Crouzeix-Velte Decompositions and the Stokes Problem

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1 Crouzeix-Velte Decompositions and te Stokes Problem PD Tesis Strauber Györgyi Eötvös Loránd University of Sciences, Insitute of Matematics, Matematical Doctoral Scool Director of te Doctoral Scool: Dr. Laczkovic Miklós Professor, Member of te Hungarian Academy of Sciences Applied Matematics Doctoral Program Director of te program: Dr. Micaletzky György Professor, Doctor of te Hungarian Academy of Sciences Tesis supervisor: Dr. Stoyan Gisbert Professor, Eötvös Loránd University of Sciences, Faculty of Informatics, Department of Numerical Analysis 2008

2 1 Abstract In our dissertation te staggered grid approximation of te Stokes problem on several special domains and wit dierent boundary conditions is investigated, using nite dierence and box metods. (for nite element metod see: Girault-Raviart, 1986) We deal wit metods were te discrete Crouzeix- Velte decomposition (see: Stoyan-Strauber-Baran, 2004; or Subsection 1.2.) exists: we prove te existence of tis decomposition and sow numerical results to prove te eectiveness of tese discretizations. 1.1 Te Stokes problem In our dissertation we consider at rst te following rst-kind Stokes problem in a cartesian coordinate system: u + grad p = f, in Ω, (1.1) div u = 0, in Ω, (1.2) were Ω IR n, n = 2, 3 is a bounded, simply-connected open domain, wit Lipscitz continuous boundary Ω, and u(x) = (u 1 (x),..., u n (x)) T, f(x) = (f1 (x),..., f n (x)) T, dened for x = (x 1,..., x n ) Ω. On te boundary, omogeneous Diriclet boundary conditions are imposed: u = 0, on Ω. (1.3) Te problem consists in nding a vector-function u(x) and a scalar function p(x) tat satisfy te system of partial dierential equations above. Te function u is te velocity of uid and p is te (kinematic) pressure. For a constant also appearing in te rst equation, te kinematic viscosity, we ave cosen te value 1. Finally, f describes accelerations caused by an external force eld, and te boundary conditions mean tat te walls are impermeable and at rest. A unique weak solution u V and p P exists wen, for example, f (L 2 (Ω)) n, (see, e.g.,varnorn,1994), were P := L 2,0 (Ω) is te subspace of L 2 (Ω) of square integrable functions wit zero integral over Ω, wit te Hilbert space L 2 (Ω) = {φ (φ, φ) < }, (φ, ψ) = φψdω and were V := (H 1 0(Ω)) n is te well-known Sobolev space, wit generalized derivatives in (L 2 (Ω)) n and wit zero boundary values in te sense of traces on te boundary Ω. Ω 1

3 1.2 Te CrouzeixVelte decomposition For an n > 2, let (, ) denote te Euclidean scalar product in IR n, moreover, let A, B, C IR n n be matrices satisfying A = B + C, (1.4) A = A T > 0, (1.5) B = B T 0, C = C T 0, (1.6) δ := dim ker B 1, ρ := dim ker C 1. (1.7) Ten, wit a suitable subspace W IR n (wic may turn out to be empty) and wit ortogonality to be understood in te sense of te scalar product (A, ), te following ortogonal decomposition of IR n can be derived: IR n = ker B ker C W. (1.8) Tis decomposition is called te algebraic CrouzeixVelte decomposition of IR n. (Te denition of te analytical CrouzeixVelte decomposition is described in te dissertation.) For te eigenvalues λ i of te generalized eigenvalue problem we ave λax = Bx (1.9) λ i [0, 1] for all i (1.10) and also using te eigenvectors x (i), te subspaces in (1.8) can be caracterized as follows: ker B = span(x (i), λ i = 0), ker C = span(x (i), λ i = 1), W = span(x (i), λ i (0, 1)). We can use te following correspondences connected wit te matrices in (1.4): A, B grad div, C curl rot, were is te (vector) Laplace operator and were te sign expresses only an analogy between a dierential operator and a matrix, and is not necessarily a (good) approximation. In tis sense (1.4) corresponds to te well known identity = grad div + curl rot (1.11) 2

