Capter Grid Transfer Remark. Contents of tis capter. Consider a grid wit grid size and te corresponding linear system of equations A u = f. Te summary given in Section 3. leads to te idea tat tere migt be an iterative metod for solving tis system efficiently, wic uses also coarser grids. In order to construct suc a metod, one needs mecanisms tat transfer te information in an appropriate way between te grids.. Te CoarseGrid SystemandteResidual Equation Remark. Basic idea for obtaining a good initial iterate wit a coarse grid solution. One approac for improving te beavior of iterative metods, at least at te beginning of te iteration, consists in using a good initial iterate. For te model problem, one can try to find a good initial iterate, e.g., by solving te problem approximately on a coarse grid, using only a few iterations. Te application of only a few iterations is called smooting, and te iterative metod itself smooter, since only te oscillating error modes (on te coarse grid) are damped. Te solution from te coarse grid can be used as initial iterate on te fine grid. Remark.3 Study of te discrete Fourier modes on different grids. Given a grid Ω. Inpractice, auniformrefinementstepconsistsindividinginalvesallintervals of Ω, leading to te grid Ω. Ten, te nodes of Ω are te nodes of Ω wit even numbers, see Figure.. 0 3 5 6 7 8 Ω 0 3 Ω Figure.: Coarse and fine grid. Consider te k-t Fourier mode of te fine grid Ω. If k N/, ten it follows for te even nodes tat ( ) ( ) jkπ jkπ wk,j = sin = sin = w N N/ k,j, j =,..., N. 9
Hence, te k-t Fourier mode on Ω is te k-t Fourier mode on Ω. From te definition of te smoot and oscillating modes, Remark 3.7, it follows tat by going from te fine to te coarse grid, te k-t mode gets a iger frequency if l N/. Note again tat te notion of frequency depends on te grid size. Te Fourier mode on Ω for k = N/ is represented on Ω by te zero vector. For te transfer of te oscillating modes on Ω, i.e., for N/ < k < N, one obtains a somewat unexpected results. Tese modes are represented on Ω as relativelysmootmodes. Tek-tmodeonΩ becomestenegativeofte(n k)- t mode on Ω (exercise?): w k,j w N k,j ( ) ( ) jkπ jkπ = sin = sin, N N/ ( ) ( ) j(n k)π j(n k)π = sin = sin N/ N ( jnπ = sin N jkπ ) N ( ) ( ) jkπ jkπ = sin(jπ) cos +cos(jπ) sin }{{} N }{{} N =0 = ( ) jkπ = sin, N i.e., wk,j = w N k,j. Tis aspect sows tat it is necessary to damp te oscillating error modes on Ω before a problem on Ω is considered. Oterwise, one would get additional smoot error modes on te coarser grid. Remark. Te residual equation. AniterativemetodfortesolutionofAu = f can be applied eiter directly to tis equation or to an equation for te error, te so-called residual equation. Let u (m) be an approximation of u, ten te error e (m) = u u (m) satisfies te equation Ae (m) = f Au (m) =: r (m). (.) Remark.5 Nested iteration. Tis remark gives a first strategy for using coarse grid problems for te improvement of an iterative metod for solving Au = f. Tis strategy is a generalization of te idea from Remark.. It is called nested iteration: solve A 0 u 0 = f 0 on a very coarse grid approximately by applying a smooter,. smoot A u = f on Ω, solve A u = f on Ω by an iterative metod wit te initial iterate provided from te coarser grids. However, tere are some open questions wit tis strategy. How are te linear systems defined on te coarser grids? Wat can be done if tere are still smoot error modes on te finest grid? In tis case, te convergence of te last step will be slowly. Remark.6 Coarse grid correction, two-level metod. A second strategy uses also te residual equation (.): Smoot A u = f on Ω. Tis step gives an approximation v of te solution wic still as to be updated appropriately. Compute te residual r = f A v. 0
