Finite Element Approximation for Grad-Div Type Systems in the Plane. SIAM Journal on Numerical Analysis, Vol. 29, No. 2. (Apr., 1992), pp

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1 Finite Element Approximation for Grad-Div Type Systems in the Plane Ching Lung Chang SIAM Journal on Numerical Analysis, Vol. 29, No. 2. (Apr., 1992), pp Stable URL: SIAM Journal on Numerical Analysis is currently published by Society for Industrial and Applied Mathematics. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. Wed Feb 13 08:13:

SIAM J. NUMER. ANAL. Vol. 29, No. 2, pp. 452-461, April 1992 01992 Society for Industrial and Applied Mathematics 00s FINITE ELEMENT APPROXIMATION FOR GRAD-DIV TYPE SYSTEMS IN THE PLANE* Abstract.

2 SIAM J. NUMER. ANAL. Vol. 29, No. 2, pp , April Society for Industrial and Applied Mathematics 00s FINITE ELEMENT APPROXIMATION FOR GRAD-DIV TYPE SYSTEMS IN THE PLANE* Abstract. This paper studies a 3 x 3 linear system of differential equations of grad-div type with appropriate boundary conditions. In constructing numerical approximate solutions to this boundary value problem it applies a least squares finite element method to an extended system of four equations and three unknowns. Analysis of the rate of convergence of the numerical solutions and estimates of the error involved in approximating exact solutions of the considered boundary value problems are given. The method of this paper achieves optimal rates of convergence both in the HI-norm and in the no norm. Numerical examples employing piecewise-linear elements for directional triangulation and bilinear elements in a quadrilateral grid are also presented. Key words. Sobolev spaces, a priori estimate, least squares method, Fredholm operator, 11. I/ -1 norm AMS(M0S) subject classifications. 65N30, 35F15 1. Introduction. This paper treats both theoretical and computational aspects of the problem of solving a 3 x 3 linear system in two variables of the form grad(g) = u and div u = f. The classical finite element method is used for Poisson's equation Ag = f, but there may be a less accurate approximation to its gradient grad(g). In the solution of problems concerning fluid mechanics or elasticity the values of the derivatives of g are sought rather than the values of g itself. The least-squares method and standard Galerkin finite element mcthod can lead to stable numerical results, provided we make suitable assumptions regarding the shapes of the elements. For instance, under the grid decomposition property as a condition, in [8] an optimal order estimate for piecewise polynomial approximation spaces is derived. In [ll],the authors derive an appropriate variational form for the Poisson equation, where the intent is to solve for the function and its gradient. This method performs well, given properly chosen restrictions on the finite element space. By contrast, this paper employs a more standard least-squares finite element approximation, based on an extended system which becomes a regular elliptic boundary value problem [12]. A so-called "slack" variable k is introduced; it is needed in the proof of convergence and in the error analysis; the slack variable does not play a role in our computations. We apply the least-squares method to a system with three variables and four equations. Analysis of our algorithms shows that they yield optimal rates of convergence, in both the H1 and L2 norms, for computing either g or Vg in the finite-dimensional subspace of continuous piecewise polynomials of degree r. For example, if r = 1, i.e., if a continuous piecewise linear function or piecewise bilinear function is assumed, then the resulting approximations gh and uh satisfy *Received by the editors August 11, 1987; accepted for publication (in revised form) March 28, This work was supported by a Research Challenge Grant by the Ohio Board of Regents, and it was revised while the author served as a consultant to the Institute for Computational Mechanics in Propulsion, National Aeronautics and Space Administration, Lewis Research Center, Cleveland, Ohio f~e~artment of Mathematics, Cleveland State University, Cleveland, Ohio

