A LOCAL FOURIER ANALYSIS FRAMEWORK FOR FINITE-ELEMENT DISCRETIZATIONS OF SYSTEMS OF PDES

Transcription

1 A LOCAL FOURIER ANALYSIS FRAMEWORK FOR FINITE-ELEMENT DISCRETIZATIONS OF SYSTEMS OF PDES SCOTT P. MACLACHLAN AND CORNELIS W. OOSTERLEE Abstract. Since teir popularization in te late 1970s and early 1980s, multigrid metods ave been a central tool in te numerical solution of te linear and nonlinear systems tat arise from te discretization of many PDEs. In tis paper, we present a local Fourier analysis (LFA, or local mode analysis) framework for analyzing te complementarity between relaxation and coarse-grid correction witin multigrid solvers for systems of PDEs. Important features of tis analysis framework include te treatment of arbitrary finite-element approximation subspaces and overlapping multiplicative- Scwarz type smooters. Te resulting tools are demonstrated for te Stokes, curl-curl, and grad-div equations. 1. Introduction. First envisioned as a tecnique for solving Poisson-type problems wit optimal complexity, multigrid and oter multilevel algoritms ave become te metods of coice for solving te matrix equations tat arise in a wide variety of applications. In tis paper, we are concerned wit te design and analysis of multigrid algoritms for te solution of discretized systems of partial differential equations. Wile general principles exist tat can aid in developing multigrid tecniques for an application area, te optimal coices for its components are often difficult to determine beforeand. Many approaces to multigrid teory ave been investigated in te last 30 years (see, for example, [8, 17, 22, 33]); among tese, te tecnique of local Fourier analysis (LFA, or local mode analysis), first introduced in [9], as remained very successful, providing accurate predictions of performance for a variety of problems, including for systems of PDEs. Te primary advantage of LFA is tat it allows quantitative prediction of multigrid convergence factors under reasonable assumptions. Te word, local, in LFA indicates a focus on te caracter of an operator in te interior of its domain, were it is assumed to be represented by a constant discretization stencil. Te Fourier symbols of suc operators can easily be computed. A furter insigt in [41] was tat all of te components in a multigrid metod can be analyzed in tis fasion, leading to a block-diagonal representation in a Fourier basis. LFA elped, for example, in te understanding of SOR as a smooter for moderately anisotropic and ig-dimensional problems [50] and te solution of te biarmonic equation wit efficiency similar to tat of Poisson [46]. Recent advances in LFA include LFA for ig-dimensional problems [52], for multigrid as a preconditioner [47], for triangular and exagonal meses [19, 51], optimal control problems [5], and discontinuous Galerkin discretizations [23]. Te LFA monograp and related software of Wienands and Joppic [46] focus on LFA for collocated discretizations, providing an excellent tool for experimenting wit Fourier analysis. In tis paper, we present a framework for performing te local Fourier analysis for state-of-te-art finite-element discretizations of systems of PDEs. In particular, In preparation. Tis researc was supported by te European Community s Sixt Framework Programme, troug a Marie Curie International Incoming Fellowsip, MIF1-CT Te work of SM was partially supported by te National Science Foundation, under grant DMS Delft University of Tecnology, Faculty of Electrical Engineering, Matematics and Computer Science, Mekelweg 4, 2628 CD Delft, Te Neterlands and CWI, National Researc Institute for Matematics and Computer Science, Amsterdam, Te Neterlands. c.w.oosterlee@cwi.nl Department of Matematics, Tufts University, 503 Boston Avenue, Medford, MA 02155, USA. Formerly at TU-Delft and CWI. scott.maclaclan@tufts.edu 1

2 we sow tat te LFA ansatz is still valid wen using overlapping multiplicative smooters, suc as were proposed in [2], for te grad-div and curl-curl equations, and in [28,43] for te Stokes equations. Analysis of te additive versions of tese smooters was conducted in [4, 39]; owever tis form of analysis does not extend to cover te multiplicative case. LFA for overlapping multiplicative smooters as been, to our knowledge, performed in only two cases, for te staggered finite-difference discretization of te Stokes and Navier-Stokes equations [40], and for a mixed finite-element discretization of Poisson s equation [34]. We also apply tis analysis to te well-known multigrid solvers for te grad-div, curl-curl, and Stokes equations, providing quantitative predictions of te performance of multigrid metods based on tese smooters, in contrast to te non-predictive proofs of convergence offered in [2,32]. In te case of te Stokes equations, in particular, quantitative estimates ave been notably missing from te literature [31]. Te remainder of tis paper is organized as follows. First, in Section 2, we provide some background on te motivating PDE systems for tis work. LFA smooting analysis is discussed in Section 3, wit a focus on te treatment of overlapping multiplicative smooters. A detailed example is presented in Section 4. Section 5 presents two-grid LFA, focusing on te issue of multigrid grid transfers for staggered discretizations. Finally, Section 6 presents te application of tese tecniques to appropriate discretizations of te Stokes, grad-div, and curl-curl equations. Tere, we focus on te impact of te coice of transfer operators and on te coice of smooter and under-relaxation parameters on te two-grid LFA convergence. 2. Multigrid and Finite Elements for PDE Systems. Te discretization of systems of PDEs must be done wit care, to avoid te introduction of unstable modes in te resulting discrete system. In finite elements, tis typically results in coosing different finite-element subspaces for different components of te system, to satisfy known inf-sup conditions, leading to te use of Raviart-Tomas, Nédélec, or Taylor-Hood elements, for example. For a toroug treatment of tese issues in te finite-element context, see [6, 12]. We sall refer to te finite-element discretizations tat we treat ere collectively as staggered discretizations, indicating tat te nodes associated wit te discrete degrees of freedom are not aligned on te same grid for eac component of te PDE system. Te tecniques developed ere are applicable to arbitrary staggered discretizations of systems of PDEs, including te trivial case of a collocated discretization Multigrid for Systems of Equations. Many standard discretizations of systems of PDEs (including tose described below) do not guarantee tat te resulting matrices are diagonally dominant (or even tat tey are definite) eiter because of te properties of te continuum operators temselves, or because of necessary constraints on te discretizations. In tese cases, expensive relaxation tecniques may be used to reestablis effective multigrid convergence. Unfortunately, tese relaxation tecniques are no longer algebraic black boxes, like te Jacobi and Gauss-Seidel iterations. Instead, te details of tese tecniques are determined by tose of te underlying PDEs. A first indication for te appropriate coice of relaxation metod for a system of equations can be derived from te systems determinant. Interestingly, te determinant of te discrete operator may also give us valuable information about te stability of te discretizations used for a system. Te direct relation between effectiveness of smooting and te determinant of te discrete system is by means of te -ellipticity 2