4 of vector analysis. Te algebraic CrouzeixVelte decomposition is proper in case dim W = n δ ρ > 0. In te discrete case, te velocity space (wic approximates (H 1 (Ω)) n or a subspace of te latter) will be denoted by V, te pressure space will be denoted by P, and div and rot will be written for te discrete equivalents of te divergence and rotation operator, will denote te discrete Laplace operator. Te matrix corresponding to te mapping div from te velocity space into te pressure space is denoted by B and we introduce te following notations: C for te matrix of te operator rot and A for te matrix of te operator. If A, B, C matrices satisfy te following: A = B + C, A = A T > 0, (1.12) were B = B T B and C = C T C and ker B and ker C, ten a discrete Crouzeix-Velte decomposition exists and (1.8) takes te form V = ker div ker rot W = V,0 V,1 V,β. P is decomposed similarly into tree ortogonal subspaces: P = ker grad div ker rot div V,β = P,0 P,1 P,β. After discretization by te nite element or nite dierence metods, te Stokes problem takes te following form: ( ) ( ) ( ) A BT u f =. (1.13) B 0 p 0 Wit tese matrices (1.9) corresponds to λ A x = B x, wic is transformed by p := B x into λ p = S p, were S is te discrete Scur complement operator of te Stokes problem, S := B A 1 B T. If te discrete Crouzeix-Velte decomposition exists ten λ [0, 1]. In our dissertation we deal wit approximations of te Stokes problem on several domains, were te discrete Crouzeix-Velte decomposition exists. In 3

5 tis case in te spectrum of te discrete Scur complement operator, tere is an eigenvalue 1 of ig multiplicity (of te order of inner grid points). Ten, te error components lying in te eigensubspace corresponding to tis eigenvalue can be removed by one step of a simple damped Jacobi iteration or Uzawa-algoritm. Te remaining error lies in an eigensubspace of muc smaller dimension connected only wit boundary eects. Moreover, te conjugate gradient iteration automatically takes advantage of suc a spectrum and converges faster tan for discrete spaces witout a decomposition: as it is proved by te numerical results. In our dissertation te following metods are used: nite dierence and box metods for approximation of Stokes-problem, and Uzawa-algoritm (as outer iteration) in te numerical experiments, moreover te conjugate gradient metod (as inner iteration) wit an eective preconditioning matrix and FFT algoritm and alternatively multigrid metod solving te discrete Poisson equations. 2 Results 2.1 First order staggered grid approximation on nonequidistant rectangular grids First we consider te well-known staggered-grid approximation were Ω is a rectangle subdivided by a non-equidistant grid into (n 1)(m 1) rectangular cells. Te boundary conditions are omogeneous Diriclet boundary conditions. For pressure vectors p and velocity vectors u = (u, v ) T suitable discrete scalar products and te corresponding norms are introduced. Using tese scalar products and norms te following Teorem is proved, wit te notations A, B, C for te matrices of te operators, div ill rot : Teorem 1 (A u, u ) 0, = B u 2 0, + C u 2 0, (2.1) olds for all vectors u := (u, v ) T V if and only if 1,i+1/2 = 1,i 1/2 + 1,i+3/2 2 and 2,j+1/2 = 2,j 1/2 + 2,j+3/2. 2 4

6 Remark 1. It is proved tat dim(v 0 ) = (n 2) 2, dim(v 1 ) = (n 3) 2 and dim(v,β ) = 4n 9, were n denotes te number of grid points (including corner points) along a side of te square. Remark 2. We prove tat A is symmetric in te sense of te scalar product ( u, v ) 0, above, and te following equation olds: D A A = B T D B B + C T D C C, (2.2) were D A, D B, D C are diagonal matrices corresponding to te adequate norms. Tat is D A A =: Â = B + C, (2.3) were B = ˆB T ˆB és C = ĈT 1/2 Ĉ wit te notations: ˆB = D B, Ĉ = D 1/2 C. Remark 3. We prove tat D A A = Â matrix is positive denite, tat is Â > 0. Togeter wit remark 1 and 2, it means tat a proper CrouzeixVelte decomposition of te velocity and te pressure space into tree nontrivial parts exists, if n > First order staggered grid approximation based on te nite volume (box) metod on non-equidistant rectangular grids Ten we apply nite volume metod on te staggered grid, were Ω is also a rectangle subdivided by a non-equidistant grid into (n 1)(m 1) rectangular cells. Te boundary conditions are omogeneous Diriclet boundary conditions. We prove tat using tis approximation, te discrete CrouzeixVelte decomposition exists, if n > 3, witout any restriction of grid spacing. Remark Te results above (using eiter nite dierence or nite volume metods) old if Ω is a union of rectangles suc tat all te boundary lines of te dierent rectangles t on te same global grid wit grid spacing 1 and 2. 5