3 FINITE ELEMENT APPROXIMATION FOR GRAD-DIV SYSTEM 453 and Ilu - uhllo + hllu - uhiii I Ch211u112. Throughout this paper C > 0 is a constant independent of h with different values in different places in the paper. Our results hold without any restriction other than those stated and the regularity requirements for the triangulation (cf. [4]). The numerical examples given in the case of H1 spaces with respect to piecewise linear elements in directional triangles and piecewise bilinear elements in quadrilaterals support our theoretical analysis. In some sense we can say that we have found a "superconvergent" approximation to a solution of the Poisson equation, since with piecewise linear element the L2 errors both in (g - gh) and in (u - uh) are O(h2). 2. Some notation and the formulation of the problem. The two-dimensional Poisson equation can be transformed into a first-order linear system by regarding the functiori itself and the components of t,he gradient as variables. Consequently, we obtain the grad-div 3 x 3 linear system: grad(g) - u = 0 in R, div u = f in R, Ru = 0 on I?. In the above, we assume that R is a simply connected bounded domain in W2 with a Holder continuously differentiable, positively oriented boundary I?. We also assume that f is a smooth function defined on R. To simplify the numerical scheme and the analysis, we actually assume that R is a convex polygon in W2. The boundary conditions are assumed to be of the form Ru = n x u or Ru = n.u; these conditions are associated with the homogeneous Dirichlet and Neumann boundary conditions of the Poisson equation, respectively. In the case Ru = n. u, f has to satisfy the compatibility condition Jfif = 0. Throughout, we will employ standard notations from the theory of Sobolev spaces, including the usual norms. In particular, we let Hs(R) denote the Sobolev space of functions which have square integrable derivatives of order up to s on R; we write II.II, for the standard Sobolev norm associated with Hs(R). H,j(R) denotes the closure of V(R) with respect to the norm ,. H-Y(R) denotes the dual space of H[(R) and is given the structure of a normed linear space by setting, for ally p E H-s(R), the sup being over the set of all cp E H[(R) $ 0. For subsets Rh C R we may also consider the Sobolev space Hs(Rh) together with its associated norm /I - Ils,fih. For any R, I?, and s as above let Next let rh denote a family of regular triangulations [4] of the polygonal domain Rh into triangles Rj, j = 1,2,...,T, where h is the maximum diameter of these

4 454 CHING LUNG CHANG triangles. Let PT(aj)denote the space of polynomials of degree less than or equal to r defined on Rj, and let (2.3) S?={(;;)E[H~(R)]~; v:,ub and P~EP,(R,) j=1,2,... and Note that S? C CO(~) and V,h satisfies the approximation property [2], [4]: for t = 0,l and every (;) E [Hy(fl)13,1 4 s 4 r + 1, there exists a ($) E V,h such that the positive constants CYyt being independent of v, p, and h. 3. The extended system and least squares finite element method. Since the 3 x 3 linear grad-div system is not elliptic in the sense of Petrovski (even though it is elliptic in the sense of Douglis and Nirenberg [6]),the least squares method applied directly to (2.1) may lead to an unstable result. To avoid this problem, we extend (2.1) by setting [ ] I!] =F, grad(g) - u (3.1) L(u,g)= divu curl u = 0 where curl u = % - %. We consider the simplest formulation of our problem such that (3.1), with the boundary condition Ru = 0, has exactly one solution and is also solvable for an f which satisfies the compatibility condition. Let us define a quadratic functional It is easily seen that a solution of (2.1) minimizes (3.2)and that a sufficiently smooth solution of (3.1) also satisfies (2.1). For arbitrary (;:) E v?, as in (2.3), will not vanish in general. The choice of (uh,gh) E Kh which results in a minimum value for (3.3)depends on the metric chosen for (3.3).In this paper we consider an L2- norm; in this case the problem of minimizing (3.2) becomes a problem of Chebyshev approximation [12]. Observe that (;:) minimizes (3.2) if and only if a (uh,gh; vh, ph) = b (vh, ph) for all (;) E vjl,

5 FINITE ELEMENT APPROXIMATION FOR GRAD-DIV SYSTEM 455 where = l(grad(g) - u). (grad(p) - v) + div u div v + curl u curl v and P Choosing a basis for V,h reduces (3.4) to a set of algebraic linear systems. Evidently, the corresponding matrix of this system is symmetric; it will be proved to be positive definite in the next section. 4. Convergence and error estimate. To prove that the bilinear form in (3.4) is coercive, we introduce a "slack" variable k which is smooth in R, vanishes on the boundary r,and satisfies grad (g)- curl (5)- u (4.1) Lf(u, g, k) = div u curl u where curl (k) = (g,-g) T. The corresponding extended boundary conditions are R1(u, g, k) = 7 ;I :I [y] :TO onr, if Ru = n x u. and if Ru=n.u. Equation (4.1) can be rewritten as where and

