The Stability in- L and W^ of the L2-Projection onto Finite Element Function Spaces

The stability of the Z.2-projection onto some standard finite element spaces Vh, considered as a map in Lp and W^, 1 ^ p < oo, is shown under weaker regularity requirements than quasi-uniformity of the triangulations underlying the definitions of the Vh. 0. Introduction. The purpose of this paper is to show the stability in Lp and Wp, for 1 < p < oo, of the L2-projection onto some standard finite element subspaces. Special emphasis is placed on requiring less than quasi-uniformity of the triangulations entering in the definitions of the subspaces. In the one-dimensional case, which is discussed in Section 1 below, we first give a new proof of a result of T. Dupont (cf. de Boor [2]) showing Lœ stability without any restriction on the defining partitions, thus extending an earlier result by Douglas, Dupont and Wahlbin [6] for the quasi-uniform case. We then use the technique developed to show the stability in Wp, in the case p > 1, under a quite weak assumption on the partition, depending on p. We also show that some restriction on the partition is needed for stability if p > 1. We remark that the known Lp stability result has been extended to higher degrees of regularity of the subspaces; see de Boor [3] and references therein. In the case of a two-dimensional polygonal domain, discussed in Section 2, we demonstrate Lp and Wp stability results for the L2-projection onto standard piecewise polynomial spaces of Lagrangian type. The requirements on the triangulations involved are more severe than in the one-dimensional case, but allow nevertheless a considerable degree of nonuniformity. The proofs are based on a technique used by Descloux [5] to show Lx stability in the quasi-uniform case (cf. also Douglas, Dupont and Wahlbin [7]). Results such as the above are of interest, for instance, in the analysis of Galerkin finite element methods for parabolic problems. Thus Bernardi and Raugel [1] use the IV2 stability of the L2-projection to prove quasi-optimality of the Galerkin solution with respect to the energy norm, and Schatz, Thomée and Wahlbin [8] apply the Lx stability in a similar way (in the quasi-uniform case). 1. The One-Dimensional Case. In this section we shall study the orthogonal projection m = mh with respect to L2(0,1) onto the subspace Vh = (x e C(0,1); xl,, e Pk, j = 0,...,N; x(0) = X(l) = 0}, Received January 7, 1986. 1980 Mathematics Subject Classification. Primary 41A15, 41A50, 41A63. ©1987 American Mathematical Society 0025-5718/87 $1.00 + $.25 per page 521 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 522 M. CROUZEIX AND V. THOMEE where 0 = x0 < xx < ■ ■ ■ < xN+x = 1 is a partition of [0,1] and I, = (x¡, x,+1). We shall first demonstrate the following result, in which || • || denotes the norm in M0,i). Theorem 1. There is a constant C depending only on k such that hu\\P^ C\\u\\p VueL,(0,l),l < oo. We shall then turn to estimates in Wpl(0,l) ={ve L,(0,1); v' = dv/dx e Lp(0,l); v(0) = v(l) = OJ and show, with h¡ = xi+x x¡, Theorem 2. Let 1 < p < oo and assume, for p > 1, that the partition is such that h,/hj < C0a^-J\ where! < a < (k + l)^"1'. Then ¡Wl^CWu'Wp VueWpl(0,l), where C depends on k, and for p > 1 also on C0, a, and p. For the proofs of these results we introduce the spaces Vt={x^Vh;X(x,) = 0,i = l,...,N} and Vk, the orthogonal complement of Vh2 in Vh with respect to the usual inner product in L2(0,1). For k = 1 we have Vh2 = {0} and Vk = Vh. We also introduce the orthogonal projections itj onto Vfc, j = 1,2, and obtain at once (1.1) ir = irx + tt2 (tr = wx for k = 1). We note that m2 is determined locally on each L by the equations (1.2) (v2v,q)fj -(v,q)rj for q e P»(/y) = [q e PK; q(Xj) = q(xJ+1) = 0), where (•,•)/ lS the standard inner product in L2(Ij), and that a function in Vl is completely determined by its values at the interior nodes, so that dim Vk = N. For v e C[0,1] with v(0) = v(l) = 0 we shall also use the piecewise linear interpolant rhv e Vh and note that, for 1 < p < oo, (1.3) |(^)'|,<ikiu and, denoting the norm in Lp(I¡) by || • \\p ,., (!-4) Ikrhv\\pJ.K jh^v'Wp. Lemma 1. There is a constant C depending only on k such that, for 1 < p < oo, (1.5) K"M C\\u\[p, ueLp(Q,l), and (1.6) |(«2(u rhu))'\\p < C||«'|U, u e Wp\Q,\). Proof. We consider first (1.5) for p = 1 and set wA = m2u. It follows, by taking q = üh in (1.2), that n II «A Ik/,<ll"lll./JI"Jloo,/,License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use