An observation concerning Ritz-Galerkin methods with indefinite bilinear forms

Existence, uniqueness and error estimates for Ritz-Galerkin methods are discussed in the case where the associated bilinear form satisfies a Garding type inequality, i.e., it is indefinite in a certain way. An application to the finite element method is given. In this note, we would like to discuss existence, uniqueness and estimates over the whole domain for some Ritz-Galerkin methods where the bilinear form satisfies a Guarding type inequality, i.e., it is indefinite in a special way. We shall first illustrate the problem by an example. For simplicity, let &7 be a simply connected convex region in the plane with a polygonal boundary Mi and consider the Dirichlet problem 2 2 (1) Lu I D.(a..(x)DU) + ?Zb(x)Diu + c(x)u =f in Q2, u = O on am, i~= I ifJ i where L is uniformly elliptic in 2; for simplicity, we assume that the coefficients belong to C1(-Q). Let us suppose that for each f C L2(f2), the problem (1) has a unique solution u. It is then well known that u C W2(i) 2 W2(i). Suppose that we wish to approximate u using the finite element method. For this, we subdivide Q into triangles with largest side h and smallest angle a > ao > 0 and define a finite-dimensional subspace Sh C to be, for example, the set of piecewise linear functions on this triangulation which vanish on MQ. We then seek to determine an approximate solution uh C Sh from the Ritz-Galerkin equations B(Uh, o) =B(u, p) = Jjfdx, for all p C Sh, (2) /2 B(u, ~) = (i1 ai.(Diu)(D j) + 2 bi(Diu) p + cu a dx. Let us note the inequality 0 (3) jB(u, p)I 0 and 11IS denotes the norm on the Sobolev space W~s(Q) with W(?(Q) = L In the case that C2 0, then B may be indefinite and the existence and uniqueness of uh does not immediately follow. In fact, simple examples show that a solution of (2) need not existdepending on the subspace Sh used. It will turn out however that, if h is sufficiently small, then existence and uniqueness of uh can be guaranteed and quasi-optimal error estimates hold. Let us show how this may be done very simply. We shall first derive an a priori estimate. If e = u uh satisfies (2), then using (3) and (4) we obtain C lie 12I C2 le ll2 0, we obtain (7) liu Uh 111 0 and C2, both independent of 7? and h, such that (10) c "In -1 C2 II"HO 0. THEOREM. Suppose that the above conditions hold. Then there exists an ho > 0 such that, for all h C (0, ho], Eq. (9) has a unique solution uh C Sh for each u C H1. Furthermore, (13) HelH < CIIUIIH and IHellH < Cw(h)IuIIH , where C is independent of h and u. Proof We first remark that, if C2 < 0, then the result easily follows. Suppose e = u uh satisfies (9). Then, from (10), we have for f C se, IPI<IH1 1, that This content downloaded from 157.55.39.212 on Wed, 08 Jun 2016 05:56:14 UTC All use subject to http://about.jstor.org/terms 962 ALFRED H. SCHATZ C1 lIt "1 H -C2 IIuhIIH < sup[B(Uh, p)I < suplB(u, p)l < CIIuIIH 1 ~~~0 or IIuh IIH (C2/Cl) IIuh IIH < (C/C1) IIU IIH 1 01 Using the triangle inequality, we obtain (14) IHellH (C2/C,) I1eIIH O HI 1U where C is a new constant independent of u, e and h. The inequality (14) is analogous to (5) and the proof now proceeds in the same way as in the example given except that we use (12) instead of (6). The essential point here is that an inequality of the type (12) may be effectively used to treat indefinite bilinear forms where the indefiniteness is caused by lower order terms. For many methods, this inequality can be established using the previously mentioned technique of Nitsche. Department of Mathematics Cornell University Ithaca, New York 14850 1. I. BABUSKA, "Error-bounds for finite element method," Numer. Math., v. 16, 1971, pp. 322-333. MR 44 #6166. 2. I. BABUSKA, The Mathematical Foundations of the Finite Element Method, With Applications to Partial Differential Equations (edited by A. D. Aziz), Academic Press, New York and London, 1972. 3. J. BRAMBLE & M. ZLAMAL, "Triangular elements in the finite element method," Math. Comp., v. 24, 1970, pp. 809-820. MR 43 #8250. 4. J. NITSCHE, "Lineare Spline-Funktionen und die Methode von Ritz fuir elliptische Randwertprobleme," Arch. Rational Mech. Anal., v. 36, 1970, pp. 348-355. MR 40 #8250. 5. M. ZLAMAL, "A finite element procedure of the second order of accuracy," Numer. Math., v. 14, 1969/70, pp. 394-402. MR 41 #1233. This content downloaded from 157.55.39.212 on Wed, 08 Jun 2016 05:56:14 UTC All use subject to http://about.jstor.org/terms