6 456 CHING LUNG CHANG It is easy to verify that (4.1), (4.3) is strongly elliptic, i.e., det(ea + qb) = (52 + q2)2, SO [A + qb is a nonsingular matrix for each real pair (E, q) # (0,O). After elementary operations, following the procedure given by [12, pp ], we can check that the corresponding boundary condition (4.2), (4.2)' satisfies the Lopatinski condition. Thus, (4.1), (4.2) (or (4.2)') is a regular elliptic boundary value problem. It follows that the operator (L', R') is a Fredholm operator, which implies that it has a finite-dimensional nullspace. Indeed, the nullity of (4.1), (4.2) (or (4.2)') is one. Let (4.4) A(u, g, k) = g(xo) = 0, xo is a point on n. Then (4.1), (4.2) admits at most one solution. The a priori estimate for (4.1) and (4.2) can be written as: There exists Cp > 0 for p > 1such that Clearly, (4.5) remains valid for k r 0. Furthermore, if we consider the problem with respect to V1 and let g(xo) = 0, we obtain since Ll (u, g, 0) = L(u, g) H1-norm convergence. The coercivity of the bilinear form a(u, g; v, p) in (3.4) is implied by (4.6) when p = 1. It follows that the linear algebraic system generated by (3.4) is positive definite and symmetric; we obtain the following restilt. THEOREM 4.1. For each h > 0 there exists uniquely an element (If: ) E V$ such that IlL(uh, gh) - FIIi assumes a minimum value in V$. The matrix corresponding to the numerical scheme that determines this value of (u~)is symmetric and positive 9h. definite. Moreover, there exists a constant C > 0 that is zndependent of h such that (4.7) IIu - uhiil + Ig - ghill 5 c (IIU- vhlli + llg - phli) for any (it) E lib. COROLLARY 4.2. Let (If) be the solution of (2.1) and let (If:) E V$ be the solution of (3.4). If (If) E VP, where 1 5 p < r + 1, then there exists a constant C > 0, independent of h, such that Proof. (If)E VP satisfies (2.1) and ( ) must satisfy (4.1) and (4.2) in the case k = 0 in 0. Combining (2.5) and (4.7), we obtain (4.8) L2-norm convergence. Givsn the H1-norm error estimate (4.8), the proof of the L2-norm error estimate depends essentially on a duality argument. The inequalities (4.5) were shown by [l]and [12] to hold for the case p > 1. Dikanskij [5] extended these results, by an interpolation argument, to the case p 2 0. If p = 0, then

7 FINITE ELEMENT APPROXIMATION FOR GRAD-DIV SYSTEM 457 where Co > 0 is independent of u, g, and k. The boundary value problem (4.1), (4.2), or (4.2)' may not be solvable for all F E [HP-~(R)]~. Let N* denote the codimension of the range of L1 in the space [~p-l(r)]~. Since (L1, R1)in (4.1) and (4.2) or (4.2)' is regular elliptic, it is a Fredholm operator. The adjoint form of (L1, R1) satisfies an a priori estimate and N* is finite [12]. In fact, N* is the number of compatibility conditions required to ensure the existence of a solution of (4.1) and (4.2) or (4.2)'. Therefore, generally, there exist N* linear independent orthonormal function vectors Ij (j= 1,.+., N*) such that The extended problem of (3.1), (3.2), or (3.2)' can be written as L1(u,g, k) + xea1aiii = 4! in R, (4.11) R1(u, g: k) = 0 on I', ~(xo)= 0 for an xo E n; it admits exactly one solution (u, g, k; a) for any given 4! E [Hp(R)14. We define, and choose (;: ) with the property that (4.12) assumes its minimal value in and only if if where bl(v, p, 7) = p div v. The coercivity of al(u, g, Ic; u, g, k) follows by (4.5). We also have the orthogonality relation (4.14) a'(u - uh, g - gh, k - kh; vh, ph, yh) = 0 for any (Z) vih.

8 458 CHING LUNG CHANG For any E [HS-~(R)]~, there is a unique (w, 4, $)T E V1",a E RN* such that LI(w,+,$)+~~*~aiIi=@inR, (4.15) Rw=O on I?, ~(xo)= 0 for an xo E a. Applying the orthogonality (4. lo), (4.14), approximation property (2.5), and inequalities (4.5) and (4.9), we have = I(Liei, Li(w - vh, 4 - ph, + - yh))l for any (vh, ph, yh) E v:~ -< IILieillo. IIL1(w- vh, 4 - ph, lci -rh)llo Therefore, In the special case in which k = kh = 0, we have U - uh IILell-1 5 Chllelll, where e = ( 9 -gh ). From (4.8) and (4.9) the next theorem follows. THEOREM 4.3. Under the same assumptions as in Theorem 4.1 and Corollary 4.2, where C > 0 is independent of h. 5. Numerical experiments. We have considered the general case of the graddiv linear system (2.1) in the previous paragraphs. In this section we give some numerical examples illustrating the theory developed above. Two kinds of elements will be applied for the unit square, 0 < x,y < 1. In the first instance, piecewise linear elements with support in directional triangles are used as in Fig. l(a); the square is subdivided into N2 squares of side h = and then subdivided further, each small

