An introduction to the analysis and implementation of sparse grid finite element methods

Transcription

1 An ntroducton to the analyss and mplementaton of sparse grd fnte element methods Stephen Russell Nall Madden Verson 11: 8 th August, 015 Abstract Our goal s to present an elementary approach to the analyss and programmng of sparse grd fnte element methods. Ths famly of schemes can compute accurate solutons to partal dfferental equatons, but usng far fewer degrees of freedom than ther classcal counterparts. After a bref dscusson of the classcal Galerkn fnte element method wth blnear elements, we gve a short analyss of what s probably the smplest sparse grd method: the two-scale technque of Ln et al. [13]. We then demonstrate how to extend ths to a multscale sparse grd method whch, up to choce of bass, s equvalent to the herarchcal approach, as descrbed n, e.g., [3]. However, by presentng t as an extenson of the two-scale method, we can gve an elementary treatment of ts analyss and mplementaton. For each method consdered, we provde MATLAB code for the mplementaton, and a comparson of accuracy and computatonal costs. 1 Introducton Sparse grd methods provde a means of approxmatng functons and data n a way that avods the notorous curse of dmensonalty : for fxed accuracy, the computatonal effort requred by classcal methods grows exponentally n the number of dmensons. Sparse grd methods hold out the hope of retanng the accuracy of classcal technques, but at a cost that s essentally ndependent of the number of dmensons. An mportant applcaton of sparse grd methods s the soluton of partal dfferental equatons (PDEs) by fnte element methods (FEMs). Naturally, the soluton to a PDE s found n an nfnte dmensonal space. A FEM frst reformulates the problem as an ntegral equaton, and then restrcts ths problem to a sutable fnte-dmensonal subspace; most typcally, ths subspace s comprsed of pecewse polynomals. Ths new restrcted problem can be expressed as a matrx-vector equaton; solvng t gves the fnte element approxmaton to the soluton of the PDE. For many FEMs, ncludng those consdered n ths artcle, the FEM soluton s the best possble approxmaton (wth respect to a certan norm) of the true soluton that one can fnd n the fnte-dmensonal subspace. Therefore, the accuracy of the FEM soluton depends on the approxmaton propertes of the fnte dmensonal space. The man computatonal cost ncurred by a FEM s n the soluton of the matrx-vector equaton. Ths lnear system s sparse, and amenable to soluton by drect methods, such as LU-factorsaton, f t s not too large, or by hghly effcent teratve methods, such as multgrd methods, for larger systems. The order of the system matrx s the dmenson of the fnte-element space, and so great computatonal effcences can be ganed over classcal FEMs by constructng a fnte-dmensonal space of reduced sze wthout compromsng the approxmaton propertes: ths s what s acheved by sparse grd methods. We wll consder the numercal soluton of the followng partal dfferental equaton Lu := u + ru = f(x, y) n Ω := (0, 1), (1.1a) u = 0 on Ω. (1.1b) Here r s a postve constant and so, for smplcty, we shall take r = 1, but f s an arbtrary functon. Our choce of problem s motvated by the desre, for the purposes of exposton, to keep the settng as smple as possble, but wthout trvalsng t. The man advantages of choosng r to be constant are that no quadrature 1

2 1.1 MATLAB code 1 INTRODUCTION s requred when computng the system matrx, and that accurate solutons can be obtaned usng a unform mesh. Thus, the constructon of the system matrx for the standard Galerkn fnte element method s reduced to several lnes of code, whch we show how to do n Secton. In Secton 3 we show how to develop the two-scale sparse grd method and, n Secton 4 show how to extend ths to a multscale settng. A comparson of the accuracy and effcency of the methods s gven n Secton 5. We stress that the choce of constant r n (1.1) s only to smplfy the mplementaton of the standard Galerkn method: our mplementaton of the sparse grd methods s dentcal for varable r, and would requre only mnor modfcatons for nonunform tensor product meshes. We am to provde an ntroducton to the mathematcal analyss and computer mplementaton of sparse grds that s accessble to readers who have a basc knowledge of fnte element methods. For someone who s new to FEMs, we propose [0, Chapter 14] as a prmer for the key concepts (and sutable for an undergraduate audence). An extensve mathematcal treatment s gven n [1]: we use results from ts early chapters. We do not am to present a comprehensve overvew of the state of the art sparse grd methods: there are many mportant contrbutons to ths area whch we do not cte. However, we hope that, havng read ths artcle, and expermented wth the MATLAB programs provded, the reader wll be motvated to learn more from mportant references n the area, such as [3]. 1.1 MATLAB code All our numercal examples are mplemented n MATLAB. Snppets are presented n the text, and full source code s avalable from nall/sparsegrds. Detals of ndvdual functons are gven n the relevant sectons. For those usng MATLAB for the frst tme, we recommend [6] as a readable ntroducton. Many of the fner ponts of programmng n MATLAB are presented n [16] and [11]. A detaled study of programmng FEMs n MATLAB may be found n [8]. We make use of the freely-avalable Chebfun toolbox [5] to allow the user to choose ther own test problem, for whch the correspondng f s automatcally computed. We also use Chebfun for some computatons related to calculatng the error. (We hghly recommend usng Chebfun Verson 5 the most recent verson at tme of wrtng as some of the operatons we use are sgnfcantly faster than n earler versons). Only Test FEM.m, the man test harness for the functons that mplement the methods, uses Chebfun. That scrpt contans comments to show how t may be modfed to avod usng Chebfun, and thus run on Octave [17]. We have found the drect lnear solver ( backslash ) to be less effcent n Octave than MATLAB, and thus tmng may be qualtatvely dfferent from those presented here (though no doubt some optmsatons are possble). 1. Notaton We use the standard L nner product and norms are: (u, v) := uv dω, Ω u 0,Ω = (u, u). The blnear form, B(, ) assocated wth (1.1) s B(u, v) = ( u, v) + (ru, v), whch nduces the energy norm u B = { u 0,Ω + u 0,Ω} 1/. (1.) The space of functons whose (weak) k th dervatves are ntegrable on a doman ω s denoted by H k (ω), and f these functons also vansh on the boundary of ω t s H k 0 (ω). See, e.g., [1, Chap. ] for more formal defntons. Throughout ths paper the letter C, wth or wthout subscrpt, denotes a generc postve constant that s ndependent of the dscretzaton parameter, N, and the scale of the method k, and may stand for dfferent values n dfferent places, even wthn a sngle equaton or nequalty.