3 concept [10,42]. For unstable discretizations, wic give rise to unpysical oscillations in te numerical solutions, tere is no cance tat we can set up efficient local, i.e., pointwise, smooting metods. An obvious coice in te case of strong off-diagonal operators in te differential system (also indicated by te determinant) is collective smooting: All unknowns in te system at a certain grid point or grid cell are updated simultaneously. Wile te use of tese smooters leads to efficient multigrid approaces for systems of PDEs, collective relaxation is not te only possible approac. Te main alternative is te use of distributive smooters [10, 20, 24], wic take teir name from a distribution operation; te discrete (or continuum) equations are transformed by rigt matrix multiplication into a block triangular matrix tat is amenable to pointwise relaxation. Simple pointwise relaxation is performed on tis block triangular system and, ten, te resulting update is distributed (based on te transformation matrix) back to te original matrix problem. Wile distributed smooters are often found to be more efficient tan overlapping smooters [20], teir applicability is limited by te need to find an effective distribution matrix; tis is often difficult to do for problems wit unstructured grids or variable coefficients. Tus, distributed relaxation may be difficult to implement in a general and purely algebraic fasion, as would be necessary for use witin an algebraic multigrid iteration. Furtermore, te proper treatment of boundary conditions in distributive relaxation may not be trivial, as typically te operator of te preconditioned system is of iger order tan te original operator, tus requiring additional (possible nonpysical) boundary conditions witin smooting. Substantial effort as owever been put into successfully extending te distributed smooter of Hiptmair [24] to algebraic multigrid algoritms for te curl-curl equation [3, 27, 38]. In tis paper, we focus exclusively on collective relaxation approaces Discretizing Systems of PDEs. As a first pair of examples, we consider te gradient-divergence (grad-div) and curl-curl equations, and (a U) + U = F, in Ω, (2.1) (a U) + U = F, in Ω, (2.2) wit parameter a > 0, were Ω is an open domain in R d. Tese operators appear frequently in te formulation of matematical models in pysics and engineering, particularly for problems related to electro-magnetics or fluid and solid mecanics (see, for example, [2, 20, 25] for more details). In te finite-element framework, face elements, suc as te Raviart-Tomas elements, ave been proposed for accurate discretization of (2.1) [37], wile edge elements, suc as te Nédélec elements [35] are commonly used for (2.2), see Figure 2.2. Te difficulty in acieving efficient multigrid treatment of te resulting discrete linear systems comes from te fact tat te eigenspace associated wit te minimal eigenvalue of te discrete operator contains many eigenvectors (for large enoug parameter a). For Equation (2.1), tis arises because any divergence-free vector is an eigenvector corresponding to tis minimal eigenvalue, wile a similar difficulty occurs wit curlfree vectors in (2.2). In bot cases, tese components can be arbitrarily oscillatory and can neiter be reduced by standard (pointwise) smooting procedures, nor be well represented on coarse grids [2]. 3

4 (i,j) (i,j) Fig Placement of unknowns wit Raviart-Tomas FE for grad-div operator (left), and Nédélec edge elements for curl-curl (rigt). A remedy for (2.1) proposed in [45] builds upon local div-free functions and teir ortogonal complements in te finite-element space. In [1], a multigrid preconditioner was presented for a discretization wit te lowest-order Raviart-Tomas finite-element spaces on triangles. A distributive smooting tecnique in multigrid to andle te troublesome div-free components was proposed in [26]. Tese tecniques were extended for te curl-curl equations in [2, 25]. Here, we will quantitatively analyze te multiplicative collective smooter introduced in [2], known as te AFW smooter. Tis smooter can be motivated by tinking about te different treatment given by te grad-div or curl-curl operator to components of U tat look like gradients and tose tat look like curls. As te dominant part of tese operators doesn t act on one component of te solution, it is important tat te relaxation tecnique can accurately resolve tese components on all scales. Te Stokes equations are central to te simulation of certain viscous fluid-flow problems. Tey are represented by a saddle point problem, U + P = F (2.3) U = 0 (2.4) for velocity vector, U, and scalar pressure, P, of a viscous fluid.te weak form of te Stokes equations is found by multiplying by test functions, V and Q, and integrating by parts. Writing tis system in terms of te bilinear forms, a 11 (U, V) = d Ω i=1 ( U i) ( V i )dω and a 12 (V, P) = ( V)PdΩ, we ave Ω a 11 (U, V) + a 12 (V, P) = F VdΩ (2.5) Ω a 21 (U, Q) = 0, (2.6) wit a 12 (, ) = a 21 (, ), and vector-valued functions U, V : R d R d. Notice tat U and P lie in different spaces; typically, U V (H 1 (Ω)) d, wile P W L 2 (Ω). In proving uniqueness of te pressure component of te solution, P, a natural condition [6, 12, 14] arises, inf sup a 12 (V, P) = β > 0. (2.7) P W V V P V Tis condition is known by many names, including te Ladyzenskaya-Babuška-Brezzi (or LBB) condition, and te inf-sup condition. Similar considerations apply to te discrete problem attained by restricting te functions to finite-dimensional subspaces U, V V and P, Q W, leading to a 4

5 discrete version of te inf-sup condition. A natural discretization, representing bot U and P wit bilinear basis functions does not satisfy te necessary inf-sup condition [18] and, so, we are forced to consider iger-order basis functions for Equations (2.3) and (2.4), suc as te Taylor-Hood elements [6, 21] were U is represented by biquadratic basis functions and P is represented by bilinears. Te development of efficient smooters for te Stokes equations was originally performed in te staggered finite-differences setting. Tere, te concepts of collective and distributive relaxation were developed, by Vanka [43] and Brandt and Dinar [10, 11], respectively. Tese smooters were later accompanied by quantitative analysis, based on LFA. For FE discretizations, work on efficient smooters for te Stokes and Navier-Stokes equations includes tat by Braess and Sarazin [7], wic is based on an approximate factorization of (2.3) and (2.4), as well as tat by Jon and oters [28,29, 31, 49], wic focuses on a variety of smooters including tose of collective (Vanka) type. It was te FE setting especially tat drove te rapid development of te algebraic multigrid metod in te nineties (of te last century), wit te recognition of its impressive efficiency, often for completely unstructured meses. Quantitative teory for metods on tis type of meses is not available, unfortunately. To bridge te gap between te fully understood case of finite differences on structured grids and te case of finite elements on completely unstructured grids, we develop ere quantitative analysis for FE discretizations on structured quadrilateral meses in 2D, still leading to stencil-based discretizations. 3. Analysis of relaxation wit LFA. A quantitative, predictive teoretical framework, suc as LFA, allows significant algoritmic development independent of an implementation. Here and in Section 5, we review te ideas beind two-grid local Fourier analysis; first, we focus on te analysis of te smooting step in Fourier space. We consider te solution of a linear system of equations, A u = f, were te subscript,, serves to remind us tat te origins of matrix A are in te discretization of a PDE on a uniform quadrilateral grid wit messize (or, possibly, wit messizes = ( x, y,...) T tat are not uniform across dimensions). Given an approximation, v, to te solution of A u = f, te residual equation relates te error, e = u v, in tat approximation to te residual, r = f A v, as A e = r. Tus, for a given approximation, v, we can express te true solution as u = v +A 1 r. Coosing M to be an approximation to A tat is easily inverted leads to an update iteration tat can be analyzed in terms of its error-propagation operator e ( I M 1 A ) e. A complete analysis of te convergence properties of te error propagation operator arises in terms of its eigenvectors, {φ (j) }, and eigenvalues, {λ j }. Any initial error, e (0) 1, can ten be expanded into te basis given by te eigenvectors of I M A, and te error after k iterations of relaxation is given by e (k) = j σ jλ k j φ(j), were te coefficients, {σ j }, are defined so tat te expansion is valid for te initial error, e (0). Te effectiveness of te relaxation on te component of te error in te direction of a given eigenvector, φ (j), is ten given simply by te eigenvalue, λ j. If λ j is small (e.g., λ j 0.5), errors in te direction of φ (j) are quickly attenuated by te iteration. For large λ j, suc tat λ j 1, te errors in te direction of φ (j) are slow to be reduced and, after a few steps of te iteration, tese errors will dominate te remaining difference between u and v. 5

