Introduction to the Multigrid Method

Transcription

1 Semiar: Te Iterplay betwee Matematical Modellig ad Numerical Simulatio Itroductio to te Multigrid Metod Bogojeska Jasmia JASS, 005 Abstract Te metods for solvig liear systems of equatios ca be divided ito two categories: direct ad iterative metods. Te first oes ca determie te exact solutios, but are rater slow ad are restricted to a certai small set of problems for wic tey sow good performace. Te iterative metods ca be applied to a broader rage of problems, but caot damp te smoot compoets of te error ad because of tat i some cases sow a very slow covergece. Te multigrid metods ave developed from te mai idea tat te amout of computatioal work sould be proportioal to te amout of real pysical cages i te computed system. I fully developped multigrid processes te amout of computatios sould be determied oly by te amout of real pysical iformatio

2 . Model Problems Te boudary value problems give a simple testig groud for providig a basic itroductio to te multigrid metods. Altoug most of tese problems ca be adled aalytically, te umerical metods will be preseted ad tey will serve as model problems i order to preset te multigrid metod i a atural way. Te oe-dimesioal boudary value problem describig te steady-state temperature distributio i a log uiform rod is give by: u''( x) + σ u( x) = f( x) 0< x < u(0) = u() = 0 σ 0 Wit te grid poits x j = j, j = 0,,..., were =, te domai of te problem is divided ito subitervals. Te grid for tis problem sow o Figure will be deoted wit Ω. x x 0 x Figure Accordig to te fiite differece metod i te iterior grid poits te origial differetial equatios ca be replaced by a secod-order fiite differece approximatio: were v j is a approximatio to te exact solutio ( j ) j =,,, v + v v v = v = 0 j j j+ ( f x,, f x ) u x ad j K. Defiig f = ( ) K ( ) ad = ( v ) A v = f of te system above is: 0 + σ v = f j =,,..., j j,, v f is ( j ) f x for v K te matrix form + v f σ v σ f = σ v f were A is a ( ) ( ) symmetric, positive, defiite matrix. Te two-dimesioal boudary value problem as te form: u u + σ u = f( x, y), 0< x<, 0< y<, σ > 0 xx yy were u = 0 o te boudary of te uit square. Te grid sowed o Figure, is defied wit te poits ( xi, yj) = ( i x, j y), were i =,..., m, j =,..., m, x = ad y =. m

3 Figure I te same way as for te oe-dimesioal boudary value problem, replacig te derivatives by te secod-order fiite differeces leads to te system of liear equatios: vi, j + vij vi+, j vi, j + vij vi, j+ + + σ v ij = fij x y v = v = 0, i =,,..., m, j =,,..., i0 i were ij u xi, y j ad fij = f( xi, yj ). By usig T lexicograpical orderig by lies oe ca defie vi = ( vi,..., vi, ) ad T fi=( fi,..., fi, ) for i =,..., m. Accordig to tis otatio te block-tridiagoal matrix form of te system is A v = f i.e.: v is a approximatio of te exact solutio ( ) A a I v f a I A a I v f..... =..... a I A a I v f a I A v f were a =, I is a ( ) ( ) idetity matrix ad A i is a ( ) ( ) tri-diagoal y matrix give wit: + σ x + y x + σ x x + y x Ai = σ x x + y