3 STANDARD GALERKIN FEM Standard Galerkn fnte element method.1 A Galerkn FEM wth blnear elements The weak form of (1.1) wth r = 1 s: fnd u H 1 0 (Ω) such that B(u, v) := ( u, v) + (u, v) = (f, v) v H 1 0 (Ω). (.1) A Galerkn FEM s obtaned by replacng H0 1 (Ω) n (.1) wth a sutable fnte-dmensonal subspace. We wll take ths to be the space of pecewse blnear functons on a unform mesh wth M equally szed ntervals n each coordnate drecton. We frst form a one-dmensonal mesh ω x = {x 0, x 1,..., x M }, where x = /M. Any pecewse lnear functon on ths mesh can be unquely expressed n terms of the so-called hat functons x x 1 f x x x 1 x < x, 1 ψ N x (x) = +1 x f x x +1 x x < x +1, (.) 0 otherwse. Smlarly, we can construct a mesh n the y-drecton: ω y = {y 0, y 1,..., y N }, where y = /N. We can form a two-dmensonal N x N y mesh by takng the Cartesan product of ω x and ω y, and then defne the space of pecewse blnear functons, V Nx,N y H0 1 (Ω) as V Nx,N y { = span ψ Nx } =1:Nx 1 (x)ψ Ny (y), (.3) =1:N y 1 where here we have adopted the compact MATLAB notaton = 1:N x 1 meanng = 1,,..., N x 1. (Later we wll use expressons such as = 1::N x 1 meanng = 1, 3, 5,..., N x 1). For sparse grd methods, we wll be nterested n meshes where N x N y. However, for the classcal Galerkn method, we take N x = N y = N. Thus, our fnte element method for (1.1) s: fnd u V such that B(u, v ) = (f, v ) for all v V. (.4). Analyss of the Galerkn FEM The blnear form defned n (.1) s contnuous and coercve, so (.4) possesses a unque soluton. Moreover, as noted n [14, Secton 3], one has the quasoptmal bound: u u B C nf u ψ B. ψ V (Ω) To complete the analyss, we can choose a good approxmaton of u n V. A natural choce s the nodal nterpolant of u. To be precse, let I : C( Ω) V be the nodal pecewse blnear nterpolaton operator that proects onto V. Snce I u V, one can complete the error analyss usng the followng classcal results. For a dervaton, see, e.g., [9]. Lemma.1. Let τ be any mesh rectangle of sze h h. Let u H (τ). Then ts pecewse blnear nodal nterpolant, I u, satsfes the bounds u I u 0,Ω Ch ( u xx 0,Ω + u xy 0,Ω + u yy 0,Ω ), (u I u) x 0,Ω Ch( u xx 0,Ω + u xy 0,Ω ), (u I u) y 0,Ω Ch( u xy 0,Ω + u yy 0,Ω ). (.5a) (.5b) (.5c) The followng results follow drectly from Lemma.1. Lemma.. Suppose Ω= (0, 1). Let u H 0 (Ω) and I u be ts pecewse blnear nodal nterpolant. Then there exsts a constant C, ndependent of N, such that Consequently, u I u B CN 1. u I u 0,Ω CN, and (u I u) 0,Ω CN 1. 3

4 .3 Implementaton of the Galerkn FEM STANDARD GALERKIN FEM It followng mmedately from these results that there exsts a constant C, ndependent of N, such that u u B CN 1. (.6).3 Implementaton of the Galerkn FEM To mplement the method (.4), we need to construct and solve a lnear system of equatons. A useful FEM program also requres ancllary tools, for example, to vsualse the soluton and to estmate errors. Because the test problem we have chosen has a constant left-hand sde, the system matrx can be constructed n a few lnes, and wthout resortng to numercal quadrature. Also, because elements of the space defned n (.3) are expressed as products of one-dmensonal functons, t should not be surprsng that the system matrx can be expressed n terms of (Kronecker) products of matrces arsng from dscretsng one-dmensonal problems. We now explan how ths can be done. We begn wth the one-dmensonal analogue of (1.1): u (x) + u(x) = f(x) on (0, 1), wth u(0) = u(1) = 1. Its fnte element formulaton s: fnd u N V N (0, 1) such that (u N, v N) + (u N, v N ) = (f, v N ) for all v N V N (0, 1). (.7) Ths u N can be expressed as a lnear combnaton of the ψ N defned n (.): u N = µ ψ N (x). =1,...,N 1 Here the µ are the N 1 unknowns determned by solvng the N 1 equatons obtaned by takng v N = ψ N(x), for = 1,..., N 1, n (.7). Say we wrte the system matrx for these equatons as (a +a 0 ), where a (usually called the stffness matrx) and a 0 (the mass matrx) are (N 1) (N 1) matrces that correspond, respectvely, to the terms (u N, v N ) and (u N, v N ). Then a and a 0 are trdagonal matrces whose stencls are, respectvely, N ( ) 1 1 and 1 6N ( For the two-dmensonal problem (1.1) there are (N 1) unknowns to be determned. Each of these are assocated wth a node on the mesh, whch we must number unquely. We wll follow a standard conventon, and use lexcographc orderng. That s, the node at (x, y ) s labelled k = + (N 1)( 1). The bass functon assocated wth ths node s φ N k = ψn (x)ψn (x). Then the (N 1) equatons n the lnear system for (.4) can be wrtten as B(u, φ N k ) = (f, φ N k ), for k = 1,... (N 1). If r n (1.1) were varable, then each entry of the system matrx, A, would have to be computed usng a sutable quadrature rule (for a general mplementaton for arbtrary r). However, for the specal case of nterest here, where r = 1, we can express t n terms of Kronecker products of the one-dmensonal matrces descrbed above: ). A = a 0 a + a a 0 + a 0 a 0. (.8) For more on Kronecker products, see, e.g., [1, Chap. 13]. The MATLAB code for constructng ths matrx s found n FEM System Matrx.m. The man computatonal cost of executng that functon s ncurred by computng the Kronecker products. Therefore, we have reduced the number of these from three to two by codng (.8) as A = kron ( a0, a ) + kron ( a+a0, a0 ) ; The rght-hand sde of the fnte element lnear system s computed by the functon FEM RHS.m. As s well understood, the order of accuracy of ths quadrature must be at least that of the underlyng scheme, n order for quadrature errors not to pollute the FEM soluton. As gven, the code uses a two-pont Gaussan quadrature rule (n each drecton) to compute b k = (f, φ N k ), for k = 1,..., (N 1). (.9) 4