Project (restrict) te residual to Ω. Te result is called R(r ). Solve A e = R(r ) on Ω. Wit tis step, one obtains an approximation e of te error. Project (prolongate) e to Ω. Te result is denoted by P(e ). Update te approximation of te solution on Ω by v := v +P(e ). Tis approac is called coarse grid correction or two-level metod. Wit tis approac, one computes on Ω an approximation of te error. However, also for tis approac one as to answer some questions. How to define te system on te coarse grid? How to restrict te residual to te coarse grid and ow to prolongate te correction to te fine grid?. Prolongation or Interpolation Remark.7 General remarks. Te transfer from te coarse to te fine grid is called prolongation or interpolation. In many situations, one can use te simplest approac, wic is te linear interpolation. For tis reason, tis section will only consider tis approac. Example.8 Linear interpolation for finite difference metods. For finite difference metods, te prolongation operator is defined by a local averaging. Let Ω be divided into N/ intervals and Ω into N intervals. Te node j on Ω corresponds to te node j on Ω, 0 j N/, see Figure.. Let v be given on Ω. Ten, te linear interpolation I : R N/ R N, v = I v is given by vj = vj, j =,...,N/, vj+ = ( v j +vj+ ) (.), j = 0,...,N/, see Figure.. For even nodes of Ω, one takes directly te value of te corresponding node of Ω. For odd nodes of Ω, te aritmetic mean of te values of te neigbor nodes is computed. 0 3 5 6 7 8 Ω I 0 3 Ω Figure.: Linear interpolation for finite difference metods. Te linear prolongation is a linear operator, see below Lemma.0, between two finite-dimensional spaces. Hence, it can be represented as a matrix. Using te
standard basis of R N/ and R N, ten I =... R (N ) (N/ ). (.3)... Example.9 Canonical prolongation for finite element metods. Consider conforming finite element metods and denote te spaces on Ω and Ω wit V and V, respectively. Because Ω is a uniform refinement of Ω, it follows tat V V. Hence, eac finite element function defined on Ω is contained in te space V. Tis aspect defines a canonical prolongation I : V V, v v. Te canonical prolongation will be discussed in detail for P finite elements. Let {ϕ i } N/ i= be te local basis of V and {ϕ i }N i= be te local basis of V. Eac function v V as a representation of te form v (x) = N/ i= v i ϕ i (x), v i R, i =,...,N/. Tere is a bijection between V and R N/. Let j = i be te corresponding index on Ω to te index i on Ω. From te property of te local basis, it follows tat Inserting tis representation gives v (x) = ϕ i = ϕ j +ϕ j + ϕ j+. N/ i= = v +v ( vi ϕ i +ϕ i + ϕ i+ ( ϕ +ϕ + ) ϕ 3 ( ) ϕ 3 +ϕ + ϕ 5 ( +v3 ϕ 5 +ϕ 6 + ) ϕ 7 +... From tis formula, one can see tat te representation in te basis of V is of te following form. For basis functions tat correspond to nodes wic are already on Ω (even indices on te fine grid), te coefficient is te same as for te basis function on te coarser grids. For basis functions tat correspond to new nodes, te )
coefficient is te aritmetic mean of te coefficients of te neigbor basis functions. Hence, if local bases are used, te coefficients for te prolongated finite element function can be computed by multiplying te coefficients of te coarse grid finite element function wit te matrix (.3). Lemma.0 Properties of te linear interpolation operator. Te operator I : RN/ R N defined in (.) is a linear operator. It as full rank and only te trivial kernel. Proof: i) I is a linear operator. Te operator is omogeneous, since for α R and v R N/ it is vj = (αv) j = αv j, vj+ = ) ((αv) j +(αv) j+ = α Te operator is additive. Let v,w R N/, ten ( ) ( ) I(v+w) = (v+w) j = v j +w j = I(v) j ( ) I(v+w) j+ (vj +vj+). j ( ) + I(w), j = ((v+w)j +(v+w)j+) = (vj +vj+)+ ( ) ( ) = I(v) + I(w). j+ j+ (wj +wj+) An omogeneous and additive operator is linear. ii) I as full rank and trivial kernel. Since N/ < N, bot properties are equivalent. Let 0 = v = I(v ). From (.) it follows from te vanising of te even indices of v immediately tat vj = 0, j =,...,N/, i.e., v = 0. Hence, te only element in te kernel of I is te zero vector. Remark. Effect of te prolongation on different error modes. Assume tat te error, wic is of course unknown is a smoot function on te fine grid Ω. In addition, te coarse grid approximation on Ω is computed and it sould be exact in te nodes of te coarse grid. Te interpolation of tis coarse grid approximation is a smoot function on te fine grid (tere are no new oscillations). For tis reason, one can expect a rater good approximation