6 Finding te eigenvectors and eigenvalues of I M 1 A for tis analysis can be quite difficult, depending on te matrices, A and M. For general matrices tere may be little relation between te eigenvectors and eigenvalues of A and tose of relaxation, unless A and M are assumed to ave more structure tan is typically expected, suc as being circulant. Suc structure is strongly affected by boundary conditions on te PDE; wile te rows of te matrix corresponding to degrees of freedom in te interior of te PDE domain may ave a natural Toeplitz structure (representing a discrete PDE on a structured grid), imposition of boundary conditions usually results in a set of rows tat ave quite different values. Te key idea beind local Fourier analysis is to ignore te effect of tese boundary conditions, by extending te operator and relaxation stencils from te interior of te domain to infinite-grid Toeplitz matrices tat can bot be diagonalized by a Fourier basis. Any infinite-grid Toeplitz matrix is diagonalized by te matrix of Fourier modes, Φ, were we index te columns of Φ by a continuous index, θ ( π 2, ] d, 3π 2 and te rows by teir spatial location, x, and write φ (x, θ) = e ıθ x/. In tis setting, LFA as provided effective predictions of te performance of multigrid cycles based on many common smooters, including Gauss-Seidel [42], SOR [50], and ILU [48]. LFA for systems of PDEs is based on a simple extension of te assumptions of LFA for scalar PDEs. In te systems case, we assume tat te matrix, A, is now a blockmatrix, were eac block is an infinite-grid Toeplitz matrix. Under tis assumption, eac block in A may be diagonalized by left and rigt transformations wit Fourier matrices, Φ, altoug possibly using different nodal coordinates on te left and rigt for te off-diagonal blocks LFA for overlapping smooters. Here, we focus on te LFA of overlapping coupled multiplicative smooters. Overlapping smooters require only knowledge of te element structure, wic may be easily retained on coarse scale troug element agglomeration or AMGe tecniques [13, 15, 30, 44]. We identify a collective relaxation sceme as one tat partitions te degrees of freedom of A into regular subsets, S i,j, wose union provides a cover for te set of degrees of freedom. By saying tat tese subsets are regular, we mean tat tere is a one-to-one correspondence between te degrees of freedom in any two subsets, S i,j and S k,l ; eac subset as te same size, and te same number of eac type of degree of freedom tat comes from discretizing different unknowns of te continuum system. Te partitioning need not be disjoint; an overlapping coupled smooter occurs wen some collection of te degrees of freedom appears in multiple subsets, typically associated wit some adjacent indices. In collective relaxation, updates are computed (sequentially or in parallel) by solving te local system (or anoter nonsingular auxiliary system) associated wit eac subset, S i,j, wit te most recent residual restricted to S i,j as a rigt-and side. If te subsets S i,j are mutually disjoint, ten a collective relaxation sceme is simply a block-wise Jacobi or Gauss-Seidel sceme and can be analyzed as suc. If te blocks overlap, owever, so tat certain degrees of freedom are updated multiple times over te course of a single sweep of relaxation, classical LFA tecniques fail. In a relatively unknown paper [40], Sivaloganatan analyzed te Vanka smooter [43], a multiplicative form of overlapping collective relaxation for te staggered finite-difference discretization of te Stokes equation; unfortunately, tis paper includes several misprints, wic make te results difficult to appreciate. Independently, Molenaar analyzed a similar collective smooter for a mixed finite-element discretization of Poisson s equation [34]. Two important questions are, owever, left unanswered in [34,40]: 6

7 Fig Partitioning of degrees of freedom into overlapping subsets based on cells weter te Fourier ansatz is justified for coupled overlapping smooters and weter tese tecniques can be generalized for oter PDE problems and discretizations. LFA for non-overlapping relaxation succeeds because, in te infinite-grid Toeplitz setting, te matrix, A, is split into two Toeplitz pieces, A = M N, were M and N are also bot Toeplitz. Tus, all tree matrices (and, in particular, te error-propagation operator, M 1 N ) are diagonalized by a similarity transformation wit te Fourier matrix, Φ (or, in te case of systems, a block matrix consisting of disjoint Fourier matrices). It is not apparent tat te same is true for overlapping relaxations, because te error-propagation operator is not easily written in terms of a matrix splitting. To illustrate, we consider te (most common) case of cell-wise relaxation; for eac node, (i, j), associated wit te grid- mes, we define a cell of size adjacent to, or including node (i, j), and simultaneously solve for updates to all degrees of freedom tat fall witin or on te boundary of tis cell, see Figure 3.1. Relaxation is ten defined in a lexicograpical Gauss-Seidel manner, sequentially solving for te unknowns associated wit cell (i, j), going first across te mes from left to rigt, ten up te mes. Note tat, using tis definition of te collections, S i,j, eac degree of freedom located at te corner of cell (i, j) is included in four subsets, wile tose on te edges of cell (i, j) are included in two subsets, and degrees of freedom in te interior of a cell are included in only te subset corresponding to te cell. By a similar count, if tere are k degrees of freedom at eac cell corner, l x and l y degrees of freedom along te x and y edges of a cell, respectively, and m interior degrees of freedom, ten 4k + 2(l x + l y ) + m degrees of freedom are included in te subset, S i,j. Considering te (ypotetical) elements in Figure 3.1, two possible definitions of te subsets, S i,j, are igligted. One possibility, using te element boundaries (solid lines) to define te cells yields subsets tat overlap bot at te corners of te elements and along one of te element boundaries, wile tere is a unique interior node in eac subset tat belongs only to S i,j. Anoter possibility, using te dualelement boundaries (marked by te dased lines), as overlap only at te element edges, wit two interior nodes for eac subset, S i,j. In order to use te Fourier ansatz, tat te error-propagation operator for coupled (overlapping) relaxation is diagonalized by te Fourier matrix, Φ, we need to know tat tis is true, regardless of te distribution of degrees of freedom witin S i,j. Tis is proven in te following result. Teorem 3.1. Assume tat A is a block matrix wit infinite-grid Toeplitz blocks, corresponding to te discretization of a two-dimensional PDE on a regular grid wit messize,, and let k index te variables witin te collections, S i,j, of variables to be updated simultaneously. Let te initial error (before te beginning of 7