4 . Basic Iterative Scemes Te ext step is to cosider ow te model problems tat are defied i te previous sectio ca be solved usig some basic iterative or relaxatio scemes. Te problems will be give i teir matrix form Au = f, were u is te exact solutio ad v is te correspodig approximatio. Te vector orms will be used as a measure for te error tat is defied wit e= u v. Te residual equatio is Ae = r, were te residual is defied as r = f A v. Te equatio u= v+ e is te residual correctio. We usig te equatio u v= A rad ( old ) ( ) v v, u v ew ( ) ( ) ( ), a iteratio v ew = v old + B r old ca be formed were B is a approximatio to A. Te equatio for te iteratio ca take sligtly differet form ( ew ) ( old ) ( old ) v = R v + B f = R v +g, were R = I B A. Usig tis form te exact solutio will be fixed poit i.e. u= R u + g. Te error will be give by e ( ew) = R e ( old ), or = ( m) m (0) e R e if m iteratios are performed ad it ca be bouded wit ( m) m (0) e R e, were some proper vector ad matrix orms are used. From tis iequality it follows tat te error will ted to zero i te relaxatio process if R <. Defiitio Assymptotic covergece factor is te spectral radius defied as ρ R = max λ,..., λ. ( ) { } Lemma R m 0 as m if ad oly if ρ ( ) < R. (0) Usig te lemma defied above ad takig ito cosideratio tat for ay iitial vector v, ( m ) 0 ρ R <, it ca be cocluded tat te covergece of te e as m if ad oly if ( ) iteratio is give by te coditio ρ ( R ) <. 3. Jacobi Relaxatio Oe of te basic relaxatio scemes is te Jacobi Relaxatio Sceme. For simplicity te oedimesioal boudary value problem will be cosidered wit σ =0 i.e.: u + u u = f, j =,,...,, u = u = 0 j j j+ j 0 Te Jacobi relaxatio for tis problem is give by te followig system of equatios: ( ew) ( old ) ( old ) vj = ( vj + vj+ + f j), j =,..., Te correspodig matrix form is ( ew) = ( old ) + v R v D f, were = ( + ) A = D L U. Te weigted or damped Jacobi relaxatio is defied wit: J R D L U ad ( ew) ( old ) ( old ) ( old ) vj = ( ) vj + ( vj + vj+ + f j), j =,..., ( ew) ( old ) or wit te equivalet matrix form v = R v + D f, were R = I+ R ad is a weigtig factor tat is properly cose. ( ) J J

5 3. Gauss-Seidel Relaxatio Te Gauss-Seidel relaxatio is similar to te Jacobi relaxatio ad for te simplified oedimesioal model problem wit =0 σ is defied as follows: ( ew) ( ew) ( old ) vj = ( vj + vj+ + f j), j =,..., ( ) ( ) or usig a matrix form v ew = R v old + ( D L) f, were ( ) G R = D L U ad A = D L U. Te differece from te Jacobi relaxatio is tat Gauss-Seidel uses te compoets of te ew approximatio as soo as tey are calculated, wic reduces te storage requiremets for te approximatio vector v to locatios, because tere is o eed for keepig te values of tis vector for te old ad te ew iteratio. 3.3 Fourier Modes For simplicity we will cosider te omogeeous liear system Au = 0. We immediately ca see tat te exact solutio to tis system is u= 0 ad te error is e= u v = v. j k p Defiitio Te vectors vj = si, 0 j, k, were k is frequecy or waveumber idicatig te umber of alf-sie waves tat costitute v o te domai are called Fourier modes (Figure 3). G Figure 3 Defiitio 3 Te waveumbers i rage k are called smoot or low-frequecy modes, ad tose i rage k are called oscillatory or ig-frequecy modes. If we take te Fourier modes give i Figure 3 as iitial iteratio ad we perform 00 sweeps of te weigted Jacobi iteratio, we will get te results for te error sow i Figure 4. As we ca see o te figure te error decreases wit eac iteratio, but te iger wave umbers sow muc larger rate of decrease.

6 Figure 4 I order to see a more realistic case we take a iitial guess tat does ot cotai oly sigle mode, but a combiatio of a low-frequecy, medium-frequecy ad ig-frequecy wave, j π 6 j π 3 j π i.e.: v j = si si si As we ca see o Figure 5 te error decreases very fast oly i te first five iteratios. After tat te decrease of te error becomes very slow. Figure 5 Te quick elimiatio of te ig-frequecy modes of te error gives te fast iitial decrease. Te presece of te low-frequecy modes results i a very slow error decrease as we cotiue wit te iteratios ad sigificatly degrades te performace of te stadard iteratio metods. Te iteratios would coverge fast oly if te error cotais ig-frequecy modes, wic are damped very fast. I order to see wy tis appeed we must examie te problem a bit more formally. At first it sould be poited out tat te weigted Jacobi metod preserves modes, i.e. performig te relaxatios oly te amplitude of te modes is caged. Havig te fact tat R = I A λ( R) = λ( A ), we get tat R ad A ave te same eigevectors:

7 j k π wk,j = si, k. Te eigevalues of A are give wit: k π λk ( A ) = 4si, k ad te eigevalues of R are: k π λk ( R ) = si, k. Havig te eigevectors of A, we ca expad te error vector e (0) i te form: e (0) c k k= = w. Usig te formula for te error after m iteratios ad te fact tat te k eigevectors of m m m m R ad A are te same we get: = ck k = ck λk ( ) k k= k= e R w R w. I te last formula we ca clearly see tat te kt mode of te error after m iteratios is reduced m m by a factor of λ k ( R ). If 0< te λ k ( R ) < ad we will ave a coverget Jacobi iteratio. But for all, 0 < we get tat: π π π λ = si = si. Accordig to tis formula te eigevalue tat correspods to te smootest mode will always be close to oe for ay coice of ad terefore te smoot compoets of te error coverge very slowly. If we wat to improve te accuracy of te solutio by takig smaller grid spacig te λ will be eve more close to. No value of ca reduce te smoot compoets of te error. We ca oly fid te value of tat provides us wit te best λ R = λ R for dampig of te oscillatory modes of te error. Solvig te equatio ( ) ( ) te weigted Jacobi metod leads to = ad λk, for k, wic tells us tat 3 3 te oscillatory compoets of te error will be reduced at least by a factor of tree i eac iteratio sweep. Tis brigs us to a importat caracteristic of eac stadard relaxatio sceme. Defiitio 4 Te largest absolute value amog te eigevalues i te upper alf of te spectrum (te oscillatory modes) of te iteratio matrix is called smootig factor. 3. Te Multigrid Metod 4. Coarse Grids Providig a good iitial guess ca improve te performace of a relaxatio sceme i te iitial iteratio sweeps. A good way for gettig a better iitial guess is takig a coarse grid ad performig a certai umber of iteratios. O Figure 6 a smoot wave (waveumber 4) is sow o a grid wit poits ad o a coarse grid wit 6 poits. We see tat te smoot wave o te fie grid looks more oscillatory we pojected o te coarse grid i.e. te smootig property we usig coarse grids becomes a advatage. Moreover, te relaxatio o a coarse grid is less expesive because tere are less poits tat sould be kept i memory ad te coarse grid as a margially improved covergece rate te covergece factor is Ο. ( ) /

8 Figure 6 Let us see te projectio of te smoot wave o te coarse grid ito more detail. Te kt mode o Ω becomes kt mode o Ω for k < i.e. : j k j k wk, j si π si π w = = = k, j. Because of aliasig, for k > te kt mode / o Ω becomes ( k)t mode o Ω ad te oscillatory modes will be misiterpreted as relatively smoot: w ( ) ( ) π j k π j π k j k = si = si = si = w / k, j k, j Te cocept of coarse grid ad its mai property of makig smoot modes to look more oscillatory, gives te idea to move to coarser grid we te relaxatio begis to stall because te relaxatio will be more effective i dampig te oscillatory compoets of te error. 4. Nested Iteratio Te ested iteratio is based o te idea of performig a certai umber of prelimiary iteratios i order to get a better iitial guess for te fie-grid iteratio. It ca be described as follows: Relax o Au = fo a very coarse grid to obtai a iitial guess for te ext fier grid... 4 Relax o Au = fo Ω to obtai a iitial guess for Ω Relax o A u= f o Ω to obtai a iitial guess for Ω Relax o Au = fo Ω to obtai a fial approximatio to te solutio.

9 4.3 Correctio Sceme Te correctio sceme uses te idea tat we ca relax directly o te error by usig te residual equatio: Ae = r= f Av wit iitial guess e= 0. Additioally tis previously described relaxatio is equivalet to a relaxatio o te equatio Au = fwit a arbitrary iitial guess v. Te correctio sceme ca be described wit: Relax o Au = fo Ω to obtai a approximatio v Compute te residual r = f A v Relax o te residual equatio Ae = ro Ω to obtai a approximatio to te error e Correct te approximatio obtaied o Ω wit te error estimate obtaied o Ω : v v + e Te mai idea ere is tat at first we relax o te fie grid. We te covergece becomes slow we relax o te residual equatio o a coarser grid were we obtai a approximatio to te error. Te we retur back to te fie grid usig te obtaied approximatio to te error. 4.4 Iterpolatio Operator I te previous two subsectios we gave two scemes tat ca potetially improve te performace of te relaxatio metods. But some of te steps, like ow do we trasfer a vector from te coarse grid to te fie grid ad vice versa, still eed to be specified ito more details. We sould also poit out tat we will cosider oly te case were te coarse grid as twice as less poits compared to te precedig fie grid. Tis is doe because of simplicity ad also because we will get te same coclusios usig differet grid spacigs. Te iterpolatio operator is based o a commo procedure i umerical aalysis called iterpolatio or prologatio ad provides us wit te ecessary tecique for trasferrig te error approximatio e from te coarse grid Ω to te fie grid Ω. Practise as sow tat for most multigrid implemetatios te liear iterpolatio gives very good results, so we will also use it ere. Te iterpolatio operator I is a liear operator from to, wit a full rak ad a trivial ull-space. It ca be see as a mappig I : Ω Ω, tat trasforms coarse-grid vectors ito fie-grid vectors usig te formula I v = v, were v j = vj, v j+ = ( vj + vj+ ), 0 j. Tis procedure is illustrated o Figure 7. Figure 7 It is importat to see ow tis operator works we we ave smoot ad we we ave oscillatory vector o te fie grid. Te iterpolatio process we te real vector is smoot is illustrated o Figure 8.