5 .4 Numercal results for the Galerkn FEM STANDARD GALERKIN FEM It can be easly adapted to use a dfferent quadrature scheme. See., e.g., [9, 4.5] for hgher order Gauss-Legendre and Gauss-Lobatto rules one can use wth these elements. To compute the error, we need to calculate u u B = B(u u, u u ). However, usng Galerkn orthogonalty, snce u solves (.1) and u solves (.4), t s easy to see that B(u u, u u ) = (f, u) (f, u ). (.10) The frst term on the rght-hand sde can be computed (up to machne precson) usng Chebfun Energy = n t e g r a l ( u. f, [ 0, 1, 0, 1 ] ) ; where u and f are chebfuns that represent the soluton and rght-hand sde n (1.1). We estmate the term (f, u ) as Galerkn Energy = un b ; where un s the soluton vector, and b s the rght-hand sde of the lnear system, as defned n (.9)..4 Numercal results for the Galerkn FEM The test harness for the code descrbed n Secton.3 s named Test FEM.m. As gven, t mplements the method for N = 4, 5,..., 10, so the problem sze does not exceed what may be reasonably solved on a standard desk-top computer. Indeed, on a computer wth at least 8Gb of RAM, t should run wth N = 11 ntervals n each coordnate drecton, resolvng a total of 4, 190, 09 degrees of freedom. We present the results n Table 1, whch were generated on a computer equpped wth 3Gb of RAM, allowng to solve problems wth N = 1 ntervals n each drecton (.e., 16, 769, 05 degrees of freedom). One can observe that the method s frst-order convergent, whch s n agreement wth (.6). Ths s verfed n the dagram on the left of Fgure 1, whch compares the error for varous values of N wth the lne y = N 1. Fgure 1b shows u u for N = 64. N Errors 6.869e e e e e e e e-04 Table 1: Errors u u B, for the classcal Galerkn method appled to (1.1) 10 1 Error N 1 3 x (a) Rate of convergence y (b) u u 64, x Fgure 1: The convergence of the classcal Galerkn method (left), and the dfference between the true and computed solutons wth N = 64 (rght) 5

6 3 TWO-SCALE SPARSE GRID FEM 3 A two-scale sparse grd fnte element method The man purpose of ths artcle s to ntroduce sparse grd FEMs, whch can acheve accuracy that s comparable to the classcal Galerkn FEM of Secton, but wth much fewer degrees of freedom. As we descrbe n Secton 4, most of these methods are descrbed as multscale or mult-level. However, the smplest sparse grd method s, arguably, the two-scale method proposed by Ln et al. [13]. Those authors were motvated to study the method n order to prove certan superconvergence results. For us, ts appeal s the smplcty of ts mplementaton and analyss. By extendng our MATLAB program from Secton.3 by only a few lnes of code, we can obtan a soluton that has, essentally, the same accuracy as the classcal FEM, but usng N 4/3 rather than N degrees of freedom. Moreover, havng establshed some basc prncples of the method n ths smple settng, we are well equpped to consder the more complcated multscale method of Secton 4. Recallng Secton., the error analyss of a FEM follows drectly from establshng the approxmaton propertes of the fnte dmensonal space and, typcally, ths follows from an analyss of a partcular canddate for an approxmaton of the true soluton: the nodal nterpolant. Therefore, n Secton 3.1 we descrbe a twoscale nterpolaton operator. Ths naturally leads to a two-scale FEM, the mplementaton of whch s descrbed n Secton 3.3. That secton also contans numercal results that allow us to verfy that the accuracy of the soluton s very smlar to the FEM of Secton. Although the whole motvaton of the method s to reduce the computatonal cost of mplementng a classcal Galerkn method, we postpone a dscusson of the effcency of the method to Secton 5, where we compare the classcal, two-scale and multscale methods drectly. We wll make repeated use of the followng one-dmensonal nterpolaton bounds. Ther proofs can be found n [0, Theorem. 14.7]. Theorem 3.1. Suppose that u H (0, 1) H 1 0 (0, 1). Then the pecewse lnear nterpolant I N u satsfes u I N u 0,Ω Ch u 0,Ω, (3.1a) and u (I N u) 0,Ω Ch u 0,Ω, (3.1b) for = 1,,..., N, and h = x x The two-scale nterpolant Let V Nx ([0, 1]) be the space of pecewse lnear functons defned on the one-dmensonal unform mesh wth N x ntervals on [0, 1]. Defne V Ny ([0, 1]) n the same way. Then we let V Nx,N y be the tensor product space V Nx ([0, 1]) V Ny ([0, 1]). Let I Nx,N y : C( Ω) V Nx,N y be the nodal pecewse blnear nterpolaton operator that proects onto V Nx,N y. Wrte I Nx,0 for the nterpolaton operator that nterpolates only n the x-drecton, so I Nx,0 : C( Ω) V Nx ([0, 1]) C([0, 1]). Smlarly, let I 0,Ny : C( Ω) C([0, 1]) V Ny ([0, 1]) nterpolate only n the y-drecton. Then clearly I Nx,N y = I Nx,0 I 0,Ny = I 0,Ny I Nx,0, (3.a) x I N x,n y = I 0,Ny x I N x,0, (3.b) y I N x,n y = I Nx,0 y I 0,N y. (3.c) In ths subsecton we present an nterpretaton of the two-scale technque outlned n [14]. The two-scale nterpolaton operator Î : C( Ω) V s defned as Î u = I N,σ(N) + I σ(n),n I σ(n),σ(n), (3.3) where σ(n) s an nteger that dvdes N. The followng dentty appears n both [7] and [14], and s an ntegral component of the followng two-scale nterpolaton analyss: I u Îu = (I N,0 I σ(n),0 )(I 0,N I 0,σ(N) ). (3.4) 6

7 3. Two-scale sparse grd fnte element method 3 TWO-SCALE SPARSE GRID FEM Theorem 3.. Suppose Ω = (0, 1) and u H 1 0 (0, 1). Let I u be the blnear nterpolant of u on Ω, and Î u be the two-scale blnear nterpolant of u descrbed n (3.3). Then there exsts a constant C ndependent of N, such that Îu I u B Cσ(N) 3. Proof. Frst from (3.4), and then by (3.1a) of Theorem 3.1, one has Îu I u 0,Ω = (I N,0 I σ(n),0 )(I 0,N I 0,σ(N) )u 0,Ω Cσ(N) (I 0,N I 0,σ(N) ) u x Cσ(N) σ(n) 4 u 0,Ω x y 0,Ω = Cσ(N) 4 4 u x y Cσ(N) 4. (3.5) 0,Ω Now, followng the same reasonng, but ths tme usng (3.1b) from Theorem 3.1 we have x (Îu I u) Cσ(N) 1 (I 0,N I 0,σ(N) ) u 0,Ω x Cσ(N) 3. (3.6) 0,Ω Usng the same approach, the correspondng bound on / y(î u I u) 0,Ω s obtaned, and so (Îu I u) 0,Ω Cσ(N) 3. To complete the proof, usng the defnton of the energy norm and the results (3.5) and (3.6) one has Îu I u B Îu I u 0,Ω + (Îu I u) 0,Ω Cσ(N) 4 + Cσ(N) 3 Cσ(N) 3. Ths result combned, va the trangle nequalty, wth Theorem., leads mmedately to the followng theorem. Theorem 3.3. Let u and Î be defned as n Theorem 3.. Then there exsts a constant, C, ndependent of N, such that u Îu B C(N 1 + σ(n) 3 ). Remark 3.4. When makng a choce for σ(n), we wsh for the sparse grd method to be as economcal as possble whle stll retanng the accuracy of the classcal scheme. The result of Theorem 3.3 suggests that we can choose σ(n) such that N 1 = Cσ(N) 3. Thus we choose σ(n) = N 1/3. Corollary 3.5. Takng σ(n) = N 1/3 n Theorem 3.3, there exsts a constant, C, such that u Îu B CN Two-scale sparse grd fnte element method Let ψ N(x) and ψn (y) be defned as n (.). We now let V H0 1 (Ω) be the fnte dmensonal space gven by { } =1:N 1 { } =1:σ(N) 1 V = span ψ N (x)ψ σ(n) (y) + span ψ σ(n) (x)ψ N (y). (3.7) =1:σ(N) 1 =1:N 1 Now the FEM s: fnd û V such that B(û, v ) = (f, v ) v V. (3.8) The same reasonng that lead to (.6), Theorem 3.3 can be appled to gve the followng error estmate. Theorem 3.6. Let u be the soluton to (1.1), and û the soluton to (3.8). Then there exsts a constant C, ndependent of N, such that u û B C(N 1 + σ(n) 3 ). In partcular, takng σ(n) = N 1/3, u û B CN 1. 7

8 3.3 Implementaton of the two-scale method 3 TWO-SCALE SPARSE GRID FEM 3.3 Implementaton of the two-scale method At frst, constructng the lnear system for the method (3.8) may seem somewhat more dauntng than that for (.4). For the classcal method (.4), each of the (N 1) rows n the system matrx has (at most) nne non-zero entres, because each of the bass functons shares support wth only eght of ts neghbours. For a general case, where r n (1.1) s not constant, and so (.8) cannot be used, then the matrx can be computed ether from a 9-pont stencl for each row, whch ncorporates a sutable quadrature rule for the reacton term, or by teratng over each square n the mesh, to compute contrbutons from the four bass functons supported by that square. In contrast, for any choce of bass for the space (3.7), a sngle bass functon wll share support wth O(σ(N)) others, so any stencl would be rather complcated (ths s clear from the sparsty pattern of a typcal matrx shown n Fgure 4). Further, determnng the contrbuton from the O(σ(N)) bass functons that have support on a sngle square n a unform mesh appears to be non-trval. However, as we shall see, one can borrow deas from Multgrd methods to greatly smplfy the process by constructng the lnear system from entres n the system matrx for the classcal method. We begn by choosng a bass for the space (3.7). Ths s not qute as smple as takng the unon of the sets { ψ N (x)ψ σ(n) (y) } =1:N 1 =1:σ(N) 1 and { ψ σ(n) (x)ψ N (y) } =1:σ(N) 1 =1:N 1 snce these two sets are not lnearly ndependent. There are several reasonable choces of a bass for the space. Somewhat arbtrarly, we shall opt for { } =1:N 1 { } =1:σ(N) 1 ψ N (x)ψ σ(n) (y) ψ σ(n) (x)ψ N (y). (3.9) =1:σ(N) 1 =(1:N 1)/(σ(N):σ(N):N σ(n)) Ths may be nterpreted as takng the unon of the usual blnear bass functons for an N σ(n) mesh, and a σ(n) N mesh; but from the second of these, we omt any bass functons assocated wth nodes found n the frst mesh. For example, f N = 7, these bass functons can be consdered to be defned on the meshes shown n Fgure, where each black dot represents the centre of a blnear bass functon that has support on the four adacent rectangles., (a) A sutable grd for { ψ 7 (x)ψ 3 (y) } =1:6 =1: (b) A sutable grd for { ψ 3 (x)ψ 7 (y) } =1: =(1:6)/{9,18} Fgure : Meshes for the two-scale FEM, for N = 7 To see how to form the lnear system assocated wth the bass (3.9) from the entres n (.8), we frst start wth a one-dmensonal problem. Let V σ(n) be the space of pecewse lnear functons defned on the mesh ω σ(n) x, and whch vanshes at the end-ponts. So any v V σ(n) can be expressed as v(x) = v ψ σ(n) (x), =1:σ(N) 1 8

9 3.3 Implementaton of the two-scale method 3 TWO-SCALE SPARSE GRID FEM Suppose we want to proect v onto the space V N. That s, we wsh to fnd coeffcents w 1,..., w N 1 so that we can wrte the same v as v(x) = w ψ N (x), =1:N 1 Clearly, ths comes down to fndng an expresson for the ψ σ(n) n terms of the ψ N. Treatng the coeffcents (v 1,..., v σ(n) ) and (w 1,..., w N ) as vectors, the proecton between the correspondng spaces can be expressed as a (N 1) (σ(n) 1) matrx. Ths matrx can be constructed n one lne n MATLAB: p = sparse ( nterp1 ( x, eye ( length ( x ) ), x ) ) ; where x s a unform mesh on [0, 1] wth N ntervals, and x s a unform mesh on [0, 1] wth N ntervals, where N = σ(n) s a proper dvsor of N. To extend ths to two-dmensons, we form a matrx P 1 that proects a blnear functon expressed n terms of the bass functons n { } =1:N 1 ψ N (x)ψ σ(n) (y) to one n V usng =1:σ(N) 1 P1 = kron ( p ( : end 1,:end 1), speye (N 1)); Next we construct the matrx P that proects a blnear functon expressed n terms of the bass functons n { ψ σ(n) } =1:σ(N) 1 (x)ψ N (y), =(1:N 1)/(σ(N):σ(N):N σ(n)) to one n V. Part of ths process nvolves dentfyng the nodes n ωx N Therefore, we use two lnes of MATLAB code to form ths proector: whch are not contaned n ω σ(n) x. UnqueNodes=sparse ( s e t d f f ( 1 : ( N 1), N/N :N/N :N N/N ) ) ; P = kron ( sparse ( UnqueNodes, 1 : length ( UnqueNodes ), 1 ), p ( : end 1,:end 1)); The actual proector we are lookng for s now formed by concatenatng the arrays P 1 and P. That s, we set P = (P 1 P ). For more detals, see the MATLAB functon TwoScale Proector.m. In a Multgrd settng, P would be referred to as an nterpolaton or prolongaton operator, and P T s known as a restrcton operator; see, e.g., []. It should be noted that, although our smple constructon of the system matrx for the classcal Galerkn method reles on the coeffcents n the rght-hand sde of (1.1) beng constant, the approach for generatng P works n the general case of varable coeffcents. Also, the use of the MATLAB nterp1 functon means t s a trval exercse to adust t to work for non-unform meshes. Equpped wth the matrx P, f the lnear system for the classcal Galerkn method s Au = b, then the lnear system for the two-scale method s (P T AP )û = P T b. The soluton can then be proected back onto the orgnal space V by evaluatng P û. To use the test harness, Test FEM.m, to mplement the two-scale method for our test problem, set the varable Method to two-scale on Lne 18. In Table we present results for the two-scale method that correspond to those n Table 1 for the classcal method of Secton. Snce n Theorem 3.6 we take σ(n) = N 1/3, for smplcty we have chosen the values of N so that they are perfect cubes. Hence they don t all correspond to values used n Table 1. However, from the log-log plot of Fgure 3a we can see that the result of Theorem 3.6 s verfed n practce: the method s ndeed frst-order convergent n the energy norm. Moreover, the errors for the two-scale method are very smlar to those of the classcal (the dfference s to the order of 1%) even though far fewer degrees of freedom are nvolved. For example, when N = 1 the classcal FEM nvolves 16,769,05 degrees of freedom, compared wth 1,65 for the two-scale method. However, comparng Fgure 1b and Fgure 3b, we see that the nature of the pont-wse errors are very dfferent. We defer further comparsons between the methods to Secton 5, where effcency of the varous methods s dscussed n detal. 9