8 te relaxation sweep) for eac unknown, U (k), be given by a single Fourier mode, E (k) i,j = α (k) e ıθ x(k) i,j /, were x (k) i,j is te location of te discrete node corresponding to unknown U (k) associated wit relaxation subset S i,j. Let te update for te degrees of freedom in eac subset S i,j be calculated as U new i,j = U old i,j + B 1 R old i,j, were R old i,j is te residual at te nodes in S i,j evaluated before tese unknowns are updated by te relaxation for cell S i,j, Ui,j old and Ui,j new are te approximations to U i,j before and after te relaxation sweep, and B is some nonsingular approximation of A i,j, te diagonal block of A corresponding to te subset S i,j. Consider a partial lexicograpical relaxation sweep, at te stage were te correction to cell S i,j is to be computed. Suppose tat, for all degrees of freedom, k n, located at nodes of te cells, S l,m (so tat x (kn) l,m is te lower-left corner of te cell associated wit S l,m ), te once-corrected, twice-corrected, tree-times-corrected, and four-times-corrected errors satisfy E (kn,1) l,m E (kn,2) l,m E (kn,3) l,m E (kn,4) l,m = α(kn,1) e ıθ x(kn) l,m / for m j, or m = j + 1 and l i, = α(kn,2) e ıθ x(kn) l,m / for m j, or m = j + 1 and l < i, = α(kn,3) e ıθ x(kn) l,m / for m < j, or m = j and l i, = α(kn,4) e ıθ x(kn) l,m / for m < j, or m = j and l < i. Furter, suppose tat for all degrees of freedom, k, located on te orizontal edges of te cells, S l,m (so tat x (k ) l,m lies on te bottom edge of te cell associated wit S l,m), te once-corrected and twice-corrected errors satisfy E (k,1) l,m = α(k,1) e ıθ x(k ) l,m / for m j, or m = j + 1 and l < i, E (k,2) l,m = α(k,2) e ıθ x(k ) l,m / for m < j, or m = j and l < i. Similarly, suppose tat for all degrees of freedom, k v, located on te vertical edges of te cells, S l,m (so tat x (kv) l,m lies on te left edge of te cell associated wit S l,m), te once-corrected and twice-corrected errors satisfy E (kv,1) l,m = α(kv,1) e ıθ x(kv ) l,m / for m < j, or m = j and l i, E (kv,2) l,m = α(kv,2) e ıθ x(kv ) l,m / for m < j, or m = j and l < i. Finally, suppose tat for all degrees of freedom, k i, located strictly in te interiors of te cells, S l,m, te once-corrected errors satisfy E (ki,1) l,m = α(ki,1) e ıθ x(k i) l,m / for m < j, or m = j and l < i. 8

9 Ten, after te corrections ave been computed for te degrees of freedom in S i,j, E (kn,1) i+1,j+1 = α(kn,1) e ıθ x(kn) i+1,j+1 /, E (kn,2) i,j+1 = α(kn,2) e ıθ x(kn) i,j+1 /, E (kn,3) i+1,j = α (kn,3) e ıθ x(kn) i+1,j /, E (k,1) i,j+1 = α(k,1) e ıθ x(k ) i,j+1 /, E (kv,1) i+1,j = α (kv,1) e ıθ x(kv ) i+1,j /, E (ki,1) i,j = α (ki,1) e ıθ x(k i) i,j /. E (kn,4) i,j = α (kn,4) e ıθ x(kn) i,j /, E (k,2) i,j = α (k,2) e ıθ x(k ) i,j /, E (kv,2) i,j = α (kv,2) e ıθ x(kv ) i,j /, Proof. Consider a single degree of freedom, k, in S i,j. Te residual, r (k) i,j, associated wit k before te relaxation on cell S i,j can be expressed as a function of te Fourier coefficients corresponding to te errors (bot original and updated), as r (k) i,j = (A E) (k) i,j. In particular, for any i, j, we can write r(k) i,j = f (k) ({α})e ıθ x(k) i,j /, were {α} denotes te set of all Fourier indices, as described in te statement of te teorem. Note tat f (k) ({α}) depends only on te Fourier coefficients and te identity of te degree of freedom, k, witin S i,j and, in particular, is independent of te cell indices, i, j, under consideration. Tis is because te update states of te variables around eac cell move in a consistent way as te relaxation proceeds, so tat eac cell sees te same types of updated errors in its neigborood wen te relaxation sweep for S i,j begins. Based on tis, we notice tat, before relaxation on S i 1,j, r (k) i 1,j = f (k) ({α})e ıθ x(k) i 1,j / = f (k) ({α})e ıθ x(k) i,j / e ıθ1 = e ıθ1 r (k) i,j. As tis is true for all degrees of freedom, k, in S i,j, we can write e ıθ1 R old i,j Now consider te update equations, B ( U new i 1,j U old B ( U new i,j = R old i 1,j. (3.1) i 1,j U old i,j ) = R old i 1,j, (3.2) ) = R old i,j. (3.3) After relaxing on cell S i 1,j and before relaxing on cell S i,j, Equation (3.2) olds, giving a relationsip between te various Fourier coefficients (as given in te statement of te teorem). During relaxation over cell S i,j, Equation (3.3) is solved to update te solution at te nodes in S i,j. In Equation (3.2), we can express te difference, Ui 1,j new U i 1,j old, for eac node k in terms of te Fourier coefficients for te errors, E (k) i,j, based on te expansions given in te assumptions. For example, a degree of freedom, k i, in te interior of te cell associated wit S i 1,j, U (ki,new) i 1,j U (ki,old) i 1,j = E (ki,old) i 1,j E (ki,new) i 1,j = (α (ki) α (ki,1) )e ıθ x(k i) i 1,j /, = e ıθ1 (α (ki) α (ki,1) )e ıθ x(k i ) i,j /. (3.4) In Equation (3.3), we can express only one term in te difference, U new i,j Ui,j old, in terms of te Fourier coefficients for te errors, E i,j. If we again consider te node, k i, in te interior of te cell associated wit S i,j, we can express te error in te approximation 9

