The element separation property and parallel finite element methods for the Navier-Stokes equation

Appl. Math. Left. Vol. 8, No. 4, pp. 97-102, 1995 Pergamon 0893-9659(95)00056-9

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T h e E l e m e n t Separation P r o p e r t y and Parallel Finite E l e m e n t M e t h o d s for the N a v i e r - S t o k e s E q u a t i o n W . LAYTON Department of Mathematics, University of Pittsburgh Pittsburgh, PA 15260 wj l~vms, cis. pitt. edu

P. RABIER Department of Mathematics, University of Pittsburgh Pittsburgh, PA 15260, U.S.A. Rabier©vms. cis. pitt. edu

(Received July 1993; accepted August 1993) A b s t r a c t - - T h i s report considers the amount of parallelism attainable in certain robust domain decomposition methods for linearizations of the Navier-Stokes equations. The connection between parallelism and the finite element basis is shown in terms of the separation property of that basis, introduced in Section 2. Section 3 studies these questions in particular for the Navier-Stokes equations. Section 3 resolves the separation properties of common, low order spaces which satisfy the inf-sup condition, as well as introduces a new, low order space which has the (optimal) element separation property.

1. I N T R O D U C T I O N In the parallel solution of nonlinear systems arising from finite element discretizations of the boundary value problem, iterative methods which uncouple the global problem into subproblems upon smaller subdomains are frequently the methods of choice. Unfortunately, popular domain decomposition methods for symmetric problems are often not sufficiently robust for application to the Navier-Stokes equations at a moderate to high Reynolds number. "Robustness," as usual, means uniformity in the problem parameters--in this case uniform--in-Re-convergence of the iterative domain decomposition method. Such Uniform ,'. ;. Robust methods are very scarce. Kellogg [1] gives one for the model convection-diffusion (C-D) equation which uses precise enclosure regions for the spectrum of the discretization matrix. Axelsson, Eijkhout, Polman and Vassilevski [2] give a block preconditioning technique for a special discretization of the model C-D equation on a structured mesh which appears uniform in extensive experiments. Layton and Rabier [3-5] give a novel domain decomposition method which is observed (and proven in [6]) to be uniform for general finite element discretizations, general unstructured meshes, etc., as well as being element-wise data-parallel. These techniques were extended to the Navier-Stokes equations in [4], which also introduced a new family of stable, element-wise parallel, low-order elements for Navier-Stokes problems. Implicit in this work on domain-decomposition procedures is the limitations imposed by the finite element space chosen. This report examines those limitations by formulating "separation" Partially supported by NWO Grant B 61-222. "Pypeset by ~4.A/cS-TEX 97

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W . LAYTON AND P. RABIER

properties of the chosen basis for a finite element space; Definitions 2.1-2.3. The separation property of the finite element space determines the grain fineness of the parallelism attainable in the algorithms of [3-5]. Section 3 applies these ideas to the Navier-Stokes equations, the main thrust of this report. A new, low order element is announced which has the (optimal) element separation property. Other popular, low order elements are also considered from this point of view, all satisfy at least a strip-separation property. As for comparing the elements, the nonconforming linear, constant pair of Crouzeix and Raviart [7] is the simplest for incompressible flows as well as the most highly parallel. The "MINI" element of Arnold, Brezzi and Fortin [8] is the simplest conforming element which may be used for both compressible and incompressible flow problems. If nonconforming elements are allowed, the element of Layton and Rabier [4] may be used for both compressible and incompressible flow problems and allows a degree of parallelism comparable to the Crouzeix-Raviart element.

2. F O R M U L A T I O N

OF THE SEPARATION PROPERTY

will denote a polyhedral domain in ]~d (d = 1, 2, 3) covered with a finite element mesh ~h(~). e shall denote a generic (closed) element (triangle of quadrilateral in ~2, tetrahedron or brick in ~3) in 7rh(~-~). sh(~~) denotes a finite element space with basis (I) = { ¢ 1 , . . . , (~N}DEFINITION 2.1. Let w be the interior of a collection of elements and let w l , . . . ,O)L be the connected components of w. We say that the basis (I) separates the connected components of w if, for every i, 1 < i < N , one has ¢i[~l ~ 0

implies

¢i[~z' - 0,

forallw~,w~,,1 < l' < L , V ¢1. EXAMPLE 2.1. S h : = C o piecewise linears on D -- (0,1). Let D -- (0,1) with elements ej ( x j , x j + l ) where x0 = 0 and xg+l = 1. Color these alternately "RED" if j is even and "BLACK" if j is odd. Let 0.)RED/BLACK : {ej : color (ej) is R E D / B L A C K (resp.)}, and let (I) = { ¢ 0 , . . . , ¢ N + 1 } be the standard nodal basis, Cj(xi) = 5ij (Kronecker delta). It then readily follows that (I) separates the connected components of WRED as well as those of ¢dBLACK. =

EXAMPLE 2.2. S h := C O piecewise bilinears on ~ = (0, 1) x (0, 1). Let D = (0, 1) × (0, 1) be divided into a uniform square mesh with Ax = Ay = 1 / ( N + 1), giving 7rh(f~). Let S h := C o piecewise bilinears on 7rh(12) and color the horizontal strips alternately RED and BLACK. T h a t is, an element e = ( x j , x j + l ) x (Yk,Yk+l) has color RED if k is even and BLACK if k is odd. Let ~dRED/BLACK :={e E 7i'h(~'~) : e is R E D / B L A C K (resp.)}, the connected components are then strips wz :=(0,1) x (Yk,Yk+l) for 0 < k < N. If (I) = {¢jk,0 < j < N + 1,0 < k < N + 1} is the standard nodal basis for S h, Cjk (x j,, Yk') ----6jj,6kk, , the (I) separates the connected components of WRED as well as those of 5dBLACK. DEFINITION 2.2. A two-coloring of ~rh(f~) is an association of each e E 7rh(D) with one of two colors, traditionally "RED" or"BLACK". Let ~rh(f~) be two colored, with LdRED = interior {Ue : e 7rh(~'~) and color (e) = RED}, analogously for wBLACK Let w R, wa wlB, WBLB be the connected components of cdRED and 0.) B L A C K respectively. Then, the two coloring oflrh(f~) is said to be a C

.

.

.

.

'

L R '

" " " '

(i) strip coloring of ~rh(D) if for all j, k with j # k and colors R / B , respectively, dimension (W~/ B IIw - " "---~' k ) =dim(g):---I, (ii) element coloring of 7rh(f~) if for ali j, k with j ~ k and colors R / B , respectively, dimension ~ R / B A ,.,---h-~ ~'k ] < _ d - 2 .

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99

As examples in 2-D, a "checkerboard" coloring is an element coloring while stripes would be a strip coloring. See [9] for more on optimal coloring of unstructured meshes. DEFINtTION 2.3. A basis ¢ has the strip separation property if for every strip coloring of Trh(~), separates the connected components of both WRED and WBLACK. ~ has the element separation property if for every element coloring of ~rh(~), q) separates the connected components of both 09RED and 0)BLAC K.

Element-wise parallel algorithms have long been recognized as offering the ultimate in flexibility, storage efficiency and speed of execution, see, e.g., the pioneering work of Hughes et al. [10] on the "element-by-element" preconditioning method. The possibility of formulating the element-wise data parallel algorithms of Layton and Rabier [3-5] depends critically upon • possessing the element separation property. These algorithms are fast, data-parallel [5,11,12] easy to program and robust [5,6]. Similarly, stripwise data parallel versions of the algorithms in [3-5] depend upon the basis • possessing the strip separation property. 3. A P P L I C A T I O N

TO

THE

NAVIER-STOKES

EQUATIONS

Consider the solution of the linearized Navier-Stokes equations: given U such that V o U = 0, -Re-1A

u + U o V u + a u + V_p = f , inl2,

Vou=0,

i n ~ , u = 0 o n c~t, [ p d x = O , Ja

(3.1)

where a >_ 0 is nonzero for time dependent problems. For example, if u = u(tn+l) in (3.1). U is frequently and extrapolation over previous time levels, U = (3/2)u(tn) - (1/2)U(tn_l), and au = At-lu(t~+l). The standard Galerkin FEM formulation of (3.1) is: choose finite element spaces (X_h, M h) and calculate (uh,p h) • (X h, M h) satisfying

[a~(uh,v)+(Ph,V°V)e--(f,v)e]

=0,

f o r a l l v E X h,

(q, V o uh)¢ = 0,

for all q • M h,

(3.2)

where (q,V o w)~ = f~qV o wdx, and (choosing, for illustration, the convective form of the nonlinear term)

dx.

ae(u,v):--jf[Re-lVu:Vv+UoVuov+auov]

The stability of the method (3.2) in the graph norm is equivalent, [13], to the spaces (X__h, M h) satisfying the celebrated "inf-sup" or LBB condition: for some constant ~ independent of h, inf

qeMh

sup

,ex"

Ilqlioivll

>/~ > O,

-

Ilqllo 2 :=

Iql2 dx,

Iv_IT:=

e "~

IVvl 2dx.

(3.3)

Thus (3.3) is an essential constraint upon the spaces (X__h, M h) chosen for (3.2), see [13] for more detail. Given the finite element triangulation r h ( ~ ) , let pk(e) denote the set of all polynomials of degree < k on e. Define Sh(~):={¢(x) E C°(~'~) : (~ I e • p l ( e ) , for all e • 71"h(~'~)}. ~1,)~2 and "~3 will denote the barycentric coordinates of e a n d Ce°(x) -- )~1)~2)~3 the cubic bubble function on e [8,14]. Further, s h c ( g t ) shall denote the space of nonconforming, linear Crouzeix-Raviart [7] elements. The basis functions of Shc(~) are associated with mid-edge nodes on 7rh(~t).

100

w . LAYTON AND P. RABIER

Define

oh

SNC := {¢ • Shc(f~) : ¢(N) = 0 for all mid-edge nodes N on cOD},

and ~h ( ~ ) : = ( ¢ • sh(D) : ¢(N) = 0 for all vertex nodes N on cOD}. Note that functions in S h c ( D ) have mean value zero on each edge on cOD but are not generally pointwise zero on 0f~. We consider the following low order elements ( X h, M h) for (3.2) with regard to their separation properties. Here Z h :~(~h

~ B3) 2, M

h := S h f~ L~(f~) are associated with the nodes: X h ~

,

Qh ~

(3.4a)

The "MINI" element, of Arnold, Brezzi and Fortin [8]. Here __Xh

"--k'--[~hNc/h2,M h : = { ¢ :

¢]~ • P ° ( e ) for all e • 71"h(D)}CI L02(D) are associated with the

nodes: X h ~

,

Qh ~

(3.4b)

The Crouzeix-Raviart [7] pair. Here, let zrh(D) be constructed from 7rh°(~) by one uniform triangulation refinement (connecting mid-edges of all triangles). Define X h :~_~(~h (~'~))2 and M h :-- {¢(x) : ¢(X)ie0 • P ° ( e o ) for all e0 • 7rh°(e)} (1 L02(~). These are associated with the nodal selection ( - : ~rh°(D),- and : ~h(~)). X h ~ , Qh N (3.4c) _

.

_

The conforming linear-macro element constant pair (see, e.g., [13,15,16]). Here X h :=(SNc oh @ B3)2, Qh := S h c N Lo2(D) are associated with the nodal choices X h ~

,

Qh ~

(3.4d)

The new element pair of [4]. Here X h :~-~({¢ E C°(~) : ¢[e • P 2 ( e ) and ¢ = 0 on 0~}) 2, Qh :_ Sh(D) AL2(~) are associated with the nodal choices: X h ~ , Qh ~ (3.4e) The Hood-Taylor element. Equations (3.4a)-(3.4e) have all been proven to satisfy the inf-sup condition (3.3), see, e.g., [4,7,8,13-16]. The element (3.4d) shares the nice property of (3.4b) that discretely divergence free basis functions can be constructed for general, unstructured meshes [17,18]. Since (3.4d) is a new element, we include for completeness a sketch of the proof that (3.4d) satisfies the inf-sup condition. THEOREM 3.1.

PROOf.

[4; Prop. 2.1] (3.4d) satisfies t h e inf-sup c o n d i t i o n (3.3).

[Sketch] The proof uses Fortin's method (see, e.g., [13,15] for the general technique),

of constructing ~ - : (/~/1 (f~))2 __+ X h satisfying ]]~'[[ < C and ~-~(q, V o Iv - ~'(v)]) -- 0, for

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101

all q E Qh. The explicit construction of 5r will be given here; see [4] for verification that this construction satisfies these two conditions, vh:7:v is determined by its nodal values (controids and mid-edges). Thus, define V_h by: v h ( y ) -~ 0 for all mid-edge nodes N on 0f~. For all nodes N on an edge f not on Of~, define v h ( N ) by f f ( v - v_h)(s)ds = 0, where s denotes arclength on f . For each controid node N (associated with a cubic bubble function) v_h(N) is defined by the requirements that: for j = 1,2,

fe (Vj -- vh)(x) dx = ~oe Xj(V -- v_h)(x) ° ne ds. T h e p r o o f t h a t V_h is well defined and 5r verifies the needed conditions is given in detail in [4]. Concerning the separation properties of the s t a n d a r d nodal bases for (3.4a)-(3.4e), we state the following. W h e n referring to an element (3.4a)-(3.4e) it will be understood t h a t the s t a n d a r d nodal basis for t h a t element is intended. THEOREM 3.2. Let 7rh(~) be an edge-to-edge triangulation of f~. Concerning (3.4a)-(3.4e): The "MINI" element (3.4a) satisfies the strip separation property but not the element separation

property. The Crouzeix-Raviart element (3.4b) satisfies the element separation property. The conforming linear-macro constant element (3.4c) satisfies the strip separation property on the macro dement mesh 7rh°(~) but not on 7rh(~). The element (3.4d) of [4] satisfies the element separation property. The Hood-Taylor element, (3.4e) satisfies the strip separation property but not the element separation property. PROOF. This follows by considering the supports of the corresponding basis functions for (3.4a)(3.4e). Using vertex nodes as degrees of freedom limits one to strip separation while using only edge and interior nodes allows element-wise separation.

4. C O N C L U S I O N S Robustness, meaning uniform convergence in the Reynolds number, is often lacking in iterative m e t h o d s for solving discretized Navier-Stokes systems yet it is much more i m p o r t a n t t h a n optimal order complexity in the meshwidth, for example. For one robust, d a t a parallel m e t h o d of [3-6,11,12,19] the limiting factor in the attainable parallelism is the separation properties of the finite element spaces chosen for the discretization. It is shown t h a t most popular spaces satisfy at least a strip separation p r o p e r t y and the new element of [4] as well as the Crouzeix-Raviart element satisfy the element separation property. This allows their use in robust element-wise, d a t a parallel m e t h o d s following [4].

REFERENCES 1. R.B. Kellogg, Spectral bounds and iterative methods in convection dominated flow, (preprint), (1991). 2. O. Axelsson, V. Eijkhout, B. Polman and P. Vassilevski, Iterative solution of singular perturbation 2 nd order boundary value problems by use of incomplete block factorization methods, BIT 29, 867-889 (1989). 3. W. Layton and P. Rabier, Domain decomposition via operator splitting for non-symmetric problems, Appl. Math. Lett. 5 (2), 67-70 (1992). 4. W. Layton and P. Rabier, An element-by-element parallel, low order element for the Stokes problem, ICMA Report 92-173, Univ. of Pittsburgh, (1992). 5. W. Layton and P. Rabier, Peaceman-Rachford procedure and domain decomposition for finite element problems, ICMA Report 91-166, Univ. of Pittsburgh, (1991). 6. W. Layton, J. Maubach and P. Rabier, Uniform convergence estimates for an elementwise parallel finite element method, ICMA Report, Univ. of Pittsburgh, (1992). 7. M. Crouzeix and P.A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numer. T, 33-76 (1973). 8. D. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21, 337-344 (1984).

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9. R. Jeurissen and W. Layton, Load balancing by graph coloring: An algorithm, Comput. Math. Applic. 27, 27-32 (1994). 10. T.J.l:t. Hughes, I. Levit and J. Winget, Element-by-element implicit algorithms for heat conduction, J. Eng. Mech. 109, 576-585 (1983). 11. W. Layton, J. Maubach, P. Ftabier and A. Sunmonu, Parallel finite elements methods, Proc. Fifth ISMM Conf. on Parallel and Distributed Computing and Systems, (1992). 12. A. Sunmonu, Multi-tasking an element-by-element finite element method on the CRAY YMP, ICMA Report, Univ. of Pittsburgh, (1991); Parallel Computing (submitted). 13. M. Gunzburger, Finite Element Methods for Vicous Incompressible Flows, A Guide to Theory, Practice and Algorithms, Academic Press, Boston, (1989). 14. L. Mansfield, Finite element spaces with optimal rates of convergence for the stationary Stokes problem, R.A.I.R.O. Num. Anal. 16, 49-66 (1982). 15. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin, (1991). 16. V. Giranlt and P.A. Raviart, Finite Element Methods for the Navier-Stokes Equations, Springer-Verlag, Berlin, (1986). 17. X. Ye, The construction of null basis for the discrete divergence operator, J. Comp. and Appl. Math. (to appear) (1995). 18. X. Ye and G. Anderson, The derivation of minimal support basis functions for the discrete divergence operator, J. Comp. Appl. Math. (to appear) (1995). 19. W. Layton, J. Maubach and P. l=tabier, Parallel algorithms for maximal monotone operators of local type, Numer. Math. (to appear) (1995). 20. O. Axelsson and V. Barker, Finite Element Solution of Boundary Value Problems, Academic Press, Orlando, (1984).