10 4 MULTISCALE SPARSE GRID FEM N Errors 6.1e e e e e e e e-04 Table : Errors u û B, for the two-scale method appled to (1.1) 10 1 Classcal FEM Error Two scale FEM N x (a) Rate of convergence y 0 0 (b) u û 64,64 x Fgure 3: The convergence of the classcal and two-scale methods (left), and the dfference between the true and computed two-scale soluton wth N = 64 (rght) nz = nz = 87 Fgure 4: Sparsty patterns for the two-scale method wth N = 7 ordered as (3.9) on the left, and by lexcographcal orderng on the rght. 4 A multscale sparse grd fnte element method We have seen that the two-scale method can match the accuracy of the classcal FEM, even though only O(N 4/3 ) degrees of freedom are used, rather than O(N ). We shall now see that t s possble to further reduce the requred number of degrees of freedom to O(N log N), agan wthout sacrfcng the accuracy of the method very much. The approach we present s equvalent, up to the choce of bass, to the herarchcal sparse grd method descrbed by, for example, Bungartz and Grebel [3]. But, because we present t as a generalsaton of the two-scale method, we lke to refer to t as the multscale method. Informally, the dea can be summarsed n the followng way. Suppose that N = k for some k. For the two-scale method, we solved the problem (n a sense) on two overlappng grds: one wth N N 1/3 ntervals, and one wth N 1/3 N. Instead, we could apply the same algorthm, but on grds wth N (N/) and (N/) N ntervals respectvely. Next we apply ths same two-scale approach to each of these two grds, gvng three overlappng grds wth N (N/4), (N/) (N/), and (N/4) N ntervals. The process s repeated recursvely, untl the coarsest grds have ntervals n one coordnate drecton the smallest feasble number. 10

11 4.1 The multscale nterpolant 4 MULTISCALE SPARSE GRID FEM More rgorously, we begn n Secton 4.1 by constructng a multscale nterpolant, whch s then analysed n Secton 4.. As wth the two-scale method, ths leads to a FEM, descrbed n 4.3. In Secton 4.4 we show how ths can be programmed, and present numercal results for our test problem. Throughout the analyss, we use the followng denttes, whch are easly establshed usng, for example, nductve arguments. Lemma 4.1. For any k 1 we have the followng denttes = k 1, and = 1 k. 4.1 The multscale nterpolant Much of the secton s taken from [15, 3.1]. Let I denote the pecewse blnear nterpolaton operator on V. Consder the followng two-scale nterpolaton technque (3.3): I (1) = I N,σ(N) + I σ(n),n I σ(n),σ(n), (4.1) where σ(n) s an nteger that dvdes N. In partcular, n Corollary 3.5, σ(n) = CN 1/3 s presented as a sutable choce. However, the same approach can be appled to, say, the terms I N,σ(N) and I σ(n),n, and agan recursvely to the terms that emerge from that. Followng the approach that s standard for multscale sparse grd methods, we let σ(n) = N/. Let I Nx,N y denote the pecewse blnear nterpolaton operator on V Nx,N y. The correspondng Level 1 operator s I (1) N x,n y := I Nx, Ny + I Nx,N y I Nx, Ny. (4.a) The postvely sgned terms of (4.a) are assocated wth spaces of dmenson O(N /) whle the negatvely sgned term s assocated wth a space of dmenson O(N /4). The Level operator s obtaned by applyng the Level 1 operator to the postve terms n (4.a), gvng I () = I(1) N, N + I (1) N,N I N, N = I N, N 4 + I N, N + I N 4,N I N, N 4 I N 4,. (4.b) N Note that the rght-hand sde of ths expresson features three (postvely sgned) operators that map to subspaces of dmenson O(N /4), and two (negatvely sgned) operators that map to subspaces of dmenson O(N /8). To obtan the Level 3 operator, we agan apply the Level 1 operator to the postve terms n (4.b), because they are the ones assocated wth the larger spaces. In general, the Level k operator s obtaned by applyng the Level 1 operator of (4.a) to the postve terms n I (). Ths leads to the followng defnton. Defnton 4. (Multscale Interpolaton Operator). Let I (0) = I I I N,N N. For k =, 3,..., let I (k), N sgned terms n I (). and, from (4.a), let I(1) = I + N, N be obtaned by applyng the Level 1 operator n (4.a) to the postvely We now provde an explct formula for I (k). Ths recovers a standard expresson used, for example, n the defnton of the combnaton technque outlned n [18]. Lemma 4.3. Let I (k) by the multscale nterpolaton operator constructed n Defnton 4. above. Then I (k) = k k I N, N I N k =1, N, for k = 0, 1,,.... (4.3) k+1 Proof. It s easy to check that the formula (4.3) s consstent wth the constructon gven n Defnton 4. for k = 0 and k = 1. Next, assume that (4.3) holds for an arbtrary k = n; that s, I (n) = n n I N, N I N n =1, N. (4.4) n+1 11

12 4.1 The multscale nterpolant 4 MULTISCALE SPARSE GRID FEM Followng Defnton 4., apply the Level 1 operator (4.a) to each term n the frst sum of the rght-hand sde of (4.4). Ths gves n = [ I (n+1) I N, N + I N n+1 +1, N n I N n = I N n+1,n + [ That s, (4.3) holds for k = n + 1, as requred. n ] +1, N I N n+1 =1, N n+1 ] I N, N I N n+1 +1, N n+1 n+1 = n+1 I N, N I N n+1 =1, N. n+ Although (4.3) s a succnct representaton of the multscale nterpolaton operator, for the purposes of analyss we are actually nterested n the dfference between operators at successve levels. Frst consder the dfference between the nterpolaton operators at Levels 0 and 1: I (0) I(1) = I I N, N I N,N + I N,. N Recallng (3.a) and (3.4) we can that ths expresson can be wrtten as I (0) I(1) = (I N,0 I 0,N ) (I N,0 I 0, N ) (I N,0 I 0,N ) + (I N,0 I ) 0, N ) ( = (I N,0 I N,0 an dentty used repeatedly n [14]. One can derve a smlar expresson I (1) I() : I (1) I() = I N, N Invokng (3.a) and smplfyng we get I (1) I() =(I N,0 I 0, N + I N,N I N, N I N, N 4 I N, N ) + (I N,0 I 0,N ) (I N,0 I 0, N I N 4,N + I N, N 4 ) (I N,0 I 0, N ) 4 I 0,N I 0, N + I N 4,. N (I N,0 I ) (I 0, N N 4,0 I 0,N ) + (I N,0 I ) + (I 0, N N 4 4,0 I ), 0, N ) ( ) ) ( ) = (I N,0 I N,0 I 0, N I 0, N + (I N 4,0 I N4,0 I 0,N I 0, N. ), (4.5) We now gve the general form for the representaton of the dfference between operators at successve levels n terms of one-dmensonal operators. The resultng dentty s an mportant tool n our analyss of the bound on the error for the multscale nterpolaton operator. Lemma 4.4. Let I (k) be the multscale nterpolaton operator defned n (4.3). Then, for k = 0, 1,,..., I () I(k) = ( ) ( I N,0 I N +1,0 ) I 0, N I 0, N k Proof. From the expresson for the multscale operator n (4.3) we can wrte I () I(k) = I N, N I N =1, N I N, N + I N k =1, N k+1 ( ) ( = I N, N I N, N I N, k N I N k, N ( ) = I N, N I N, N + I N k +1, N I N k, N ( ) = I N, N I N, N + I N k +1, N k whch, recallng (4.5), gves the desred expresson. k ( = k k + I N k,n I N, N I N, N ), (4.6) + I N +1, N k + I N, k N I N k k,n I N +1, N k + I N +1, N k I N ) +1, N, 1

13 4. Analyss of the multscale nterpolaton operator 4 MULTISCALE SPARSE GRID FEM 4. Analyss of the multscale nterpolaton operator In ths secton we provde an analyss for the error ncurred by the multscale nterpolant. Standard fnte element analyss technques then provde a full analyss of the underlyng method. We begn by establshng a bound, n the energy norm, for the dfference between nterpolants at successve levels. Theorem 4.5. Suppose Ω = (0, 1). Let u H0 1 (Ω) and I (k) be the mult-scale nterpolaton operator defned n (4.3). Then there exsts a constant, C, ndependent of N and k, such that I (k) u I() u B C(k4 k+1 N k+1 N 3 ). Proof. By statng I (k) u I() u 0,Ω n the form of Lemma 4.4 and applyng the trangle nequalty, along wth (3.1a) of Theorem 3.1 and Lemma 4.1 we see I (k) u I() u 0,Ω = ( ) N ( C +1 I 0, (I N,0 I N +1,0)(I 0, N I 0, N )u k 0,Ω ) N I u N 0, k x C 0,Ω ( ) ( ) N N 4 u +1 k x y 0,Ω C ( + )( k )N 4 = Ck4 k+1 N 4. (4.7) Usng a smlar argument to (4.7), except usng (3.1b) rather than (3.1a) of Theorem 3.1 we have x ( (I N,0 I N +1,0)(I 0, N I 0, N k )u ) 0,Ω ( ) 1 N C 0 +1 (I 0, N I 0, N ) u k x C4 k+1 N 3. (4.8) 0,Ω The correspondng bound on the y-dervatves s obtaned n a smlar manner. Combnng these results gves (I (k) u I() u) 0,Ω C4 k+1 N 3. Now followng from the defnton of the energy norm and results (4.7) and (4.8) one has I (k) u I() u B (I (k) u I() u) 0,Ω + I (k) u I() u 0,Ω C4 k+1 N 3 + Ck4 k+1 N 4. We now want to show that I (k) u I() u B s an upper bound for I () u I u B, thus showng that n, order to estmate I u I (k) u B, t s enough to look at I (k) u I() u B. Lemma 4.6. Let u and I (k) N and k, such that be defned as n Theorem 4.5. Then there exsts a constant, C, ndependent of I () u I u B I (k) u I() u B. Proof. Takng the result of Theorem 4.5 and by applyng an nductve argument one can easly deduce that 4 +1 k4 k+1 and = k+1. =1 The result then follows drectly by applyng the trangle nequalty and observng that I () u I( 1) u B (4 +1 N N 3 ). =1 =1 13

14 4.3 Multscale sparse grd fnte element method 4 MULTISCALE SPARSE GRID FEM Lemma 4.7. Let u and I (k) N and k, such that be defned as n Theorem 4.5. Then there exsts a constant, C, ndependent of I (k) u I u B C4 k+1 N 3. Proof. By the trangle nequalty and the results of Theorem 4.5 and Lemma 4.6 one has I (k) u I u B I (k) u I() u B + I () u I u B C(k4 k+1 N k+1 N 3 ). The result now follows by notng that, for k N (whch wll always be the case), 4 k+1 N 3 k4 k+1 N 4. Usng the trangle nequalty, we combned ths result wth Theorem. and 4.7 to establsh the followng error estmate. Theorem 4.8. Let u and I (k) N and k, such that be defned as n Theorem 4.5. Then there exsts a constant, C, ndependent of u I (k) u B C(N k+1 N 3 ) Remark 4.9. Our prmary goal s to construct an effcent fnte element method, by takng the smallest possble space on whch I (k) u can be defned. Thus we want to take k as large as s possble whle retanng the accuracy of the underlyng method. On a unform mesh the coarsest grd must have at least two ntervals n each coordnate drecton. From (4.3) t s clear that the coarsest grd we nterpolate over has N/ k ntervals. Thus the largest and most useful value of k we can choose s k = log N 1. In lght of ths, we have the followng result. Corollary Let u and I ( k) be defned as n Theorem 4.5. Takng k := k = log (N) 1 n Theorem 4.8, there exsts a constant, C, ndependent of N, such that u I ( k) u B CN Multscale sparse grd fnte element method Let ψ N and ψ N be as defned n (.) We now defne the fnte dmensonal space V (k) H1 0 (Ω) as V (k) = span { ψ N (x)ψ N/k } =1:N 1 { (y) + span =1:N/ k span {ψ N/ (x)ψ N/ ψ N/ (y) (x)ψ N/ } =1:N/ 1 =1:N/ 1 } =1:N/ 1 (y) =1:N/ 1 + span {ψ N/k (x)ψ N (y) } =1:N/ k 1 =1:N 1 Note that each two dmensonal bass functon s the product of two one-dmensonal functons that are of dfferent scales. Ths descrpton nvolves (k + 1) k N + ( k+1 4)N + k + 1 functons n the spannng set, whch are llustrated n the dagrams n Fgure 5. The left most dagram shows a unform mesh wth N = 16 ntervals n each coordnate drecton. Each node represents a bass functon for the space V (0) = V. In the centre column of Fgure 5 we show the grds assocated wth V (1). Notce that the spaces assocated wth these two grds are not lnearly ndependent. To form an nvertble system matrx, n Fgure 5 we hghlght a partcular choce of non-redundant bass functons by sold crcles: { ψ N (x)ψ N/ (y) } =1::N 1 =1:N/ 1 and { ψ N/ (x)ψ N (y) The rght-most column of Fgure 5 show the grds assocated wth V () represent our choce of bass: { ψ N (x)ψ N/4 (y) } =1::N 1 =1:N/4 1, { ψ N/ (x)ψ N/ (y) } =1:N/ 1 =1:N/ 1, and { } =1:N/ 1. ψ N/4 =1:N 1. Agan we use sold crcles to (x)ψ N (y) } =1:N/4 1 =1::N 1 There are many ways n whch one can choose whch bass functons to nclude. The way we have chosen, excludng the boundary, s as follows:.. 14