10 to U (ki) i,j before relaxation as α (ki) e ıθ x(k i) i,j /. Let te error in te approximation after relaxation be β (ki) e ıθ x(k i ) i,j /. Ten, U (ki,new) i,j U (ki,old) i,j = E (ki,old) i,j E (ki,new) i,j = (α (ki) β (ki) )e ıθ x(k i) i,j /, Substituting (3.1) and (3.4) into (3.2), te terms of e ıθ1 tat appear on bot sides of te equation for eac vector component cancel fully. We can ten rewrite (3.2) as B ( V i,j Ui,j old ) = R old i,j, were V i,j is determined by te updated Fourier coefficients, as appear in Equation (3.4). As tis system as te same system matrix and rigt-and side as (3.3), it must also ave te same solution, wic implies tat V i,j = Ui,j new (and, for example, β (ki) = α (ki) ), wic gives te result stated in te teorem. Teorem 3.1 states, in essence, tat te Fourier modes tat are eigenfunctions of any pointwise relaxation tat updates all nodes in te same pattern are also te eigenfunctions for any coupled relaxation (overlapping or not) tat partitions te degrees of freedom into self-similar collections of degrees of freedom tat are treated consistently. Tis, in turn, means tat we can attempt to analyze tese tecniques using classical multigrid smooting and two-grid Fourier analysis tools to measure te effectiveness of te resulting multigrid cycles. Te generalization of tis result to 3D is straigtforward. Analysis of te error-propagation operators in tis context was done by Sivaloganatan [40] for Vanka relaxation [43] for te standard, staggered finite-difference discretization of te Stokes Equations in two dimensions and by Molenaar for te mixed finite-element discretization of Poisson s equation using Raviart-Tomas elements [34]. Tis tecnique can be generalized to apply to any overlapping relaxation tat satisfies conditions suc as tose in Teorem 3.1: tat A is a block-diagonal matrix wit infinite-grid Toeplitz blocks (corresponding to te discretization of a twodimensional PDE on a regular grid wit messize, ), tat te relaxation subsets, S i,j are determined also by an infinite-grid wit messize, and tat te update matrix, B, is nonsingular. Under tese conditions, Equation (3.3) can be rewritten to give te transformation of te Fourier coefficients troug te relaxation sweep. Eac equation in (3.3) can be rewritten to give an equation relating te set of updated Fourier coefficients to te Fourier coefficients before te sweep (by moving te appropriate terms from te residual to te left-and side, and tose from te update equations to te rigt). Te resulting system of equations can be written as Lα new = Mα old, were L is a (4k + 2(l x + l y ) + m) (4k + 2(l x + l y ) + m) matrix, wile M is (4k + 2(l x + l y ) + m) (k + l x + l y + m) matrix. Computing L 1 M gives te error propagation operator tat maps from te error before te sweep to eac of te partially updated Fourier coefficients, as well as to te fully updated coefficients, α (kn,4), α (k,2), α (kv,2), and α (ki,1). Taking te (k + l x + l y + m) (k + l x + l y + m) submatrix tat corresponds to te rows of L 1 M associated wit te fully updated Fourier coefficients gives te error-propagation operator for relaxation as a wole. Te next section gives a detailed example of tis approac. 4. Overlapping-Scwarz relaxation for te Poisson equation. We explain te LFA for multiplicative smooters in detail for Poisson s equation wit a bilinear finite-element discretization, using an element-wise overlapping-scwarz relaxation. 10

11 For tis discrete operator, tis (somewat involved) smooter is not really necessary, as basic pointwise relaxation is sufficient. Tis smooter could, owever, be useful for te Poisson operator in a discontinuous Galerkin context. For te bilinear (Q1) discretization, a typical equation of te linear systems is 9 3 u i,j α= 1 β= 1 u i+α,j+β = f i,j. By element-wise overlapping, we mean tat te relaxation traverses te grid element by element, updating te four nodes at te corners of te element at eac step. Subset S i,j is taken to be te four nodes to te Nort and East of (i, j): S i,j = {(i, j), (i + 1, j), (i, j + 1), (i + 1, j + 1)}. Tus, before we relax on S i,j, te variables tat appear in te equations for S i,j are in te following states, gatered by te number of times tey ave been updated prior to considering S i,j : Four times: (i 1, j 1), (i, j 1), (i + 1, j 1), (i + 2, j 1), (i 1, j) Tree times: (i, j) Twice: (i + 1, j), (i + 2, j), (i 1, j + 1) Once: (i, j + 1) Not updated: (i + 1, j + 1), (i + 2, j + 1), (i 1, j + 2), (i, j + 2), (i + 1, j + 2), (i + 2, j + 2) At tis stage, we introduce te Fourier expansions for eac mode, in terms of te number of updates: e k,l = α e ıθ x k,l/, e k,l = α e ıθ x k,l/, e k,l = α e ıθ x k,l/, e k,l = α e ıθ x k,l/, e k,l = αe ıθ x k,l/, for four, tree, two, one, and no updates, respectively. We substitute tese expansions into te residual equations associated wit te four nodes before te relaxation on S i,j, were r k,l and e k,l are te residual and error, respectively, in te approximation to u k,l for eac node (k, l), ( 8 r i,j = 3 α 1 ( α e ı(θ1+θ2) + α e ıθ2 + α e ı(θ1 θ2) + α e ıθ1 + α e ıθ1 3 +α e ı( θ1+θ2) + α e ıθ2 + αe ı(θ1+θ2))) e ıθ xi,j/ ( 8 r i+1,j = 3 α 1 ( α e ı(θ1+θ2) + α e ıθ2 + α e ı(θ1 θ2) + α e ıθ1 + α e ıθ1 3 +α e ı( θ1+θ2) + αe ıθ2 + αe ı(θ1+θ2))) e ıθ xi+1,j/ ( 8 r i,j+1 = 3 α 1 ( α e ı(θ1+θ2) + α e ıθ2 + α e ı(θ1 θ2) + α e ıθ1 + αe ıθ1 3 +αe ı( θ1+θ2) + αe ıθ2 + αe ı(θ1+θ2))) e ıθ xi,j+1/ ( 8 r i+1,j+1 = 3 α 1 ( α e ı(θ1+θ2) + α e ıθ2 + α e ı(θ1 θ2) + α e ıθ1 + αe ıθ1 3 +αe ı( θ1+θ2) + αe ıθ2 + αe ı(θ1+θ2))) e ıθ xi+1,j+1/ A weigted overlapping multiplicative Scwartz relaxation sweep can be written in terms of its update equation. Substituting te appropriate Fourier expansions for 11