of te smoot error on te fine grid. If te error on te fine grid is oscillating, ten eac interpolation of a coarse grid approximation to te fine grid is a smoot function and one cannot expect tat te error on te fine grid is approximated well, see Figure.3. Altogeter, te prolongation gives te best results, if te error on te fine grid is smoot. Hence, te prolongation is an appropriate complement to te smooter, wic works most efficiently if te error is oscillating..3 Restriction Remark. General remarks. For te two-level metod, one as to transfer te residual from Ω to Ω before te coarse grid equation can be solved. Tis transfer is called restriction. Example.3 Injection for finite difference scemes. Te simplest restriction is te injection. It is defined by I : R N R N/, v = I v, v j = v j, j =,..., N, see Figure.. For tis restriction, one takes for eac node on te coarse grid simply te value of te grid function at te corresponding node on te fine grid. 3
0.5 oscillating error interpolant for exact grid function 0. 0.3 0. 0. value 0 0. 0. 0.3 0. 0.5 0 0. 0. 0.6 0.8 x Figure.3: Oscillating error and interpolation. 0 3 5 6 7 8 Ω I 0 3 Ω Figure.: Injection. Itturnsout tat teinjectiondoes not lead to anefficient metod. Ifoneignores every oter node on Ω, ten te values of te residual in tese nodes, and wit tat also te error in tese nodes, do not possess any impact on te system on te coarse grid. Consequently, tese errors will generally not be corrected. Example. Weigted restriction for finite difference scemes. Te weigted restriction uses all nodes on te fine grid. It is defined by an appropriate averaging I : R N R N/, v = I v, v j = ( v j +v j +v j+), j =,..., N, (.) see Figure.5. For finite difference scemes, only te weigted restriction will be considered in te following. IftespacesR N andr N/ areequippedwittestandardbases, tematrix
0 3 5 6 7 8 Ω I 0 3 Ω Figure.5: Weigted restriction. representation of te weigted restriction operator as te form I = R (N/ ) (N ). (.5)... Wit tis representation, one can see an important connection between weigted restriction I and interpolation I : I = ( I ) T. Lemma.5 Properties of te weigted restriction operator. Let te restriction operator I given by (.). Tis operator is linear. Te rank of tis operator is N/ and te kernel as dimension N/. Proof: i) Linearity. exercise. ii) Rank and kernel. From linear algebra, it is known tat te sum of te dimension of te kernel and te rank is N. Te rank of I is equal to te dimension of its range (row rank). Te range of I is equal to R N/, since every vector from R N/ migt be te image in te space of grid functions of Ω of a vector corresponding to grid functions of Ω. Hence, te rank is N/ and consequently, te dimension of te kernel is N (N/ ) = N/. Example.6 Canonical restriction for finite element scemes. Wereas for finite difference metods, one works only wit vectors of real numbers, finite element metods are imbedded into te Hilbert space setting. In tis setting, a finite element function is, e.g, from te space V, but te residual, wic is te rigt-and side minus te finite element operator applied to a finite element function (current iterate) is from te dual space ( V ) of V. In tis setting, it makes a difference if one restricts an element from V or from its dual space. For restricting a finite element function from V to V, one can take te analogon of te weigted restriction. If local bases are used, ten te coefficients of te finite element function from V are multiplied wit te matrix (.5) to get te coefficients of te finite element function in V. In te two-level metod, one as to restrict te residual, i.e., one needs a restriction from ( V ) ( ) to V. In tis situation, a natural coice consists in using te dual prolongation operator, i.e., I : ( V ) ( V ), I = ( I ). 5
Te dual operator is defined by I v,r V,(V ) = v,i r V,(V ) v V,r ( V ). Tus, if local bases and te bijection between finite element spaces and te Euclidean spaces are used, ten te restriction of te residual can be represented by te transposed of te matrix (.3). Tis issue makes a difference of a factor of compared wit te matrix for te weigted restriction. 6