Interpolation Error Estimates for the Reduced Hsieh - