An Efficient Multigrid Solver based on Distributive Smoothing

for Poroelasticity Equations

R. Wienands, Koln, F.J. Gaspar, Zaragoza,F.J. Lisbona, Zaragoza, and C.W. Oosterlee, Delft

Received February 20, 2004; revised April 13, 2004Published online: June 21, 2004 2004

� Springer-Verlag 2004

Abstract

In this paper, we present a robust distributive smoother in a multigrid method for the system ofporoelasticity equations. Within the distributive framework, we deal with a decoupled system, that canbe smoothed with basic iterative methods like an equation-wise red-black Jacobi point relaxation. Theproperties of the distributive relaxation are optimized with the help of Fourier smoothing analysis. Ahighly efficient multigrid method results, as is confirmed by Fourier two-grid analysis and numericalexperiments.

AMS Subject Classification: 65N55, 74F10, 74S10, 65M12.

Keywords: Poroelasticity, staggered discretization, multigrid, distributive relaxation, local Fourieranalysis.

1. Introduction

Poroelasticity theory addresses the time dependent coupling between the defor-mation of porous material and the fluid flow inside. The porous matrix is sup-posed to be saturated by the fluid phase. The state of this continuous medium ischaracterized by the knowledge of elastic displacements and fluid pressure at eachpoint. A phenomenological model for a rather general situation was first pro-posed and analyzed by Biot [1], studying the consolidation of soils. Poroelasticmodels are used nowadays to study problems in geomechanics, hydrogeology,petrol engineering and biomechanics [9,4].

In this paper, we present an efficient multigrid method for the system of poro-elasticity equations. In particular, we introduce a robust point-wise smoothingmethod based on distributive iteration. In distributive smoothing the originalsystem of equations is transformed by post-conditioning in order to achievefavorable properties, such as a decoupling of the equations and/or possibilities forpoint-wise smoothing. A specialty lies in the discretization approach employed.We adopt a staggered grid for the poroelasticity equations as in [5,6]. A popularalternative is to use finite elements, see, for example, [10] for the quasi-static

Computing (2004)Digital Object Identifier (DOI) 10.1007/s00607-004-0078-y

problem, or [12] for the dynamic problem. Standard finite differences do not leadto stable solutions without additional stabilization. Throughout this paper weconcentrate on Cartesian equidistant grids.

The multigrid method is developed on the basis of Fourier analysis of increasingcomplexity [2], [14]. The h-ellipticity concept is discussed, which is fundamentalfor the existence of point-wise smoothers. The distributive smoother is developedbased on insights from the Stokes and incompressible Navier-Stokes equa-tions [2], [3], [14], [18]. Optimal relaxation parameters are obtained withsmoothing analysis, leading to a relaxation method, that is robust w.r.t. theproblem parameters like Lame coefficients, permeability of the porous medium,viscosity of the fluid, and time step and grid size. Furthermore, the multigridmethod is analyzed by Fourier two-grid analysis [2], [13], [14] demonstrating anefficient interplay between relaxation and coarse grid correction.

The outline of this paper is as follows. The model and discretization are describedin Sect. 2. In Sect. 3, the separate components of the multigrid solution methodare presented and analyzed in different subsections; in Sect. 3.2 the h-ellipticitymeasure of the discretization, in Sects. 3.3 and 3.4 the relaxation method, and inSect. 3.5 the coarse grid correction. Numerical multigrid results are presented inSect. 4, confirming the theoretical considerations.

2. Mathematical Model and Discretization

2.1. Continuous System

The poroelastic model can be formulated as a system of partial differentialequations for displacements and the pressure of the fluid. One assumes thematerial’s solid structure to be linearly elastic, initially homogeneous and iso-tropic, the strains imposed within the material are small. We denote byu ¼ ðu; v; pÞT the solution vector, consisting of the displacement vector u ¼ ðu; vÞTand pore pressure of the fluid p. The incompressible, two-dimensional variant ofBiot’s consolidation model reads

�ðkþ 2lÞuxx � luyy � ðkþ lÞvxy þ px ¼ 0;

�ðkþ lÞuxy � lvxx � ðkþ 2lÞvyy þ py ¼ 0; ð1Þ

ux þ vy� �

t�a pxx þ pyy� �

¼Q

(plus initial and boundary conditions) with k; lð� 0Þ the Lame coefficients,a ¼ j=g � 0 with j the permeability of the porous medium and g the viscosity ofthe fluid, and Q the source (representing an injection or extraction process),see [1]. Problem (1) is a limit of the compressible case. The compressible systemwill be easier to solve, however, due to an extra contribution to the main diagonalof the matrix related to this system. We concentrate on a solver for thetwo-dimensional incompressible case, and focus on a model operator L which issuitable for analysis. It reads

R. Wienands et al.

L ¼

�ðkþ 2lÞ@xx � l@yy �ðkþ lÞ@xy @x

�ðkþ lÞ@xy �l@xx � ðkþ 2lÞ@yy @y

@x @y �ea @xx þ @yy� �

0

BB@

1

CCA: ð2Þ

L can be interpreted as a ‘‘stationary variant’’ of (1), i.e., the operator after animplicit (semi-) discretization in time. For example, in case of Crank-Nicholsontime discretization we have ea ¼ 0:5adt. From (2), one may calculate the corre-sponding determinant:

det Lð Þ ¼ �lD eaðkþ 2lÞD2 � D� �

with Laplace operator D and biharmonic operator D2. The principal part ofdet Lð Þ is Dm with m depending on the choice of k, l, and ea. Due to physicalreasons, we always have l, ea, kþ 2l > 0, yielding m ¼ 3. The number ofboundary conditions that must accompany L is m [2,14].

A dimensionless version of (1) can be obtained with dimensionless parameters:

bl ¼ 1þ ðk=lÞ ð¼ 1=ð1� 2mÞ; with Poisson ratio mÞ; ð3Þ

x ¼ x=‘; y ¼ y=‘; t ¼ ðkþ 2lÞat=‘2, Q ¼ ‘2Q=ðaðkþ 2lÞÞ, and unknowns u ¼u=‘; � v ¼ v=‘;p ¼ p=ðkþ 2lÞ. Here, scaling has taken place with respect to acharacteristic length of the porous medium ‘, the Lame constant kþ 2l, time scalet0, and a in (1).

2.2. Discrete System

The time-dependent operator (2) suffers from stability difficulties. The coefficientin the L3;3-block in (2) is typically, depending on the time step, extremely small. Inorder to avoid oscillating solutions, the discretization has to be designed withcare. To overcome the stability difficulties in finite differences, a staggered gridwas proposed in [5], [6]. We adopt this methodology for system (1), using centraldifferences on a uniform staggered grid with mesh size h. Staggering is, of course,a well-known discretization technique in computational fluid dynamics, in par-ticular for incompressible flow [8], [16].

Often in poroelasticity problems pressure values are prescribed at the physicalboundary. So, pressure points in the staggered grid should be located at thephysical boundary, and the displacement points are then defined at the cellfaces. Therefore, a divergence operator is naturally approximated by a centraldiscretization of the displacements in a cell, see Fig. 1. Notice that the stag-gered placement of unknowns here is different from incompressible Navier-Stokes, because of the pressure placement. The two-dimensional (infinite)staggered grid employed is composed of three types of grid points,Gh ¼ G1

h [ G2h [ G3

h, where

An Efficient Multigrid Solver based on Distributive Smoothing

Gjh :¼ x

jh ¼ xj

h; yjh

� �:¼ kx; ky� �

hþ sj; kx; ky� �

2 Z2� �

; ð4Þ

with

u� grid points x1h 2 G1h with s1 ¼ ðh=2; 0Þ;

v� grid points x2h 2 G2h with s2 ¼ ð0; h=2Þ;

p � grid points x3h 2 G3h with s3 ¼ ð0; 0Þ

and a uniform mesh size h, see Fig. 1.

The discrete system

Lhuh ¼L1;1

h L1;2h L1;3

h

L2;1h L2;2

h L2;3h

L3;1h L3;2

h L3;3h

0

B@

1

CAuhðx1hÞvhðx2hÞphðx3hÞ

0

B@

1

CA ¼ fh: ð5Þ

based on (2) reads at u-grid points:

�ðkþ 2lÞð@xxÞhuh � lð@yyÞhuh � ðkþ lÞð@xyÞh=2vh þ ð@xÞh=2ph ¼ 0; ð6Þ

at v-grid points:

�ðkþ lÞð@xyÞh=2uh � lð@xxÞhvh � ðkþ 2lÞð@yyÞhvh þ ð@yÞh=2ph ¼ 0; ð7Þ

and at p-grid points:

ð@xÞh=2uh þ ð@yÞh=2vh � eað@xxÞhph � eað@yyÞhph ¼ Q: ð8Þ

Here, the following discrete operators on the staggered grid (4) are used (given instencil notation):

Fig. 1. Staggered location of unknowns for poroelasticity

R. Wienands et al.

ð@xÞh=2 ¼^ 1

h�1 ? 1½ �h=2; �ð@xxÞh ¼

^ 1

h2�1 2 �1½ �h;

ð@xyÞh=2 ¼^ 1

h2

�1 1

?

1 �1

2

664

3

775

h=2

:

The ‘‘?’’ denotes the position on the shifted grids G1h and G2

h at which the stencil isapplied, compare with Fig. 1. ð@yÞh=2 and �ð@yyÞh are given by analogous stencils.

We choose the Crank-Nicolson discretization in time direction, with Oðdt2Þaccuracy. Second-order accuracy has been obtained for reference problems withsmooth solutions (not shown here).

3. Multigrid Solution Method

In this context, an efficient solver for the system of poroelasticity equationsdiscretized on staggered grids is necessary. Multigrid methods (see, for exam-ple, [2,7,14]) are motivated by two basic observations: Firstly many iterativemethods have a strong error smoothing effect if they are applied to discreteelliptic problems Lhuh ¼ fh. Secondly, a smooth error term can be well repre-sented on a coarser grid where its approximation is substantially less expensive.These observations suggest the following structure of a two-grid cycle for alinear problem, called the correction scheme: Perform n1 steps of an iterativerelaxation method Sh on the fine grid (pre-smoothing), compute the defect ofthe current fine grid approximation, restrict the defect to the coarse grid usinga restriction operator Rh;2h, solve the coarse grid defect equation, interpolatethe correction using a prolongation operator P2h;h to the fine grid, add theinterpolated correction to the current fine grid approximation (coarse gridcorrection), perform n2 steps of an iterative relaxation method on the fine grid(post-smoothing). Hence, the two-grid error transformation operator is givenby

Mh;2h :¼ Sn2h Ih � P2h;h L2hð Þ�1Rh;2hLh

� �Sn1

h ¼ Sn2h Ch;2hSn1

h ; ð9Þ

where Ih denotes the identity and Ch;2h is called the coarse grid correction oper-ator. Instead of inverting L2h, the coarse grid equation can be solved by arecursive application of this procedure, yielding a multigrid method. We assumestandard coarsening here, i.e., the sequence of coarse grids is obtained byrepeatedly doubling the mesh size in each space direction. This is indicated by thesubscript ‘‘2h’’.

The crucial point for any multigrid method is to identify the multigrid compo-nents yielding an efficient interplay between relaxation and coarse grid correction.A useful tool for a proper selection is local Fourier analysis.

An Efficient Multigrid Solver based on Distributive Smoothing

3.1. Basic Elements of Local Fourier Analysis for Multigrid

Classical Fourier analysis [2], [13], [14] is often applied to develop efficient mul-tigrid methods for linear elliptic equations with constant (or frozen) coefficients. Itis based on the simplification that boundary conditions are neglected and alloccurring operators are extended to an infinite grid. On an infinite grid, thediscrete solution, its current approximation and the corresponding error orresidual can be represented by linear combinations of certain exponential func-tions - the Fourier components - which form a unitary basis of the space ofbounded infinite grid functions. On the staggered grid Gh under consideration, aunitary basis of vector-valued Fourier components is given by

uh h; xhð Þ :¼exp ih � x1h=h� �

exp ih � x2h=h� �

exp ih � x3h=h� �

0

B@

1

CAwith h 2 H :¼ ð�p; p�2;

xh :¼ ðx1h; x2h; x3hÞ; xjh 2 Gj

h; ðj ¼ 1; 2; 3Þ

and complex unit i ¼ffiffiffiffiffiffiffi�1p

yielding the Fourier space

F Ghð Þ :¼ span uh h; xhð Þ : h 2 Hf g:

(For scalar equations defined, for example, on G3h, the corresponding Fourier

components read uh h; x3h� �

:¼ exp ih � x3h=h� �

.) Then, the main idea of local Fou-rier analysis is to analyze different multigrid components by evaluating their effecton the Fourier components.

If standard coarsening in two dimensions is selected, each ‘‘low-frequency’’

h ¼ h00 2 H2hlow :¼ ð�p=2; p=2�2

is coupled with three ‘‘high-frequencies’’

h11 :¼ h00 � sign h1ð Þ; sign h2ð Þð Þp; h10 :¼ h00 � sign h1ð Þ; 0ð Þp;

h01 :¼ h00 � 0; sign h2ð Þð Þp h11; h10; h01 2 H2hhigh :¼ H nH2h

low

� �

in the transition from Gh to G2h. That is, the related three high-frequencycomponents are not visible on the coarse grid G2h as they coincide withthe coupled low-frequency component. Now, the Fourier space can be subdi-vided into the corresponding four-dimensional subspaces, known as 2h-har-monics:

F2hðhÞ :¼ span uh h00; xh� �

;uh h11; xh� �

;uh h10; xh� �

;uh h01; xh� �� �

ð10Þ

with h ¼ h00 2 H2hlow.

R. Wienands et al.

3.2. Measure of h-Ellipticity for the Fine Grid Discretization Lh

The h-ellipticity measure is often used to decide whether or not a certain dis-cretization is appropriate for a multigrid treatment. A ‘‘sufficient’’ amount ofh-ellipticity (some form of ‘‘ellipticity’’ in the discretization) indicates that point-wise error smoothing procedures can be constructed [2], [3], [14]. The measure ofh-ellipticity for the ð3� 3Þ-system of equations is defined by

EhðLhÞ :¼min det ~LhðhÞ

� ��� �� : h 2 H2hhigh

n o

max det ~LhðhÞ� ��� �� : h 2 H

� � ;

where the complex ð3� 3Þ-matrix ~LhðhÞ is the Fourier symbol of Lh, i.e.,

Lhuhðh; xhÞ ¼ ~LhðhÞuhðh; xhÞ:

The determinant of the discrete version of (2) is given by

det Lhð Þ ¼ �lDh ~aðkþ 2lÞD2h � Dh

� �;

where the discrete Laplacian and the discrete biharmonic operator are representedby the following stencils

� Dh¼^1

h2

�1�1 4 �1

�1

2

4

3

5

h

; D2h¼^ 1

h4

1

2 �8 2

1 �8 20 �8 1

2 �8 2

1

2

666664

3

777775

h

: ð11Þ

Theorem 1. The measure of h-ellipticity for the discrete system of poroelasticityequations ((6), (7), (8)) is given by

EhðLhÞ ¼2eaðkþ 2lÞ þ h2

128eaðkþ 2lÞ þ 16h2:

Proof: The Fourier symbols of the discrete scalar operators (which are analo-gously defined as for systems above, see [2], [14] for details) read,

e@x

� �

h=2ðhÞ ¼ 1s1; � f@xx

� �

hðhÞ ¼ s21; �eDhðhÞ ¼ s21 þ s22 ð12Þ

f@xy

� �

h=2ðhÞ ¼ � s1s2; eD2

hðhÞ ¼ ðs21 þ s22Þ2; ð13Þ

where s1 :¼ 2h sin h1=2ð Þ and s2 :¼ 2

h sin h2=2ð Þ. (The operators in the y-direction gosimilarly.) The Fourier symbol of the system and its determinant read

An Efficient Multigrid Solver based on Distributive Smoothing

eLhðhÞ ¼

ðkþ 2lÞs21 þ ls22 �ðkþ lÞs1s2 is1

�ðkþ lÞs1s2 ls21 þ ðkþ 2lÞs22 is2

is1 is2 ea s21 þ s22� �

0

BB@

1

CCA;

det eLhðhÞ� �

¼ l s21 þ s22� �

eaðkþ 2lÞ s21 þ s22� �2þs21 þ s22

� �: ð14Þ

Due to k, l, ea � 0 and the definition of s1 and s2 it follows from (14) that

maxh2H det eLhðhÞ� �n o

is obtained at hmax ¼ ðp; pÞ leading to

det eLhðhmaxÞ� �

¼ 64

h6l 8eaðkþ 2lÞ þ h2� �

: ð15Þ

Similarly, minh2Hhighdet eLhðhÞ� �n o

is obtained at hmin ¼ ðp=2; 0Þ, ð0; p=2Þ yielding

det eLhðhminÞ� �

¼ 4

h6l 2eaðkþ 2lÞ þ h2� �

: ð16Þ

Combining (15) and (16) concludes the proof. (

Thus, EhðLhÞ is uniformly bounded away from zero for all reasonable combina-tions of k, l, ea � 0 and h > 0. As a consequence, it should be possible to findefficient point-wise smoothers within a multigrid method. This may be surprising,because L1;1

h and L2;2h from (5) may contain grid anisotropies depending on the

choice of the Lame coefficients. That is, the size of the coefficients referring to thedifferent spatial directions (i.e., �ðkþ 2lÞ and �l) may vary considerably.Apparently, the smoothing properties of a proper point relaxation scheme for thesystem are not affected by these scalar grid anisotropies. For a vanishing mesh sizeone obtains

limh!0

EhðLhÞ ¼1

64> 0

implying that the above considerations are valid in the limit of small mesh size aswell.

3.3. Distributive Relaxation Sh

We construct a distributive relaxation for the discrete system Lh. In order to relaxLhuh ¼ fh, we introduce a new variable wh by uh ¼ Chwh and consider the trans-formed system LhChwh ¼ fh. Ideally (compare with [2]), Ch is chosen such that theresulting system LhCh is triangular and the diagonal elements of LhCh are com-posed of det Lhð Þ. Then, the resulting transformed system is suited for decoupledsmoothing, i.e., each equation can be treated separately. The new contributionhere is the following choice for the distributor

R. Wienands et al.

Ch ¼

Ih 0 � @xð Þh=20 Ih � @y

� �h=2

ðkþ lÞ @xð Þh=2 ðkþ lÞ @y� �

h=2 �ðkþ 2lÞDh

0

BBB@

1

CCCAð17Þ

with identity Ih. Then, the transformed system for the interior points (Remarks 1and 2 refer to the boundaries) reads

LhCh ¼

�lDh 0 0

0 �lDh 0

LC3;1h LC3;2

h eaðkþ 2lÞD2h � Dh

0

BB@

1

CCA with ð18Þ

LC3;1h ¼ @xð Þh=2�eaðkþ lÞ @xxxð Þh=2þ @xyy

� �h=2

� �and ð19Þ

LC3;2h ¼ @y

� �h=2�eaðkþ lÞ @xxy

� �h=2þ @yyy

� �h=2

� �; ð20Þ

where the central discrete operators read in stencil notation

@xð Þh=2¼^ 1

h�1 ? 1½ �h=2; @xxxð Þh=2¼

^ 1

h3�1 3 ? �3 1½ �h=2;

@xxy� �

h=2¼^ 1

h3

1 �2 1

?

�1 2 �1

2

6664

3

7775

h=2

:

The other discrete operators are given by analogous stencils.

For an implementation of the distributive relaxation it is convenient to considerthe correction equations

Lhdumþ1 ¼ rm

h and LhChdwmþ1 ¼ rm

h

with update dumþ1 ¼ Chdwmþ1 ¼ uh � umþ1h and residual rm

h ¼ Lhumh � fh. u

mh de-

notes the approximation after the mth iteration of the exact discrete solution uh.

The distributive relaxation consists of two steps. In the first step, a newapproximation dwmþ1 to the ‘‘ghost variable’’ dw ¼ ðdwu; dwv; dwpÞT is calculated.This will be done by decoupled red-black point relaxation, due to the structure ofthe transformed system LhCh; discussed in Sect. 4. In the second step, a newapproximation for uh is computed by

umþ1h ¼ um

h þ dumþ1h ¼ um

h þ Chdwmþ1: ð21Þ

An Efficient Multigrid Solver based on Distributive Smoothing

In detail, the new approximation in (21) is given by

umþ1h ¼ um

h þ dwmþ1u � @xð Þh=2dwmþ1

p ;

vmþ1h ¼ vm

h þ dwmþ1v � @y

� �h=2dwmþ1

p ;

pmþ1h ¼ pm

h þ ðkþ lÞ @xð Þh=2dwmþ1u þ ðkþ lÞ @y

� �h=2dwmþ1

v

� ðkþ 2lÞDh dwmþ1p :

This implementation is straightforward.

Remark 1. The distributive relaxation operations described above ((17), (18)) areoperator manipulations in which the discretization of boundary operators is nottaken into account explicitly. Experience with distributive relaxation gained incomputational fluid dynamics learns that the zero blocks in (18) may not alwaysequal zero exactly for certain boundary conditions. Therefore, it is often advisedto perform additional relaxation steps near boundaries. In the application pre-sented here, we do not need the additional treatment near the boundary.

Remark 2. A ‘‘left distributor’’ for ChLhuh ¼ Chfh may read:

Ch ¼Ih 0 ðkþ lÞð@xÞh=20 Ih ðkþ lÞð@yÞh=2

�ð@xÞh=2 �ð@yÞh=2 �ðkþ 2lÞDh

0

@

1

A:

In that case, we obtain

ChLh ¼�lDh 0 LC1;3

h0 �lDh LC2;3

h0 0 eaðkþ 2lÞD2

h � Dh

0

@

1

A

with LC1;3h ¼ LC3;1

h and LC2;3h ¼ LC3;2

h ; see (19), (20). We end up with an uppertriangular system. In a first step then, the last equation should be updated afterwhich the other two may be treated. The advantage of a left distributor may bethat we still deal with the primary unknowns uh, whereas in the right distributorcase we work with wh as the slack variable. A disadvantage of a left distributor isthat the right-hand side must also be transformed. We have chosen for the rightdistributor as we do not encounter any problems in defining boundary conditionshere. Also in the case of stress boundary conditions, treated in a future paper, it iseasily possible to set up the distributive system near the boundaries.

Remark 3. For the discrete Stokes operator

Lh;st ¼�Dh 0 @xð Þh=20 �Dh @y

� �h=2

@xð Þh=2 @y� �

h=2 0

0

B@

1

CA

R. Wienands et al.

the distributor proposed in [3], [18] is given by

Ch;st ¼Ih 0 � @xð Þh=20 Ih � @y

� �h=2

0 0 �Dh

0

@

1

A:

The transformed system then reads

Lh; stCh;st ¼�Dh 0 00 �Dh 0

@xð Þh=2 @y� �

h=2 �Dh

0

@

1

A:

Note that for the particular parameter selection (which is of no physical rele-vance) k ¼ �1, l ¼ 1, and ea ¼ 0, (5) and (17) coincide with Lh;st and Ch;st,respectively. Regarding this matter, the distributor for the poroelasticity modeloperator can be considered as a generalization of the well-known distributiverelaxation for the staggered version of the Stokes equations.

3.4. Optimal Multigrid Smoothing for the System of Poroelasticity

The smoothing method Sh in a multigrid algorithm is designed to reduce high-frequency components of the error between exact solution and current approxi-mation effectively. A quantitative measure for its efficiency represents thesmoothing factor obtained by Fourier analysis. Fourier smoothing analysis isbased on the observation that many classical relaxation methods (like Jacobi orGauss-Seidel relaxation) leave the spaces of 2h-harmonics invariant, i.e.,

ShjF2hðhÞ ¼: eShðhÞ 2 C12�12 h 2 H2hlow

� �:

Applying an ‘‘ideal’’ coarse grid correction operator

Qh;2hjF2hðhÞ ¼: eQh;2h ¼ diagf0; 0; 0; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1g 2 C12�12

which annihilates the low-frequency error components and leaves the high-fre-quency components unchanged yields the smoothing factor [2], [14]

q1 Lh; nð Þ :¼ suph2H2h

low

q eQh;2heSn

hðhÞ� �

;

i.e., the asymptotic error reduction of the high-frequency error components by nsweeps of the relaxation method. Here, qðMÞ denotes the spectral radius of thematrix M . In analogy to the two-grid factor to be defined below, it could also benamed one-grid factor as it only takes the fine grid operators-relaxation anddiscretization-into account. The subscript ‘‘1’’ refers to one-grid. For scalarequations, we have eShðhÞ; eQh;2h ¼ diagf0; 1; 1; 1g 2 C4�4.

An Efficient Multigrid Solver based on Distributive Smoothing

The smoothing factor q1ðLh; nÞ for n distributive relaxations governed by (17) isdetermined by the diagonal blocks of the transformed system (18) [2], [14]. Moreprecisely, we have

q1 Lh; nð Þ ¼ maxfq1 LC1;1h :¼ �lDh; n

� �;

q1 LC3;3h :¼ eaðkþ 2lÞD2

h � Dh; n� �

g: ð22Þ

This means that the calculation of q1 Lh; nð Þ reduces to the computation of thespectral radii of certain ð4� 4Þ-matrices. Both scalar operators LC1;1

h , LC3;3h

occurring in (22) are isotropic in the sense that the coefficients referring to differentspatial directions are of the same size. Hence, a distributive point relaxationmethod can be used for all choices of k, l, and ea as it was already anticipated bythe measure of h-ellipticity.

There are many efficient relaxation schemes known for LC1;1h . The smoothing

properties of some of these schemes are, however, not satisfactory for LC3;3h , if it is

dominated by the biharmonic term which depends on the set of parameters andthe mesh size under consideration. More precisely, the corresponding smoothingfactor increases for an increasing eaðkþ 2lÞ=h2. This can be observed for a fixedset of parameters k, l, ea and a decreasing mesh size h. In Table 1 the smoothingfactors are presented for red-black Jacobi (RB-JAC) point relaxation. Here, thecomputational grid is subdivided into red and black points in a checkerboardmanner. RB-JAC consists of a Jacobi sweep over the red points only followed bya Jacobi sweep over the black points using the updated values at the red points.

Remark 4. Note, that RB-JAC coincides with the well-known Gauss-Seidelrelaxation with a red-black numbering of grid points for 5-point discretizationslike Dh. However, this equivalence is not longer valid for discrete operators basedon ‘‘larger’’ stencils like D2

h; see Remark 5.4.5 from [14] for details.

RB-JAC is the basis for very efficient multigrid methods for the Poisson equa-tion [13], [14] which is demonstrated by the smoothing factor 0.25. However, forthe biharmonic operator a deterioration to q1 D2

h; 1� �

¼ 0:64 can be observed. ForLC3;3

h , the parameters k, l, and ea are fixed and the mesh size h varies between 1/4and 1/256. The choices for these parameters are representative for geophysicalapplications. Table 1 shows that the smoothing properties for LC3;3

h deterioratewith a decreasing mesh size (i.e., with increasing eaðkþ 2lÞ=h2) as the biharmonicterm dominates.

Table 1. Smoothing factors q1ð:; 1Þ for three operators; k ¼ 1250, l ¼ 12500, ea ¼ 10�7

h 14

18

116

132

164

1128

1256

�Dh 0.25 0.25 0.25 0.25 0.25 0.25 0.25D2

h 0.64 0.64 0.64 0.64 0.64 0.64 0.64eaðkþ 2lÞD2

h � Dh 0.31 0.41 0.54 0.61 0.63 0.64 0.64

R. Wienands et al.

Improved smoothing factors can be obtained by introducing a one-stage param-eter x in RB-JAC. A one-stage variant of an arbitrary relaxation method Sh isgiven by

ShðxÞ :¼ 1� xð ÞIh þ xSh

with discrete identity Ih. To construct an optimal one-stage relaxation, we searchfor the parameter x which minimizes the corresponding smoothing factor. Thismeans that one has to solve the following minimization problem:

minx

suph2H2h

low

q eQh;2heShðx; hÞ� �

ð23Þ

with eShðx; hÞ :¼ ð1� xÞeIh þ xeShðhÞ and identity matrix eIh 2 C4�4 (for the scalarcase). The situation is particularly transparent, if we assume a non divergingrelaxation Sh equipped with a real-valued ‘‘high-frequency spectrum’’

rS :¼ spectrum of eQh;2heShðhÞj h 2 H2hlow

n o;

i.e., rS � Smin; Smax½ � � ½�1; 1�. Then, (6) reduces to a classical minimizationproblem,

minx

sup�1�Smin�z�Smax�1

ð1� xÞ þ xzj j;

see, for example, [15]. The optimal smoothing one-stage parameter and the relatedsmoothing factor are given by

xopt ¼2

2� Smax � Sminand q1ð:; n ¼ 1Þ ¼ Smax � Smin

2� Smax � Smin: ð24Þ

Remark 5. Note, that the one-stage parameter is applied after a complete RB-JAC step (and not-as usual overrelexation parameters-within each half step ofRB-JAC relaxation). For Jacobi (JAC) relaxation, overrelaxation and one-stageparameter coincide since unknowns are updated after the complete relaxationsweep and not dynamically within the relaxation process (as for Gauss-Seidel orpattern relaxations like RB-JAC).

Example 1. As an example we consider Jacobi relaxation which is defined by

SJACh :¼ Ih � D�1h Lh;

where Dh denotes the diagonal part of some discrete operator Lh under consid-eration. Obviously, the Fourier components are eigenfunctions of SJAC

h yielding a‘‘diagonal’’ Fourier representation

An Efficient Multigrid Solver based on Distributive Smoothing

eSJACh ðhÞ ¼ diagfA00;A11;A10;A01g 2 C4�4 with

Aa ¼ 1� eD�1h ðhaÞeLhðhaÞ ð25Þ

for scalar operators Lh. For the Laplacian Lh ¼ �Dh (11), we have Dh ¼^ 1h2 ½4�h

leading to

Aa ¼ 1� h2

4eDhðhaÞ ¼ 1þ sin2ðha

1=2Þ þ sin2ðha2=2Þ ¼

1

2ðcosðha

1Þ þ cosðha2ÞÞ;

compare with (12). From the above Fourier representation of SJACh , we easily

obtain

rS ¼ ½Smin ¼1

2ðcosðpÞ þ cosðpÞÞ ¼ �1; Smax ¼

1

2ðcosð�p=2Þ þ cosð0ÞÞ ¼ 1=2�:

Applying (24) yields the well-known optimal damped Jacobi smoother for theLaplacian:

xopt ¼ 4=5 and q1ðDh; n ¼ 1Þ ¼ 3=5:

For RB-JAC relaxation, the situation is somewhat more difficult as the Fouriercomponents are no longer eigenfunctions of the relaxation operator. It stillleaves the spaces of 2h-harmonics invariant, but certain Fourier componentsare coupled by RB-JAC yielding off-diagonal entries in its Fourier represen-tation:

eSRBh ðhÞ ¼ eSB

h ðhÞ � eSRh ðhÞ with ð26Þ

eSRh ðhÞ ¼

1

2

A00 þ 1 A11 � 1 0 0

A00 � 1 A11 þ 1 0 0

0 0 A10 þ 1 A01 � 1

0 0 A10 � 1 A01 þ 1

0

BBB@

1

CCCA; ð27Þ

eSBh ðhÞ ¼

1

2

A00 þ 1 �A11 þ 1 0 0�A00 þ 1 A11 þ 1 0 0

0 0 A10 þ 1 �A01 þ 10 0 �A10 þ 1 A01 þ 1

0

BB@

1

CCA: ð28Þ

For the derivation of these Fourier representations for the consecutive Jacobisweeps over the red (R) and the black (B) points, respectively, we refer to [13],[14].

Example 2. The optimal one-stage parameter for RB-JAC relaxation applied toDh is given by xopt Dhð Þ ¼ 16=15 leading to q1 Dh; 1ð Þ ¼ 1=5, whereas for D2

h we

R. Wienands et al.

have xopt D2h

� �¼ 25=18 yielding q1 D2

h; 1� �

¼ 1=2; compare with Example 4.3.1 andProposition 6.6.1 from [17], respectively. These results have been derived usingeSRB

h ðhÞ with Aa from Example 1 for the Laplacian and with Aa ¼ 1� h420eD2

hðhaÞ (see(11), (13)) for the biharmonic operator.

Since LC3;3h is a combination of the two operators from Example 2, it is reasonable

to search for an optimal one-stage RB-JAC relaxation for

LC3;3h ¼ cD2

h � Dh with c ¼ eaðkþ 2lÞ � 0 ð29Þ

leading to the following theorem.

Theorem 2. The spectrum rS (w.r.t. the high-frequency error components) of pointRB-JAC relaxation applied to LC3;3

h (29) is bounded by

Smin ¼ �16c2 þ 10ch2 þ h4

8 5cþ h2ð Þ2and Smax ¼

8cþ h2� �2

4 5cþ h2ð Þ2:

Proof: The Fourier representation eSRBh ðhÞ 2 C4�4 for point RB-JAC relaxation

applied to a two-dimensional operator like LC3;3h is given in (26). After a pro-

jection onto the high frequency components using the ideal coarse grid correctionoperator eQh;2h ¼ diagf0; 1; 1; 1g one obtains

eQh;2heSRBh ðhÞ ¼

0 0 0 0

aðhÞ bðhÞ 0 0

0 0 dðhÞ eðhÞ0 0 f ðhÞ gðhÞ

0

BBBB@

1

CCCCAwith h 2 H2h

low;

aðhÞ ¼ 1

4�A2

00 þ 1þ ðA11 þ 1ÞðA00 � 1Þ� �

;

bðhÞ ¼ 1

4�ðA00 � 1ÞðA11 � 1Þ þ ðA11 þ 1Þ2� �

;

dðhÞ ¼ 1

4ðA10 þ 1Þ2 � ðA01 � 1ÞðA10 � 1Þ� �

;

eðhÞ ¼ 1

4ðA10 þ 1ÞðA01 � 1Þ � A2

01 þ 1� �

;

f ðhÞ ¼ 1

4�A2

10 þ 1þ ðA01 þ 1ÞðA10 � 1Þ� �

;

gðhÞ ¼ 1

4�ðA10 � 1ÞðA01 � 1Þ þ ðA01 þ 1Þ2� �

and Aa ¼ AðhaÞ ¼ 1� h4

20cþ 4h2ceD2

hðhaÞ � eDhðhaÞ� �

; ð30Þ

compare with (27), (28), (25), (11), (12), and (13). The eigenvalues of eQh;2heSRBh ðhÞ

read k1ðhÞ ¼ 0, k2ðhÞ ¼ bðhÞ, and

An Efficient Multigrid Solver based on Distributive Smoothing

k3=4 ¼dðhÞ þ gðhÞ

2� 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidðhÞ2 þ 4eðhÞf ðhÞ � 2dðhÞf ðhÞ þ gðhÞ2

q:

A straight-forward but lengthy analysis yields that the spectrum rS of eQh;2heSRB

h ðhÞ(h 2 H2h

low) is real-valued. One may verify that the extreme values lie at theboundary of H2h

low leading to

Smin ¼ k2ð0;�p=2Þ ¼ k2ð�p=2; 0Þ ¼ � 16c2 þ 10ch2 þ h4

8 5cþ h2ð Þ2

and Smax ¼ k3ð0;�p=2Þ ¼ k3ð�p=2; 0Þ ¼8cþ h2� �2

4 5cþ h2ð Þ2:

(

Corollary. Using the above proposition and (24) we can construct an optimal one-stage method with one-stage parameter

xopt LC3;3h

� �¼

16 5cþ h2� �2

3 96c2 þ 46ch2 þ 5h4ð Þ ð31Þ

and optimal smoothing factor

q1 LC3;3h ; 1

� �¼ 8cþ h2

16cþ 5h2: ð32Þ

Since c ¼ eaðkþ 2lÞ � 0 and h > 0 it can be easily seen from (32) that

1=5 � q1 LC3;3h ; 1

� �� 1=2 ð33Þ

for all possible choices of c ¼ eaðkþ 2lÞ and h. More precisely, the lower bound isobtained if c ¼ 0. Then LC3;3

h reduces to the Laplacian and the correspondingoptimal one-stage method is given by xðLC3;3

h Þ ¼ 16=15 and q1ðLC3;3h ; 1Þ ¼ 1=5;

see above. The upper bound is reached if the biharmonic operator dominatesLC3;3

h , i.e., c=h2 !1. For a fixed mesh size h this gives:

limc!1

xopt LC3;3h

� �¼ lim

c!1

16 25þ 10h2=cþ h4=c2� �

3 96þ 46h2=cþ 5h4=c2ð Þ ¼400

288¼ 25

18;

limc!1

q1 LC3;3h ; 1

� �¼ lim

c!1

8þ h2=c16þ 5h2=c

¼ 1

2;

recovering the optimal one-stage method for the biharmonic operator.

The smoothing strategy is that the first two equations in (18) are smoothed byone-stage RB-JAC relaxation with xoptðDhÞ, whereas for the third equation

R. Wienands et al.

xoptðLC3;3h Þ is chosen, leading to the following smoothing factor for the system of

poroelasticity:

q1 Lh; 1ð Þ ¼ max q1 Dh; 1ð Þ; q1 LC3;3h ; 1

� �n o¼ q1 LC3;3

h ; 1� �

:

From (33) it immediately follows that

1=5 � q1 Lh; 1ð Þ � 1=2

which is a strong robustness result for such a complicated system involving severalparameters (ea; k; l; h). For example, two steps of the proposed RB-JAC one-stagemethod applied to the realistic set of parameters from Table 1 yields a satisfactoryq1 Lh; 2ð Þ ¼ 0:25.

Remark 6. The efficiency of many solution methods for problems from linearelasticity depends on the Poisson ratio m defined in (3). The smoothing factorq1 Lh; 1ð Þ only depends on the ratio c=h2 and there is no particular difficulty causedby certain values for the Poisson ratio which is demonstrated by Tables 2 and 3.We use a fixed set of parameters for h, k and ea and vary l in order to analyze theeffect of the Poisson ratio. It can be clearly seen, that the smoothing factor isdetermined by c (for fixed mesh size) and is not affected by the often crucial valuem ¼ 0:5 for the Poisson ratio. For small values for c (due to ea ¼ 5�7 in Table 2) thebest possible smoothing factors are obtained independent of the Poisson ratio,whereas for large values for c (due to ea ¼ 5�2 in Table 3) the worst possiblesmoothing factors are reached, again independent of the Poisson ratio. Summa-rizing, the robust behavior of the proposed relaxation method is independent ofthe Poisson ratio. Note that xoptðLC3;3

h Þ and q1ðLh; 1Þ shown in Tables 2 and 3result from a simple evaluation of (31) and (32), respectively.

Table 2. Poisson ratio m and corresponding smoothing factor q1 Lh; 1ð Þ (up to three digits) for varying land fixed k ¼ 1, ea ¼ 5�7, h=1/64

l m c x optðLC3;3h Þ q1ðLh; 1Þ

1 0.25 1:5 � 10�6 1.072 0.20610�1 0.455 6:0 � 10�7 1.069 0.20210�2 0.495 5:1 � 10�7 1.068 0.20210�4 0.499 5:001 � 10�7 1.068 0.202

Table 3. Poisson ratio m and corresponding smoothing factor q1 Lh; 1ð Þ (up to three digits) for varying land fixed k ¼ 1, ea ¼ 5�2, h=1/64

l m c x optðLC3;3h Þ q1ðLh; 1Þ

1 0.25 0.15 1.389 0.49910�1 0.455 0.06 1.388 0.49910�2 0.495 0.051 1.388 0.49910�4 0.499 0.050 1.388 0.499

An Efficient Multigrid Solver based on Distributive Smoothing

Remark 7. Applying Smin and Smax from Theorem 2 it is possible to constructmulti-stage variants of RB-JAC relaxation (see, for example, [17]) with even betterproperties. However, it turned out in the Fourier two-grid analysis and in thenumerical tests that it does not pay off to invest to much work into smoothingbecause the coarse grid correction cannot reduce the low-frequency error com-ponents equally well. Therefore, we focus on one-stage RB-JAC smoothingmethods.

3.5. Coarse Grid Correction

An appropriate coarse grid correction on the Cartesian grid Gh consists ofstraightforward geometric transfer operators Rh;2h, P2h;h, which are well-estab-lished in the field of computational fluid dynamics and direct coarse grid dis-cretizations (i.e., coarse grid analogs of Lh). Since we use a staggered grid, we haveto distinguish the transfer operators which act on the different grids Gj

h(j ¼ 1; 2; 3), see Fig. 1. At u- and v-grid points we consider 6-point restrictions andat p-grid points a 9-point restriction. In stencil notation they are given by

Ruh;2h ¼

^ 1

8

1 1

2 ? 2

1 1

2

64

3

75

2h

h

; Rvh;2h ¼

^ 1

8

1 2 1

?

1 2 1

2

64

3

75

2h

h

; Rph;2h ¼

^ 1

16

1 2 1

2 4 2

1 2 1

2

64

3

75

2h

h

;

respectively. The restriction operator for the defect in the p-equation differs fromthe usual one in solving the incompressible Navier-Stokes equations, because ofthe placement of pressure points at the vertices, whereas a cell-centered pressuregrid is employed in fluid mechanics applications. As the prolongation operatorsP u=v=p2h;h , we apply the usual interpolation operators based on linear interpolation of

neighboring coarse grid unknowns, dictated by the staggered grid (see, forexample, sect. 8.7 in [14]). The pressure prolongation is the adjoint of itsrestriction.

3.6. Fourier Two-grid Analysis

The crucial observation in the classical Fourier two-grid analysis is that the two-grid operator (9) leaves the spaces of 2h-harmonics (10) invariant. Hence, the two-grid operator can be represented in Fourier space by a block matrix consisting ofð4� 4Þ-blocks for scalar equations and by ð12� 12Þ-blocks for our discrete sys-tem (5):

Mh;2hjF2hðhÞ ¼: eMh;2hðhÞ ¼ eSn2h ðhÞeCh;2hðhÞeSn1

h ðhÞ 2 C12�12� �

with Fourier representation eCh;2hðhÞ of the coarse grid correction operator. Fordetails on Fourier two-grid analysis and the derivation of eCh;2hðhÞ, we referto [2,13,14] and especially to [3,11] for the analysis on staggered grids.

R. Wienands et al.

From the above representation, one may easily calculate the two-grid convergencefactor as the supremum of the spectral radii from the related block matrices by acomputer program:

q2 :¼ suph2H2h

low

q eMh;2hðhÞ� �

:

4. Numerical Experiments

In this section, the robustness and efficiency of the distributive relaxation methodis investigated by comparing the theoretically predicted convergence factors withthe actually obtained numerical convergence. We choose a zero right-hand side,homogeneous boundary conditions and a random initial guess to avoid round-offerrors. Local Fourier analysis, as discussed in the previous section, yieldsasymptotic convergence estimates since it is based on certain spectral radii. Wemeasure the asymptotic numerical multigrid convergence during the first time stepby performing 100 multigrid cycles and taking the average of the last 50 defectreduction factors:

qkðnumÞ :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq100

k � q99k q51

k50

q¼

ffiffiffiffiffiffiffiffiffiffiffiffires100

res5150

r

with qmk ¼ resm=resm�1 and the maximum norm of the residual over the three

equations in the system after the mth multigrid cycle:

resm :¼ krmh;1k1 þ krm

h;2k1 þ krmh;3k1:

The subscript ‘‘k’’ denotes the number of grids involved in the multigrid solutionmethod. V(1,1) denotes a V-cycle with one pre- and one post-relaxation, F(1,1)the corresponding F-cycle.

The insensitivity of the smoothing method to critical values for the Poissonratio carries over to the complete multigrid solver. We fix parameter a ¼ 1,yielding ea ¼ 0:5dt; due to the Crank-Nicrolson time discretization. Tables 4and 5 show theoretical predictions and numerically obtained convergencefactors for the parameters k ¼ 1; h ¼ 1=64; 10�4 � l � 1, and dt ¼ 10�6

(Table 4) and dt ¼ 10�1 (Table 5). Obviously, these factors are independent ofthe varying l and thus independent of the varying Poisson ratio. Instead theyare governed by c (for fixed mesh size): the smaller c, i.e., the smaller ea, thebetter the convergence.

Results for more realistic sets of parameters are shown in Table 6. It can beclearly seen, that the two-grid analysis provides excellent estimates for thenumerically observed F-cycle convergence involving six grids. Applying thecomputationally less expensive V-cycle leads to a slight increase of the multigridconvergence.

An Efficient Multigrid Solver based on Distributive Smoothing

5. Conclusions

We provide a fast and accurate discrete solution for the incompressible variant ofthe poroelasticity equations, discretized on a staggered grid to deal with stabilitycomplications.

A robust distributive relaxation method for the system of poroelasticity equationshas been introduced. The properties of the smoother were analyzed and optimizedby Fourier smoothing analysis. With standard geometric transfer operators anddirect coarse grid discretization, an efficient multigrid method, based on point-wise smoothing methods results.

The analysis of the multigrid method has been performed with classical mul-tigrid Fourier analysis techniques. Their benefits have become clear in thiswork. The numerical multigrid results agree very well with the results from theFourier analysis. This is an important gain by the analysis. The influence ofdifferent relaxation parameters on the multigrid convergence factor can be verywell predicted. The main disadvantage of Fourier analysis may be that it isnot straightforward to apply to non–Cartesian grid applications. However, theinsights obtained for the Cartesian grid case are valuable for the develop-ment of efficient solvers for poroelasticity problems in more complicated do-mains.

Table 4. Local Fourier analysis results and numerical convergence factors for varying l and fixedk ¼ 1, dt ¼ 10�6, h ¼ 1=64

l cycle q1 q2 q6ðnumÞ1, 10�1, 10�2,10�4 V(1,1) 0.04 0.11 0.161, 10�1, 10�2, 10�4 F(1,1) 0.04 0.11 0.10

Table 5. Local Fourier analysis results and numerical convergence factors for varying l and fixedk ¼ 1, dt ¼ 10�1, h ¼ 1=64

l cycle q1 q2 q6ðnumÞ1, 10�1, 10�2,10�4 V(1,1) 0.25 0.25 0.311, 10�1, 10�2, 10�4 F(1,1) 0.25 0.25 0.25

Table 6. Local Fourier analysis results and numerical convergence factors for various parameters andfixed mesh size h ¼ 1=64

parameter set cycle q1 q2 q6ðnumÞk ¼ 1250, l ¼ 12500, V(1,1) 0.25 0.25 0.30

dt ¼ 10�6 F(1,1) 0.25 0.25 0.25k ¼ 0, l ¼ 0:5, V(1,1) 0.25 0.25 0.28

dt ¼ 10�2 F(1,1) 0.25 0.25 0.24k ¼ 0, l ¼ 0:5, V(1,1) 0.04 0.11 0.16

dt ¼ 10�6 F(1,1) 0.04 0.11 0.10k ¼ 1, l ¼ 1, V(1,1) 0.25 0.25 0.31

dt ¼ 10�1 F(1,1) 0.25 0.25 0.25k ¼ 103, l ¼ 104, V(1,1) 0.25 0.25 0.31

dt ¼ 10�1 F(1,1) 0.25 0.25 0.25

R. Wienands et al.

References

[1] Biot, M.A.: General theory of three dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941).[2] Brandt, A.: Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD-Studie

Nr. 85, Sankt Augustin, Germany 1984.[3] Brandt A., Dinar, N.: Multigrid solutions to elliptic flow problems. In: Numerical methods for

partial differential equations (Parter, S., ed.), pp. 53–147. New York: Academic Press, 1979.[4] Ehlers, W., Blum, J. (eds.): Porous media: theory, experiments and numerical applications. Berlin:

Springer, 2002.[5] Gaspar, F.J., Lisbona, F.J., Vabishchevich, P.N.: A finite difference analysis of Biot’s

consolidation model. Appl. Num. Math. 44, 487–506 (2003).[6] Gaspar, F.J., Lisbona, F.J., Vabishchevich, P.N.: Stable finite difference discretizations for soil

consolidation problems. (submitted).[7] Hackbusch, W.: Multi-grid methods and applications. Berlin: Springer, 1985.[8] Harlow F.H., Welch, J.E.: Numerical calculation of time-dependent viscous incompressible flow

of fluid with a free surface. Phys. Fluids 8, 2182–2189 (1965).[9] Mow V.C., Lai, W.M.: Recent developments in synovial joint biomechanics. SIAM Rev. 22, 275–

317 (1980).[10] Murad, M.A., Thomee, V., Loula, A.: Asymptotic behaviour of semidiscrete finite element

approximations of Biot’s consolidation problem. SIAM J. Numer. Anal. 33, 1065–1083 (1996).[11] Niestegge A., Witsch, K.: Analysis of a multigrid Stokes solver. Appl. Math Comp. 35, 291–303

(1990).[12] Santos, J.E., Douglas J. Jr., Calderon, A.P.: Finite element methods for a composite model in

elastodynamics. SIAM J. Numer. Anal. 25, 513–532 (1988).[13] Stuben K., Trottenberg, U.: Multigrid methods: Fundamental algorithms, model problem

analysis and applications. In: Multigrid methods. (eds.) Lecture Notes in Math. 960, pp. 1–176.(Hackbusch, W., Trottenberg, U. Berlin: Springer, 1982.

[14] Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. New York: Academic Press 2001.[15] Varga, R.S.: Matrix iterative analysis. Englewood Cliffs: Prentice-Hall 1962.[16] Wesseling, P.: Principles of computational fluid dynamics. Berlin: Springer 2001.[17] Wienands, R.: Extended local Fourier analysis for multigrid: optimal smoothing, coarse grid

correction, and preconditioning. Ph.D. Thesis, University of Cologne, Germany 2001.[18] Wittum, G.: Multi-grid methods for Stokes and Navier-Stokes equations with transforming

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Roman WienandsUniversity of Cologne,Mathematical InstituteWeyertal 86-9050931 Cologne, Germanywienands@mi.uni-koeln.de

Francesco J. GasparDepartamento de Mathematica Aplicada,University of ZaragozaPedro Cerbuna 1250009 Zaragoza, Spainfjgaspar@unizar.es

Francesco J. LisbonaDepartamento de Mathematica Aplicada,University of ZaragozaPedro Cerbuna, 1250009 Zaragoza, Spainlisbona@unizar.es

Cornelis W. OosterleeDelft University of Technology,Faculty of Information Technology Systems,Department of Applied Mathematical AnalysisMekelweg 4,2628 CD Delft, the NetherlandsC.W.Oosterlee@math.tudelft.nl

An Efficient Multigrid Solver based on Distributive Smoothing