10 Figure 8 From te picture above we ca see tat if te error o Ω is smoot te iterpolat will also be smoot, i.e. a iterpolat of te coarse-grid error gives a good iterpretatio of te real error. We te real error is oscillatory, Figure 9 sows tat te iterpolat is smoot, i.e. i tis case a iterpolat of te coarse-grid error may give a poor iterpretatio of te real error. Figure 9 Beig efficiet we te error is smoot, te iterpolatio operator provides a complemet to te relaxatio process. Te iterpolatio process is a part of te ested iteratio ad correctio sceme, so tey also sow best performace for smoot errors. 4.5 Restrictio Operator Te restrictio operators are used for trasferrig vectors from a fie grid to a coarse grid. Tey are liear operators from to deoted as I, are wit a full rak ad ave a ullspace of dimesio. Te restrictio operators ca be see as mappigs I : Ω Ω tat usig te formula I v = v take fie-grid vectors ad produce coarse-grid vectors. Te simplest oe is ijectio, defied wit vj = v j, were te correspodig value of te fie-grid poit is simply take as a value of te coarse-grid poit. Aoter restrictio operator is full weigtig, defied wit vj = ( v j + v j + v j+ ), j -, were we take 4 weigted averages of values at eigbourig fie-grid poits i order to get te values of te coarse-grid poits. Tis process is illustrated i Figure 0. Figure 0

11 4.6 Two-Grid Correctio Sceme Havig te detailed defiitios of te iterpolatio ad te restrictio operator we ca ow give te procedure tat describes te two-grid correctio sceme. v MG( v, f ) Relax ν times o A u = f o Ω wit iitial guess v Compute te fie-grid residual r = f A v it to te coarse grid by r Solve A e = r o Ω = I r Iterpolate te coarse-grid error to te fie-grid by e ad restrict = I e ad correct te fie-grid approxiamtio by v v + e Relax ν times o A u = f o Ω wit iitial guess v A ice illustratio is give i Figure. Figure As we ca see o te picture above, at te begiig we relax usually to 3 times o te fiegrid. After we calculate te residual of te approximatio tat we got we trasfer it by te restrictio operator to te coarse grid. Te te residual equatio is solved (or approximate solutio is foud) o te coarse grid. Te last step is trasferrig te error (or te approximated error) wit te iterpolatio operator back to te fie-grid ad correctig te fie-grid approximatio. Tis is also followed by a few iteratio sweeps. Te importat tig tat sould be oted ere is tat wit te relaxatio we elimiate te oscillatory compoets of te error, ad assumig tat we ca get a accurate solutio to te residual equatio, te iterpolatio operator will get a relatively smoot error. As we kow from before te iterpolatio operator is most effective o smoot errors, so we are supposed to get a good correctio of te fie grid approximatio. 4.7 V-Cycle Sceme Tere is oe problem i te procedure described i te previous subsectio ad tat is te solutio of te residual equatio A e = r o te coarse grid. If we ca otice tat tis