15 4.3 Multscale sparse grd fnte element method 4 MULTISCALE SPARSE GRID FEM Fgure 5: Meshes for the multscale method for N = 16, based on the spaces (left to rght) V (0), V (1) and V () when N x > N y nclude every second bass functon n the x-coordnate drecton; when N x = N y or when N x = N y nclude all bass functons; from the remanng subspaces nclude every second bass functon n the y-coordnate drecton. In general the choce of bass we make for the space V (k) s odd the bass we choose s: ()/ l=0 {ψ N/l } =1::N/ l 1 ψ N/k l =1:N/ k l 1 {ψ N/(k+1)/ And when k s even the bass we choose s: s dependent on whether k s odd or even. When k } =1:N/ (k+1)/ 1 ψ N/()/ =1:N/ ()/ 1 k l=(k+3)/ {ψ N/l } =1:N/ l 1 ψ N/k l ; (4.9a) =1::N/ k l 1 k/ 1 l=0 {ψ N/l } =1::N/ l 1 ψ N/k l =1:N/ k l 1 {ψ N/k/ } =1:N/ k/ 1 ψ N/k/ =1:N/ k/ 1 k l=k/+1 {ψ N/l } =1:N/ l 1 ψ N/k l. (4.9b) =1::N/ k l 1 Ths has dmenson k (k/ + 1)N N + 1. For a computer mplementaton one can, of course, choose alternatve ways of expressng the bass. Although these are mathematcally equvalent, they can lead to dfferent 15

16 4.4 Implementaton of the multscale method 4 MULTISCALE SPARSE GRID FEM lnear systems. We can now formulate the multscale sparse grd fnte element method: fnd u (k) V (k) such that B(u (k), v ) = (f, v ) We wll consder the analyss for the case k := k = log N 1. for all v V (k). (4.10) Theorem Let u be the soluton to (1.1), and u ( k) the soluton to (4.10). Then there exsts a constant C, ndependent of N and ε, such that u u ( k) B CN 1. Proof. The blnear form (4.10) s contnuous and coercve, so t follows from classcal fnte element analyss that u u ( k) B C nf ψ V ( k) (Ω) u ψ B. Snce I (k) u V ( k) (Ω) the result follows as an mmedate consequence of Corollary Implementaton of the multscale method We now turn to the constructon of the lnear system for the method (4.10). As wth the constructon of the lnear system for (3.8), we begn by buldng matrces that proect from the one-dmensonal subspaces of V N to the full one-dmensonal space V N. In MATLAB ths s done wth the followng lne of code px = sparse ( nterp1 ( Sx, eye ( length ( Sx ) ), x ) ) ; where x s a unform mesh on [0, 1] wth N ntervals, and S x s a submesh of x wth N/M ntervals and M s a proper dvsor of N. Next we buld a collecton of two-dmensonal proecton matrces, P. Each of these proects a blnear functon expressed n terms of each of the bass functons n (4.9), to one n V. Ths s done usng the MATLAB functon MultScale Proector.m and the followng pece of code: P = kron ( py ( :N, : s k p y :My), px ( :N, : s k p x :Mx) ) ; Here M x and M y are the number of ntervals n the coarse x- and y-drectons respectvely. The values skp x and skp y are set to ether 1 or dependng on whether we nclude all bass functons, or every other bass functon, n (4.9). The proectors n the collecton are concatenated to gve the proector from V (k) to U That s P = (P 1 P... P k P k+1 ). As s the case wth the two-scale method, havng constructed P and gven the lnear system for the classcal Galerkn method, Au = b, the lnear system for the multscale method s (P T AP )u (k) = P T b. Evaluatng P u (k) then proects the soluton back to the orgnal space V. To use the test harness, Test FEM.m, to mplement the multscale method for our test problem, set the varable Method to multscale on Lne 18. MultScale Proector.m. In Table 3 we present numercal results for the multscale method. These results correspond to those n Table 1 for the classcal Galerkn method of Secton, as well as those of Table for the two-scale method of Secton 3. From Fgure 6a we can see that, n practce, the CN 1 term from the theoretcal results of Theorem 4.11 s observed numercally. The errors for the multscale method are very smlar to those of the classcal Galerkn method and the two-scale method, however the degrees of freedom are greatly reduced. We saw n Secton 3.3, that for N = 1 the classcal Galerkn FEM nvolves 16, 769, 05 degrees of freedom, compared to 1, 65 for the two-scale method. Ths s further reduced to 45, 057 degrees of freedom for the multscale method (takng k = log N 1). Agan comparng Fgure 1b, Fgure 3b and Fgure 6b we see that although the nature of the pont-wse errors are qute dfferent, they are all smlar n magntude. 16