9 FINITE ELEMENT APPROXIMATION FOR GRAD-DIV SYSTEM 459 square into two triangles by drawing a diagonal. In the second example the bilinear element is of the form shown in Fig. l(b). FIG. 1. (a) Directional triangle; (b) Bilinear quadrilateral Numerical example 1. grad (g)- u = 0 in S1, divu=2(x-x2)+2(y-y2) inn, uxn=o on I?, g(0) = 0. Thus the exact solution of the problem is Table 1 exhibits the numerical results for the piecewise linear elements in directional triangulation and for the bilinear quadrilaterals, respectively. It indicates the size of the vector error e = (u - uh, g - gh) with respect to both H1 and L2 norms. TABLE1

10 460 CHING LUNG CHANG 5.2. Numerical example 2. grad (g)- u = 0 in 0, div u = -27r2 cos(7rx).cos(7ry) in R, u.n = 0 on I?, g(0) = 0. The exact solution of this problem is Table 2 gives the size of the vector error for Example 2. TABLE Discussion of the examples. Both Examples 1and 2 support the error estimates given in (4.8) and (4.15), i.e., the error of vector e = (u - uh,g - gh) in H1-norm and in L2-norm are proportional to h and h2, respectively. But in each of these two numerical results the constants h-lllelll and h-2llello differ. The grids for these computations do have different shape; if we choose the grids so that in each case the number of elements is the same, the above constants, which govern the rate of convergence, still differ. This is so even though the CPU times for the respective computations are almost equal. Recall that llello = (llu-uh11i+11g-ghllo)1/2.roughly speaking, this quantity may be regarded as measuring the size of the H1 error in g-gh, which error is of order O(h2). Acknowledgment. All of the computer work above was performed by the author at the Institute for Computational Mechanics in Propulsion, NASA Lewis Research Center, Cleveland, Ohio. REFERENCES [I] S. AGMON,A. DOUGLIS, L. NIRENBERG, near the boundary for solution of AND Estimates elliptic partial differential equations satisfying general boundary conditions I, 11, Comm. Pure Appl. Math., 12 (1959), pp ; 17 (1964), pp [2] I. BABUSKAND A. K. AZIZ,Mathematical foundation of the finite element method, Academic Press, New York, 1972.

FINITE ELEMENT APPROXIMATION FOR GRAD-DIV SYSTEM 461 C. L. CHANG AND M. D. GUNZBURGER, A finite element method for first order elliptic systems in three dimensions. Appl. Math. Comp., 23 (1987). P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1977.

11 FINITE ELEMENT APPROXIMATION FOR GRAD-DIV SYSTEM 461 C. L. CHANG AND M. D. GUNZBURGER, A finite element method for first order elliptic systems in three dimensions. Appl. Math. Comp., 23 (1987). P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, A. S. DIKANSKIJ, Conjugate problem of elliptic differential and pseudo-differential boundary value problems in a bounded domain, Math. USSR Sborn., 20 (1973), pp A. DOUGLIS AND L. NIRENBERG, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math., 8 (1955), pp G. J. FIX,M. D. GUNZBURGER, AND R. A. NICOLAIDES, On Fnite element methods of the least square type, Comput. Math. Appl., 5 (1979)., On mixed finite element methods for first order elliptic systems, Numer. Math., 37 (1981), pp G. 3. FIXAND M. E. ROSE,A comparative study of finite element and finite difference methods for Cauchy-Riemann type equations, SIAM 3. Numer. Anal., 22 (1985), pp P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, P. A. RAVIARTAND J. M. THOMAS, Primal hybrid finite element methods for second order elliptic problems, J. Math. Comp., 31 (1977), pp W. WENDLAND, Elliptic Systems in the Plane, Putnam, London, K. YOSHIDA, Functional Analysis, Springer-Verlag, Berlin, 1979.

12 LINKED CITATIONS - Page 1 of 1 - You have printed the following article: Finite Element Approximation for Grad-Div Type Systems in the Plane Ching Lung Chang SIAM Journal on Numerical Analysis, Vol. 29, No. 2. (Apr., 1992), pp Stable URL: This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR. References 9 A Comparative Study of Finite Element and Finite Difference Methods for Cauchy-Riemann Type Equations George J. Fix; Milton E. Rose SIAM Journal on Numerical Analysis, Vol. 22, No. 2. (Apr., 1985), pp Stable URL: NOTE: The reference numbering from the original has been maintained in this citation list.

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