12 problem is ot muc differet ta te origial problem we ca solve it recursively. Namely we ca apply te two-grid correctio procedure to te residual equatio o Ω ad te 4 move to a coarser grid i.e. Ω i order to obtai te correctio. We repeat te process util we reac a grid were we ca fid a exact solutio of te residual equatio (we ca eve reac to grids wit oe poit if it is ecessary). After tat we go up to te fier grids usig te correspodig iterpolatio operators. A otatio modificatio is eeded i order to be able to describe tis recursive procedure algoritmically. Te rigt ad-side of te residual equatio will be deoted as f, u will replace te solutio of te residual equatio e ad fially v will deote te approximatios to u. Tese cages are appropriate because solvig te residual equatio is adled te same way as te origial equatios ad we get simplified otatio for describig te wole procedure. It sould also be poited out tat as a iitial guess for te first visit to Ω we will coose v = 0, because tere o iformatio available for te solutio u. Te process described above is sow i Figure. Figure Takig ito accout tat v ad f must be stored at eac level ad tat for d dimesios te coarser grid as -d te umber of poits as te fier grid, for te storage costs of te V- d d d d Md Cycle we ave ( ) <. Te computatioal costs of a V-Cycle d wit oe pre-coarse-grid correctio relaxatio sweep ad oe post-coarse-grid relaxatio d d d Md sweep are give wit (... ) <, were te cost of oe relaxatio sweep o te fie grid is oe workig uit (WU). 4.8 Full Multigrid V-Cycle Te full multigrid V-Cycle combies te ested iteratio ad te V-Cycle. Te basic idea ere is tat a better iitial guess for te first fie-grid iteratio of te V-Cycle ca improve its performace. I te cotext of multigridig a good cadidate is te ested iteratio, wic suggests performig prelimiary iteratios o te coarse grid Ω. Now we also eed a iitial guess for te Ω problem. Te ested iteratio uses recursio for solvig tis problem. 4 Agai we move te problem to te coarser grid Ω, ad we cotiue tis process util we reac te coarsest grid were we ca solve te problem explicitly. After tat we move up to te fier grids usig te iterpolatio operator. 3 d Te full multigrid V-Cycle, were te coarse-grid rigt-sides are iitialized by trasferrig from te fie grid, ca be described wit te followig procedure: f

13 v FMG ( f ) giitialize f I f, f I f, gsolve or relax o coarsest grid giterpolate iitial guess v g I 4 4 ( ) Perform V-cycle v MV v, f ν 0 times giterpolate iitial guess v g I ( ) Perform V-cycle v MV v, f ν 0 times v v Te parameter tat specifies te umber of V-Cycles performed at eac level ν 0 is determied experimetally ad usually as te value oe. As we ca see i te described procedure eac V-Cycle is preceded by a V-Cycle performed o a coarser grid i order to provide a good iitial guess. Figure 3 bellow gives a ice scematic represetatio. Figure 3 Te recursive procedure for te full multigrid is as follows: ( f ). Iitialize f I f, f I f, If Ω coarsest grid, set v 0, te go to 3. Else f v v FMG I FMG ( f ) ( f ) 3. Correct v I v 4. v MV ( v, f ) ν 0 times

14 Takig ito accout tat te size of te workig uit for te coarse grid j is jd times of te size of te workig uit o te fie grid, te costs of full multigrid for ν 0 = ν =... = are less d d ta (... d ) =. d 4.9 Buildig A ( ) e u v, i.e. te error lies i te rage of te iterpolatio. From tis it follows tat tere exists a vector u Ω, suc tat e = I u ad for te residual equatio we get A I u = r. Figure 4 sows ow Rage. At te begiig for simplicity we assume tat for te error olds = R( I ) A acts o ( I ) Figure 4 Te values A I u are zero at te odd grid poits of Ω, so te odd rows of A I are zero ad te eve rows are actually te coarse-grid poits of Ω. Accordig to tis if we leave out te odd rows i te residual equatio we get its coarse-grid form. Tis ca be doe by applyig te restrictio operator I ad we get I I I A u = r. From ere we ca defie te coarse-grid operator as A = I A I. Te same result ca be obtaied we usig te secod-order fiite differeces we te origial problem is discretized o Ω. Te argumet tat e Rage( I ) does ot old i te geeral case, because if it olds we ca immediately solve exactly te residual equatio o Ω. However, it gives a uderstadable defiitio of A ad te two very importat variatioal properties: Galerki Coditio: A I A I T I c I, c = ( ) = tat is eoug for basic uderstadig ad a itroductio to te multigrid metod. 4.0 Spectral Aalysis Te spectral aalysis of te restrictio ad iterpolatio operator aswers te questio of ow tese two operators act o te modes of A. Te modes of A for te oe-dimesioal model problem, as we defied tem i oe of te previous sectios, are give wit j k π wk,j = si, k,0 j. We te restrictio operator acts upo te modes of A we get:

15 g g kπ w = w k kπ w( k) = w k < smoot modes: I k cos k, oscillatory modes: I si k, Accordig to tis we ca coclude tat usig te restrictio operator te oscillatory modes o Ω caot be represeted o Ω. Tis operator trasforms tis modes ito relatively smoot modes o Ω. Te kt ad (-k)t modes o Ω, bot represet te kt mode o Ω. Defiitio 5 Te pair of fie grid modes { k, k} olds tat ( ) j + w = w. k, j k, j w w is called complemetary modes. It also Deotig {, Wk = spa wk w k}, it ca be stated tat I : { Wk spa wk }. Te same aalysis, but ow performed o te iterpolatio operator I gives: kπ kπ I wk = cos wk si w k, k <, so we ca coclude tat iterpolatio of smoot modes o Ω creates oscillatory modes o Ω. For te two-grid correctio sceme (TG) we ave: ( ) ( ) ( ) ν I I ( C( )) ν ( ) I TG ν v R v + C f + A f A R v + f e I I A A R e e As we kow from before, te error ca be expressed as a liear combiatio of te modes of A. We are ow iterested i fidig out ow TG acts o te modes of A. If we cosider TG wit o iteratios ( ν = 0 ) ad te made spectral aalysis of te operators, we ave: TGw = s w + s w k k k k k TGw k = ckwk + ckw k, k. From te equatios above it ca be see tat TG witout relaxatios elimiates te smoot modes of te error ad leaves te oscillatory modes udamped. If we iclude te relaxatios, we ave: ν ν TGwk = λkskwk + λkskw k ν ν TGw k = λ kckwk + λ kckw k, k were λ k is te eigevalue of te relaxatio metod correspodig to w k. Observig te equatios above, te fact tat te relaxatios sow te best efficiecy o te oscillatory modes ad TG aloe acts o te smoot modes, we ca coclude tat tis combiatio will elimiate bot te oscillatory ad te smoot modes of te error. 4. Algebraic Aalysis From te properties of te iterpolatio ad restrictio operator, ad te ortogoality T ( ) relatiosips betwee te subspaces of a liear operator, we get: ( I ) ( I N Rage ).

16 Te secod variatioal property te gives us: N ( I ) Rage( I ) ortogoality is used it ca be obtaied tat: N ( I A ) Rage( I A ) decompose te space Ω i te followig way: Rage( I ) ( I N ) e Ω, ca ow be represeted i te form: e = s + t, were Rage( I ) N( I ) be sow tat TG is te idetity we it acts o N( I A ) Rage( I ).. If also te otio of A-. Tis allows us to Ω = A. Eac vector s ad t A. Takig ito accout tis decompositio ad te variatioal properties, it ca, ad its ull space is exactly Overall, we got a spectral decompositio Ω = L H, were L cotais te low-frequecy modes ad H cotais te ig-frequecy modes, ad a algebraic decompositio I Ω = Rage N I A of Ω. ( ) ( ) 4. How it works? Now we are ready to see wat appes beid te curtos ad ow te multigrid metod maages to elimiate te error i a very efficiet maer. Te Figure 5 gives a very good illustratio of te process. Figure 5 Te axes i te figure correspod to te two previously described decompositios. Te vector e Ω, ca ave projectios o te four axes ad tose projectios ca be furter projected. Aalysig te pictures i Figure 5 i te directio left to rigt, bottom to top, we ca see ow te error is efficietly damped. First te relaxatio sweeps elimiate te ig-frequecy compoets of te error ( e is projected oto te L-axis), te te two-grid correctio sceme

17 elimiates te compoet of e is projected oto te N( I ) ( e alog te ( ) Rage I axis, sice tat is te ull space of TG A -axis). Te o-zero compoet of te error alog te H- axis is because TG excites oscillatory modes. Repeatig tis process efficietly elimiates te error. 4. Is everytig really tat simple? Altoug te applicatio of te multigrid metod ca be very atural ad gives very good results o some basic problems, tat is ot always te case i reality. Tere are may problems tat itroduce very difficult ad ot trivial problems i coosig specifyig te grid, coosig te operators ad so o. Some of tose problems are give i te list bellow, but it is above te scope of tis itroductio to te multigrid metod to try to solve or describe tem ito more details. Aisotropic operators ad grids Discotiuous or aisotropic coefficiets Noliear problems No-scalar PDE systems Hig order discretizatio Algebraic Turbulece models Cemicaly reactig flows Socks Small-scale sigularities Refereces William L. Briggs, Va Emde Heso, ad Steve F. McCormik. A Multigrid Tutorial. SIAM, 000 A. Bradt. Multigrid Teciques: 984 Guide wit Applicatios to Fluid Dyamics. Te Weizma Istitute of Applied Sciece, Reovot, Israel, 984