17 4.5 Herarchcal bass 4 MULTISCALE SPARSE GRID FEM 10 1 Classcal FEM Error Two scale FEM Multscale FEM N x y x (a) Rate of convergence (b) u u (5) 64,64 Fgure 6: The convergence of the classcal, two-scale and multscale methods (left), and the dfference between the true soluton and multscale soluton wth N = 64 and k = 5 (rght). N Errors 7.100e e e e e-03.19e e e-04 Table 3: Errors u u (k) B, for the multscale method appled to (1.1) 4.5 Herarchcal bass The standard choce of bass used n the mplementaton of a sparse grd method s generally the herarchcal bass ([1]), whch s qute dfferent from that gven n (4.9). The most standard reference for sparse grds s the work of Bungartz and Grebel [3], so we adopt ther notaton. The herarchcal bass for the space V (Ω), whch s defned on the full grd, s wrtten as where W m,n = m,n=1,...,log N {ψ m ψ n W m,n, (4.11) } =1:: m 1 =1:: n 1 Ths s shown n Fgure 7 for the case N = 8. The herarchcal bass s constructed from 9 subspaces. Recall that each dot on these grds represents a bass functon wth support on the adacent four rectangles. For example, W 11 s shown n the top left, and represents a smple bass functon wth support on the whole doman. In the subspace W 33 n the bottom rght of Fgure 7 each bass functon has support wthn the rectangle on whch t s centred. No two bass functons n a gven subspace share support wthn that subspace. A sparse grd method s constructed by omttng a certan number of the subspaces W m,n n (4.11). For a typcal problem, one uses only those subspaces W m,n for whch m + n log N + 1: compare Fgures 3.3 and 3.5 of [3]. Ths gves the same spaces as defned by (4.9), takng k = log N 1. Expressed n terms of a herarchcal bass, the bass for V (k) s wrtten as W m,n. (4.1) m,n=1,...,log N m+n log N+1 Both choces gve rse to bass functons centred at the same grd ponts (but wth dfferent support) and have lnear systems of the same sze, but dfferent structure: see Fgure 9 and Fgure 10. It s known that ths bass leads to reduced sparsty of the lnear system (see, e.g., [10, p. 50]). Numercal experments have shown that the bass gven n (4.9) leads to fewer non-zeros entres n the system matrx, and so s our preferred bass. In Fgure 9 we compare the sparsty pattern of the system matrces for the bass descrbed n (4.9) for the case N = 3 usng a natural orderng (left) and lexcographcal orderng (rght). For the same value of N, Fgure 10 shows the sparsty patterns for the system matrces constructed usng the herarchcal. 17

18 5 COMPARISON OF THREE FEMS N = N = 4 N = 8 W 11 W 1 W 31 W 1 W 13 Fgure 7: Left: All subspaces requred to buld a herarchcal bass for a full grd where N = 8. Rght: Full grd wth bass functons for N = 8. N = N = 4 N = 8 W 11 W 1 W 31 W 1 W W 13 Fgure 8: Left: All subspaces requred to buld a herarchcal bass for a sparse grd where N = 8. Rght: Sparse grd wth bass functons for N = 8. bass (4.1), agan usng natural (left) and lexcographcal orderng (rght). The matrces of Fgure 10 contan more nonzero entres and are much denser than those of Fgure 9. 5 Comparson of three FEMs In Sectons.4, 3.3, and 4.4, we presented results that demonstrated that the three methods acheve smlar levels of accuracy. We now wsh to quantfy f, ndeed, the sparse grd methods are more effcent than the classcal method. To do ths n a thorough fashon would requre a great deal of effort to nvestgate effcent storage of matrces arsng from these specalsed grds; nvestgate the desgn of sutable precondtoners for teratve technques, or Multgrd methods. These topcs are beyond the scope of ths artcle. Instead, we apply the smplest possble black-box solvers: a drect solvers (and so avod concerns nvolvng precondtoners and stoppng crtera), we wll compare the methods wth respect to the wall-clock tme taken by MATLAB s backslash solver. Observng the dagnostc nformaton provded (spparms( spumon,1)), we have verfed that ths defaults to the CHOLMOD solver for symmetrc postve defnte matrces [4]. The results we present were generated on a sngle node of a Beowulf cluster, equpped wth 3 Mb RAM and two AMD Opterons 47, 00 MHz processors, each wth 6 cores. The effcency of parallelsed lnear algebra 18

19 6 CONCLUSIONS nz = nz = 3777 Fgure 9: Sparsty patterns for the multscale method usng the bass descrbed n (4.9) wth N = 3. On the left, unknowns are ordered as (4.9), and on the rght by lexcographcal orderng nz = nz = 5897 Fgure 10: Sparsty patterns for the multscale method usng a herarchcal bass (4.1) method wth N = 3. On the left, unknowns are ordered as n (4.1), and on the rght by lexcographcal orderng. routnes can be hghly dependent on the matrx structure. Therefore we present results obtaned both usng all 1 cores, and usng a sngle core (enforced by launchng MATLAB wth the -snglecompthread opton). All tmes reported are n seconds, and have been averaged over three runs. In Table 4 we can see that, for N = 1, over a thousand seconds are requred to solve the system for the classcal method on a sngle core, and 560 seconds wth all 1 cores enabled. (It s notable, that, for smaller problems, the solver was more effcent when usng ust 1 core). When the two-scale method s used, Table 4 shows that there s no great loss of accuracy, but solve tmes are reduced by a factor of 3 on 1 core and (roughly) a factor of 5 on 1 cores. Employng the multscale method, we see a speed-up of (roughly) 7 on one core, and 15 on 1 cores. 6 Conclusons There are many papers publshed n the lterature on the topc of sparse grd methods, partcularly over the last two decades. Often though, they deal wth hghly specalzed problems, and the analyss tends to be drected at readers that are hghly knowledgeable n the felds of dfferental equatons, fnte element methods and functonal analyss. One of the goals of ths paper has been to present a way of analysng sparse grd methods n a way that s more accessble to a more general audence. We have also provded snppets of code and detals on how to mplement these methods n MATLAB. 19

20 6 CONCLUSIONS Classcal Galerkn N Error 4.93e e e e-04 Solver Tme (1 core) Solver Tme (1 cores) Degrees of Freedom 61,11 1,046,59 4,190,09 16,769,05 Number of Non-zeros,343,961 9,406,489 37,687,31 150,87,089 Two-scale method N Error 4.668e e e e-04 Solver Tme (1 core) Solver Tme (1 cores) Degrees of Freedom 7,105 17,901 5,560 1,65 Number of Non-zeros 1,646,877 6,588,773 33,34,03 118,756,013 Multscale method N Error 4.437e-03.19e e e-04 Solver Tme (1 core) Solver Tme (1 cores) Degrees of Freedom 4,097 9,17 0,481 45,057 Number of Non-zeros 996,545 3,944,897 15,55,569 61,585,473 Table 4: A comparson of the effcency of the classcal, two-scale, and multscale FEMs Classcal FEM Error Two scale FEM Multscale FEM Classcal FEM Error Two scale FEM Multscale FEM Fgure 11: Solve tme, n seconds, for the lnear system assocated wth the classcal, two-scale, and multscale methods, usng a drect solver on 1 core (left) and 1 cores (rght) The methods presented here are not new. However, by treatng the usual multscale sparse grd method as a generalzaton of the two-scale method, we have presented an conceptually smple, yet rgorous, way of understandng and analysng the method. Here we have treated ust a a smple two-dmensonal problem on a unform mesh. However, the general approach that we present has been employed to analyse sparse grd methods for specalsed problems whose solutons feature boundary layers, and that are solved on layer-adapted meshes, see [15, 19]. More nterestng generalsatons are possble, but the most mportant of these s to hgher dmensonal problems. An exposton of the ssues nvolved s planned. 0