12 te errors before and after relaxation at te nodes in S i,j gives ω (α α )e ıθ xi,j/ ω (α α )e ıθ xi+1,j/ ω (α α )e ıθ xi,j+1/ = ω (α α )e ıθ xi+1,j+1/ r i,j r i+1,j r i,j+1 r i+1,j+1. (4.1) Now, tis system of four equations can be rearranged into a system of equations directly for te four updated Fourier coefficients, α, α, α, α. Tis is simply accomplised by expanding eac equation in terms of te Fourier expansions (using te expressions for r i,j, r i+1,j, r i,j+1, r i+1,j+1 derived above), ten collecting terms tat multiply eac of te Fourier coefficients. Te common factors of 1 3 and eıθ xk,l/ can be directly canceled to simplify te calculation. For tis example, te first equation may be rewritten as ( 8 ω + e ı(θ1+θ2) + e ıθ1 + e ıθ2 + e ı(θ1 θ2)) ( ) 8 α + α (( ) e ıθ1 + 1ω ) + e ı( θ1+θ2) α + ω eıθ2 ( ) 1 = ω 1 e ı(θ1+θ2) α, ω ω eıθ1 (( 1 1 ω ) e ıθ2 + e ı(θ1+θ2) ) α wit similar expressions resulting from te oter tree equations. Tese equations may ten be solved collectively, expressing (α, α, α, α ) T = L 1 Mα, were L is a four-by-four matrix and M is a four-by-one matrix. Te first entry in (te vector) L 1 M is te amplification factor for te complete sweep, mapping te initial error coefficient for te Fourier mode given by θ into tat after a sweep of te element-wise overlapping multiplicative Scwarz relaxation. Based on tese amplification factors, we can ten perform classical MG smooting analysis, as in [9,41], for te overlapping smooters. Figure 4.1 sows te amplification factors as a function of te Fourier angles, θ, for bot pointwise Gauss-Seidel (left) and element-wise overlapping multiplicative Scwarz relaxation (rigt). Computing te smooting factors, µ = max θ [ π 2, 3π 2 ] 2 \[ π 2, π 2 2 ] µ(θ), were µ(θ) is te amplification factor for relaxation for a given Fourier mode, θ, for tese two approaces, we see tat, for pointwise Gauss-Seidel, µ = 0.43, wile for te overlapping relaxation, µ = 0.24, or tat one sweep of te overlapping relaxation reduces ig-frequency errors about te same amount as 1.7 sweeps of pointwise relaxation. Furtermore, we can combine tis smooting analysis wit te well-known LFA two-grid analysis for scalar PDEs [41, 42] of te coarse-grid correction for tis system, using bilinear interpolation and full-weigting restriction, coupled wit a Galerkin coarse-grid operator. Te largest-magnitude eigenvalue predicted by te two-grid LFA for pointwise relaxation is 0.073, wile it is for te overlapping Scwarz relaxation in a (1,1)-cycle. One cycle of multigrid wit te overlapping relaxation brings about te same total reduction in error as 1.4 cycles using pointwise relaxation. Tus, te overlapping relaxation yields a better solver, but te extra cost of te overlapping relaxation likely doesn t pay off (unless it can be implemented very efficiently). As a comparison, we consider te true performance of multigrid V(1,1) and W(1,1) cycles using bot pointwise Gauss-Seidel and element-wise overlapping multiplicative Scwarz smooters, sown in Table 4.1. Here, we see tat te two-grid LFA accurately predicts te W-cycle multigrid convergence rates for bot smooters, 12

13 Fig Amplification factors for pointwise Gauss-Seidel relaxation (at left) and element-wise overlapping multiplicative Scwarz (at rigt) for a Q1 discretization of te Poisson equation on a mes wit = 1, as a function of te Fourier mode, θ. 128 Pointwise Gauss-Seidel Overlapping Scwarz grid V(1,1) W(1,1) V(1,1) W(1,1) Table 4.1 Average convergence factor over 50 iterations for multigrid cycles based on pointwise and overlapping relaxation scemes. but is a noticeable underestimate for te V-cycle convergence rates. Tis is typical of LFA, because te two-grid analysis is based on exact solution of te first coarse-grid problem; a multigrid W-cycle, were tis level is visited twice per iteration is a muc better approximation of tis tan a V-cycle, wic uses muc less relaxation. 5. Two-grid local Fourier analysis. We ere discuss te basics of two-grid LFA in order to deal wit systems of PDEs on staggered FEM grids. We focus on two-grid analysis, but multilevel analysis is also possible using inductive arguments. In general, two-grid metods can be represented by error-propagation operators wit form (T G) E = ( I M 1 A ) ν2 ( I P H B 1 ) ( H R HA I M 1 A ν1 ), (5.1) were H denotes te mes size of te coarse scale, RH is te restriction operator from grid to grid H, P H is te interpolation operator from grid H to grid, and B H represents some discretization on te coarse scale. Writing te eigenvector matrix for te error-propagation operator associated wit relaxation as Φ = [ φ (1), φ (2),..., φ (N)], we know tat I M 1 A is diagonalized by a similarity transformation wit Φ, (Φ ) 1 ( I M 1 A ) Φ = Λ, were Λ is te diagonal matrix of eigenvalues of relaxation, Λ ii = λ i. We can diagonalize E (T G), also using Φ in a similarity transformation. Taking Φ H to be te matrix of eigenvectors of B H, we can ten write Φ 1 E(T G) Φ = Λ ν2 ( I ( Φ 1 P H Φ H ) Γ 1 ( Φ 1 H R HΦ ) ( Φ 1 A Φ )) Λ ν 1, (5.2) 13

14 (I, J + 1) (I + 1, J + 1) (i 3, j 3 ) (i 4, j 4 ) (i 1, j 1 ) (i 2, j 2 ) (I, J) (I + 1, J) Fig Nesting of fine-grid nodes relative to te coarse grid for nested (collocated) grids. were Γ = Φ 1 H B HΦ H is also a diagonal operator. Under te LFA assumptions for smooting, tat A and M are infinite-grid Toeplitz matrices, Φ 1 A Φ is also a diagonal matrix. If we additionally take B H to correspond to te discretization of a PDE on an infinite grid wit fixed mes-size, H, wit a stencil tat does not vary wit position, ten te difficulty in analyzing (5.2) comes from te intergrid-transfer operators, Φ 1 H R H Φ and Φ 1 P HΦ H. Te transformation of RH and P H in terms of te coarse-grid and fine-grid Fourier matrices, Φ H and Φ, depends on te relationsip between te two mes sizes, H and. Taking H = 2, as is commonly te case in geometric multigrid, ten a constant-stencil restriction operator, RH, for a two-dimensional mes maps four fine-grid frequencies into one coarse-grid function. Tese four functions, known as te Fourier armonics, are associated wit some base index, θ 0,0 ( π 2, ] 2, π 2 and tree more-oscillatory modes, associated wit frequencies θ 1,0 = θ 0,0 + ( π 0 ), θ 0,1 = θ 0,0 + ( 0 π ) ( π, and θ 1,1 = θ 0,0 + π Te action of a constant-coefficient restriction operator, R2, on a fine-grid residual, r, can be written in terms of a set of restriction weigts, {w k,l }, as ( R 2 r ) I,J = k,l ). w k,l r i+k,j+l, (5.3) were (i, j) is te fine-grid index corresponding to coarse-grid index (I, J), so (i, j) = (2I, 2J). Similarly, we can write te action of a constant-coefficient interpolation operator, P 2, on a coarse-grid correction, e 2, in terms of several sets of interpolation weigts, {w (m) K,L }, ( P 2 ) e 2 = w (m) i,j K,L e I+K,J+L, (5.4) K,L were te additional superscript, m, is used to denote te position of node (i, j) relative to te coarse node, (I, J), see Figure 5.1. Notice tat tese weigts are independent of te absolute location of te underlying nodes, but do depend on te staggering of te nodes relative to te coarse grid, (i 2I, j 2J). Te set of modes, φ 2 (x, 2θ 0,0 ) for θ 0,0 ( π 2, ] 2, π 2 is a complete set of Fourier modes on te coarse grid and tus, by assumption, diagonalize te Toeplitz operator, B 2. As suc, te spaces of armonic frequencies become invariant subspaces for te 14