7 2.3 Numericul results We sow wit numerical experiments - using te staggered grid approximation based on te nite volume metod - tat bot te Uzawa-type and te conjugate gradient-type metods are faster on suc grids wic are condensing in te center and coarser near te boundary of te domain. In te case of Uzawa-type metods we determine te optimal non-equidistant grid. 2.4 Second order staggered grid approximation based on te nite dierence metod on equidistant rectangular grids Ten we apply second order nite dierence metod on te staggered grid, were Ω is a unit square subdivided by an equidistant grid. Te boundary conditions are omogeneous Diriclet boundary conditions. After dening te suitable approximations, discrete scalar products and te corresponding norms, te following Teorem is proved: Teorem For te second order staggered grid approximation, (A u, u ) 0, B u 2 0, C u 2 0, = n 1 n 1 = 2(u 2 i,1 + u 2 i,n 1) 2(v1,j 2 + vn 1,j), 2 (2.4) i=2 were A :=, B := div and C := rot. Remark We introduce te notation Ã, were j=2 (Ã u n 1 n 1, u ) 0, = (A u, u ) 0, + 2(u 2 i,1 + u 2 i,n 1) + 2(v1,j 2 + vn 1,j). 2 i=2 Ã is a symmetric positive denite matrix in te sense of te corresponding scalar product, togeter wit A. Since dim(v 0 ) = (n 2) 2, dim(v 1 ) = (n 3) 2 and dim(v,β ) = 4n 9, a proper CrouzeixVelte decomposition exists in tis case as well, for n > 3. In tis case te algebraic decomposition exists not for te matrix A,but for Ã, ence Ã can be advantageous as a preconditional matrix solving A u = b. 2.5 Finite dierence approximation in te case of nonstandard boundary conditions We investigate te nite dierence approximation of te Stokes problem on te staggered grid wit te following non-standard boundary conditions: 6 j=2

8 1 i n 1, 1 j n 1 esetén, és u 1,j = u n,j = v i,1 = v i,n = 0, (2.5) (rot u ) 0,j = (rot u ) n 1,j = (rot u ) i,1 = (rot u ) i,n = 0, (2.6) 0 i n 1, 2 j n 1, were rot is dened as usual. (Tis boundary condition satises te Lopatinski-condition.) We sow tat a proper discrete CrouzeixVelte decomposition exists in te case of tis boundary condition as well, using eiter rst or second order approximation. 2.6 Finite dierence approximation in te case of periodical boundary conditions As we sow in tis point, te results on te existence of an analytical Crouzeix-Velte decomposition for te Stokes problem along wit Diriclet and periodical boundary conditions carry over to te discrete case for te staggered grid approximation. We apply rst order nite dierence approximation on te staggered grid, were Ω is a unit square subdivided by an equidistant grid. Periodical boundary conditions are assumed on te left and rigt sides of te unit square: u 1,j = u n 1,j, u 2,j = u n,j, 0 j n, (2.7) v 1,j = v n 1,j, v 2,j = v n,j, 1 j n. On te upper and lower sides of te unit square we prescribe omogeneous Diriclet conditions: u i,0 = u i,n = 0, 1 i n, v i,1 = v i,n = 0, 1 i n 1, were u i,0, u i,n are values on two additional grid lines, wic te grid as been supplemented wit. We prove tat a proper discrete CrouzeixVelte decomposition exists in tis case as well. 7

9 2.7 Approximation on a nonequidistant grid in 3D wit omogeneous Diriclet boundary conditions Ten te well-known dierence approximation on a staggered grid is considered in 3D case. In our case Ω is a rectangular parallelepipedon subdivided by a rectangular grid into (n 1)(m 1)(l 1) cells of volume eac, 1 := 1/(n 1), 2 := 1/(m 1), 3 := 1/(l 1). We assume n, m, l 3. First we assume omogeneous Diriclet boundary conditions: u 1,j,k = u n,j,k = v i,1,k = v i,m,k = w i,j,1 = w i,j,l = 0, were u := (u, v, w ) T is te velocity vector and 1 i n 1, 1 j m 1, 1 k l 1. We prove te existence of te discrete CrouzeixVelte decomposition. 2.8 Approximation on an equidistant grid in 3D wit periodical boundary conditions Ten we consider te rst order staggered grid approximation on an equidistant, cubic grid. On te front-back sides and on te nort-sout sides of te cube we prescribe omogeneous Diriclet conditions, and periodical boundary conditions are assumed on te east-west sides. We prove te existence of te discrete CrouzeixVelte decomposition in tis case as well. 2.9 Te Stokes problem in polar coordinates for te disk domain Finally let Ω be te unit disk Ω = {(r, ϕ) 0 r < 1, 0 ϕ < 2π}, and consider te following Stokes problem: rϕ u u r 2 v 2 r 2 ϕ p r rϕ v v r + 2 u p 2 r 2 div u = 1 r were (u, v) = u and (f 1, f 2 ) = f and rϕ = 1 r = f 1, (2.8) ϕ 1 r ϕ ) = f 2, (2.9) = 0, (2.10) ( v (ru) + r ϕ r ( r r 8 ) + 1r 2 2 ϕ 2.

