., Society for Industrial and Applied Matematics, 987, pp [9] S. Scaffer, Higer-order multi-grid metods, Mat. Comp., 43 (984, pp [] U. Trottenberg, C. Oosterlee, and A. Scüller, Multigrid, Academic Press, 2. [] R. Wienands and W. Joppic, Practical Fourier Analysis for Multigrid Metods, CRC Press, 25. [2] R. Wienands and C. W. Oosterlee, On tree-grid Fourier analysis for multigrid, SIAM J. Sci. Comput., 23 (2, pp [3] I. Yavne and E. Olvovsky, Multigrid smooting for symmetric nine-point stencils, Appl. Mat. Comp., 92 (998, pp