15 (η 1, η 2 ) = ( 1 3 1, 2 3 ) (δ 1, δ 2 ) = (0, 1 2 ) Fig Nesting of fine-grid nodes relative to te coarse grid for non-nested grids, wit (δ 1, δ 2 ) = (0, 1 2 ) and (η 1, η 2 ) = ( 1 3, 2 3 ). coarse-grid correction process and for te two-grid cycle as a wole. From te similarity transformation representation given in Equation (5.2), we can ten compute te eigenvalues of te two-grid error-propagation operator by computing te eigenvalues of Φ 1 G) E(T Φ. Tis matrix is easily permuted into block-diagonal form wit, at most, 4 4 blocks corresponding to te spaces of armonic modes LFA for Systems. For systems of PDEs, te LFA analysis does not diagonalize A troug te Fourier-mode similarity transformation but, rater, transforms te m m block matrix, A, into a matrix tat can be permuted into a block-diagonal form wit dense m m blocks, by diagonalizing eac block witin A. Te block coupling of te operator resulting from te full similarity transformation will ave larger dense blocks tan 4 4, as we must also account for te m m coupling witin A (and relaxation) tat arises because of te systems form of A. Tus, LFA for systems in 2D results in coupled 4m 4m blocks of armonics (wit 4 armonics for eac of te m scalar unknowns of te PDE) for eac base frequency, θ 0,0 ( π 2, π 2 ] 2. Aside from tis difference, te analysis below proceeds in te same manner as in scalar case Grid transfers for staggered systems. An added callenge in te multigrid treatment of a system of PDEs is tat eac different variable type may be staggered in its own way. If te different variable types do not interact in interpolation and restriction (so tat eac variable type only restricts to and interpolates from a coarse-grid variable wit te same staggering pattern), ten te LFA for interpolation for te wole system can be done treating eac variable type, in turn, as a scalar problem. If, on te oter and, tere is a need for inter-variable interpolation or restriction in te treatment of te system, we must modify te metod for te scalar case to account for te different staggering on te two grid levels. As in Figure 5.2, we now take (δ 1, δ 2 ) to describe te staggering of a fine-level variable, r (eiter to be restricted from or interpolated to), and (η 1, η 2 ) to describe te staggering of a coarse-level variable, e 2. Lemma 5.1. Let θ 0,0 ( π 2, π 2 ] 2 and (I, J) be a coarse-grid node index, identified wit fine-grid node index (i, j) = (2I, 2J). Ten, any constant-coefficient restriction operator, as defined by Equation (5.3), maps te four Fourier armonic modes, φ (x, θ 0,0 ), φ (x, θ 1,0 ), φ (x, θ 0,1 ), and φ (x, θ 1,1 ), on te grid wit sift (δ 1, δ 2 ) into te single coarse-grid mode, φ 2 (x, 2θ 0,0 ), on te grid wit sift (η 1, η 2 ). Proof. Considering a fine-grid residual, r, tat is a linear combination of te 15

16 four armonic frequencies, r i,j = c 0,0 φ (((i + δ 1 ), (j + δ 2 )), θ 0,0 ) + c 1,0 φ (((i + δ 1 ), (j + δ 2 )), θ 1,0 ) + c 0,1 φ (((i + δ 1 ), (j + δ 2 )), θ 0,1 ) + c 1,1 φ (((i + δ 1 ), (j + δ 2 )), θ 1,1 ) =c 0,0 e ıθ0,0 (i+δ1,j+δ2) + c 1,0 e ıθ1,0 (i+δ1,j+δ2) + c 0,1 e ıθ0,1 (i+δ1,j+δ2) + c 1,1 e ıθ1,1 (i+δ1,j+δ2) ( = c 0,0 + c 1,0 e ıπi e ıπδ1 + c 0,1 e ıπj e ıπδ2 + c 1,1 e ıπ(i+j) e ıπ(δ1+δ2)) e ıθ0,0 (δ1,δ2) e ıθ0,0 (i,j). As te nodes are numbered based on teir cell, node (I, J) on grid 2 can be identified wit node (i, j) on grid for i = 2I and j = 2J. We seek to represent te restricted residual on te coarse grid, R2 r, as a multiple of te coarse-grid armonic function, φ 2 ((I +η 1 )(2), (J +η 2 )(2), 2θ 0,0 ) = e ı2θ0,0 (I+η1,J+η2). Tus, we can write te restriction of r to a differently staggered coarse-grid variable in cell (I, J), were (i, j) = (2I, 2J), as ( ) R 2 r = w I,J k,l r i+k,j+l k,l = e ı2θ0,0 (I+η1,J+η2) w k,l e ıθ0,0 (k,l) e ıθ0,0 (δ1 2η1,δ2 2η2) k,l (c 0,0 + c 1,0 e ıπk e ıπδ1 + c 0,1 e ıπl e ıπδ2 + c 1,1 e ıπ(k+l) e ıπ(δ1+δ2))] Lemma 5.2. Let θ 0,0 ( π 2, ] π 2 2 and (I, J) be a coarse-grid node index. Let (i 1, j 1 ), (i 2, j 2 ), (i 3, j 3 ), and (i 4, j 4 ) be four fine-grid node indices, as identified in Figure 5.2. Ten, any constant-coefficient interpolation operator on te sifted grid maps coarse-grid mode φ 2 (x, 2θ 0,0 ) into te four fine-grid armonics, φ (x, θ 0,0 ), φ (x, θ 1,0 ), φ (x, θ 0,1 ), and φ (x, θ 1,1 ). Proof. Te Fourier analysis for interpolation can be derived by similarly accounting for te different staggering of a coarse-grid variable, e 2, staggered on grid 2 wit (η 1, η 2 ), and its fine-grid interpolant, staggered on grid wit (δ 1, δ 2 ). Coosing e I,J = φ 2 (((I + η 1 )(2), (J + η 2 )(2)), 2θ 0,0 ), we get (taking (i 1, j 1 ) = (2I, 2J)) ( P 2 e 2 )i 1,j 1 = K,L w (1) K,L eı2θ0,0 (K,L) e ıθ0,0 (2η1 δ1,2η2 δ2) e ıθ0,0 (i1+δ1,j1+δ2). (5.5) Similarly, we can derive te staggered interpolation relations for te oter 3 node points in coarse-grid cell (I, J) as ( P 2 e 2 )i 2,j 2 = w (2) e ıθ0,0 (i2+δ1,j2+δ2), K,L ( P 2 e 2 )i 3,j 3 = ( P 2 e 2 )i 4,j 4 = K,L w (3) K,L w (4) K,L eıθ0,0 (2K 1,2L) e ıθ0,0 (2η1 δ1,2η2 δ2) K,L eıθ0,0 (2K,2L 1) e ıθ0,0 (2η1 δ1,2η2 δ2) K,L eıθ0,0 (2K 1,2L 1) e ıθ0,0 (2η1 δ1,2η2 δ2) e ıθ0,0 (i3+δ1,j3+δ2), e ıθ0,0 (i4+δ1,j4+δ2). 16

17 Making te ansatz tat P 2e 2 can be written as a linear combination of te four Fourier armonics, now on te sifted grid, we ave ( ) P 2 e 2 = c i,j 0,0e ıθ0,0 (i+δ1,j+δ2) + c 1,0 e ıθ1,0 (i+δ1,j+δ2) + c 0,1 e ıθ0,1 (i+δ1,j+δ2) + c 1,1 e ıθ1,1 (i+δ1,j+δ2) ( = c 0,0 + c 1,0 e ıπ(i+δ1) + c 0,1 e ıπ(j+δ2) + c 1,1 e ıπ(i+j+δ1+δ2)) e ıθ0,0 (i+δ1,j+δ2). Equating terms for (i 1, j 1 ) = (2I, 2J) gives c 0,0 + c 1,0 e ıπδ1 + c 0,1 e ıπδ2 + c 1,1 e ıπ(δ1+δ2) = c 1 (θ 0,0 ), for c 1 (θ 0,0 ) given by te expression in Equation (5.5). Wit similar equations for te oter interpolation nodes, we ave c 0, c 1,0 e ıπδ1 c 0,1 e ıπδ2 = ˆε , c 1,1 e ıπ(δ1+δ2) wit ˆε = e ıθ0,0 (2η1 δ1,2η2 δ2) /4. K,L w(1) K,L eı2θ0,0 (K,L) K,L w(2) K,L eıθ0,0 (2K 1,2L) K,L w(3) K,L eıθ0,0 (2K,2L 1) K,L w(4) K,L eıθ0,0 (2K 1,2L 1) 6. Numerical Examples. In tis section, we give several smooting and twogrid LFA estimates for systems of PDEs witin te framework of multiplicative collective smooters of Vanka-type. Tese smooters, wic explicitly deal wit te large nullspaces tat appear, are te only collective smooters tat give satisfactory performance for te operators of interest. We do not investigate te impact of te ordering of te cells in te present paper and stay wit a lexicograpical ordering of te relaxation subsets. To take LFA from its infinite-grid setting and get a predictive analysis tool, we need to introduce a second discretization into te analysis, going from a continuous parameter, θ 0,0, to a discrete mes in θ 0,0, upon wic a convergence prediction can be made. Tus, te results presented ere ave two step-size parameters:, te spatial grid size, wic is directly reflected in te coefficients of te discrete operators to wic we apply LFA, and θ, te mes size for te discrete lattice of θ used to make a quantitative prediction based on LFA. A discrete set of te 4 4 Fourier blocks is ten analyzed, corresponding to a discrete coice of angles, θ 0,0 ( π 2, ] 2, π 2 given by a tensor product of an equally spaced mes over te interval of lengt π wit itself, wit mes-spacing θ. Because te infinite-grid fine-grid and coarse-grid operators, A and B 2, are often singular, wit te constant functions, φ (x, (0, 0)) and φ 2 (x, (0, 0)), in teir nullspaces, we coose te mes in θ 0,0 so tat θ 0,0 = (0, 0) does not appear. A prediction of te performance of te multigrid algoritm is made by measuring te largest eigenvalue of te transformed operators over tis discrete space. All of te numbers quoted ere result from tis process, for = 1 64 and θ = π 32. Te impact of finer meses in eiter space or Fourier frequency was negligible in te examples considered ere Te grad-div and te curl-curl operators. We first consider te discretization of te gradient-divergence and te curl-curl equations, (2.1) and (2.2), respectively. As te stencils, multigrid metods, and smooters, as well as teir LFA 17

18 performance estimates are very similar for te two equations, we discuss tem in one section. For te grad-div equation (2.1), we use first-order Raviart-Tomas (face) elements for te vector field U = (u, v) T, as already discussed in Section 2.2. Te discrete degrees of freedom for tis discretization are te values of u at te midpoints of mes edges tat are parallel to te y-axis, and te values of v at te midpoint of mes edges tat are parallel to te x-axis. Te resulting stencil, for u, reads, Te contribution of te mass matrix is added, in te form of a one-dimensional addition [ 2 /6 2 2 /3 2 /6], wit te central stencil element incremented by 2 2 /3. Te resulting, rotated, stencil for te v components is similar. As mentioned earlier, we coose te Nédélec edge elements [35] to discretize te weak curl-curl operator in (2.2), for unknown U = (u, v) T. Te resulting stencil for te u-component is wile, for v, we find an identical, but rotated, stencil. It is clear tat te resulting stencils for te grad-div and curl-curl operators are identical, but rotated. So, we focus on te discussion of te smooter for grad-div, as te one for curl-curl is similar and produces identical LFA results. Te overlapping smooter, in particular for te grad-div operator, was proposed in [1,2], were te degrees of freedom along te faces (edges) adjacent to a node were cosen to be relaxed simultaneously, see Figure 2.2. We refer to tis as a node-wise smooting procedure. We coose S i,j = {u i,j 1 2, u i,j+ 1 2, v i 1 2,j, v i+ 1 2,j }, and introduce in te local system for smooting te Fourier expansions for te errors in u and v, before relaxation, α u e iθ x/, α v e iθ x/, after te first correction, α ue iθ x/, α ve iθ x/, and after te second correction, α ue iθ x/, α ve iθ x/ as in Section 3. We can ten write te update equations in terms of te Fourier coefficients: λ 0 λ λ λ λ λ 2 λ λ λ 0 2 λ λ λ, δu i,j+ 1 2 δu i,j 1 2 δv i+ 1 2,j δv i 1 2,j = r u i,j+ 1 2 r u i,j 1 2 r v i+ 1 2,j r v i 1 2,j and convert tis system into a system for α u, α v, α u, α v in terms of α u and α v, as explained for Equation (4.1). In te numerical LFA smooting and two-grid experiments ere, we vary te transfer operators in te algoritms. We compare te usual six-point restriction operators, based on te fine grid residuals at te six nearest fine grid locations of te corresponding unknown, wit te two-point restriction operator. Tese operators are 18,