10 On te boundary, omogeneous Diriclet boundary conditions are imposed: u = 0, on Ω. (2.11) Te problem consists in nding a vector-function u(x) and a scalar function p(x) tat satisfy te system of partial dierential equations above. In our dissertation we consider a suitable second order nite dierence approximation and introduce te corresponding discrete scalar products and norms. Te following Teorem - similarly to (2.1) - is proved: Teorem (Ã u, u ) 0, = B u 2 0,r, + C u 2 0,r, (2.12) olds for all vectors u := (u, v ) T V, were B corresponds to te negative divergence operator, C corresponds to te rotation operator and Ã corresponds to te negative Laplace operator, wic is in polar coordinates: u = ( rϕ u u 2 v r 2 r 2 ϕ rϕ v v + 2 u r 2 r 2 ϕ ). (2.13) Remark From te Teorem above we obtain tat Ã can be written in te following form: DÃÃ =: A = B + C, (2.14) were B = ˆB T ˆB, C = ĈT Ĉ and ˆB = D 1/2 B B, Ĉ = D 1/2 C C and were DÃ, D B, D C are diagonal matrices corresponding to te adequate scalar products Numerical results for approximation on te unit disk Uzawa-algoritm Using te notations (2.14) our problem consists in nding te solution of te following algebraic system: A u + ˆB T p = f, (2.15) ˆB u = g. (2.16) 9

11 We use te Uzawa-algoritm to solve (2.15), (2.16): p (0) := 0, p (i+1) := p (i) + ω( ˆB u (i) g ) (2.17) u (i) := A 1 ( f ˆB T p (i) ) i = 0, 1, 2,.... Since te discrete Crouzeix-Velte decomposition exists, using te Uzawaalgoritm we can reac te tird Crouzeix-Velte subspace after 1 step (wit ω = 1). In tis subspace te spectrum of te Scur complement is closer, and te algoritm sows eective convergence. Te optimal iteration parameter is also calculated using te smallest and te largest of te eigenvalues dierent from 0 and 1 of te discrete Scur complement. Te numerical results sow tat te discretization obeying a discrete CrouzeixVelte decomposition leads to eectively solvable systems of algebraic equations. Te average number of te iterations using te Uzawa algoritm is 3-5, and te speed of convergence is growing togeter wit te renement of te grid Fourier transformation and conjugate gradient metod Instead of te calculation of A 1 in (2.17) in te rst case te fast Fourier transformation is used in combination wit te preconditioned conjugate gradient metod. In te numerical experiments several preconditioning matrices are investigated: te number of inner iterations needed to reac te stopping criterion of te conjugate gradient metod, using te best preconditioning matrix, is only 4-5, and te speed of convergence is growing togeter wit te renement of te grid Multigrid metod In te second case te multigrid metod is used to calculate A 1. We describe te restriction operator and optimize te prolongation operator wit numerical experiments. We compare several pre- and post-smooting iterations. Te optimal iteration parameter of te damped Jacobi iteration - as smooting iteration - is also calculated. Te results sow tat te Gauss- Seidel iteration combined wit block Gauss-Seidel iteration - as smooting iteration - is te most eective: te multigrid metod using tis iteration is on average 5-8 times faster tan using te Gauss-Seidel iteration and tis advantage is growing wit te renement of te grid. Te less eective smooting iteration is te Jacobi iteration, it results on average tree times slower convergence tan te Gauss-Seidel iteration. 10

12 3 Publications Strauber, Gy.: Discrete Crouzeix-Velte decompositions on nonequidistant rectangular grids. Annales Univ. Sci. Budapest.,44 (2002), Stoyan, G., Strauber, Gy., Baran, A.: Generalizations to discrete and analytical Crouzeix-Velte decompositions. Numer. Lin. Algebra wit Appls., 11, (2004), Strauber, Gy.: Discrete Crouzeix-Velte decomposition for te disk domain. Miskolc Matematical Notes, Vol.6 (2005), No. 1, pp

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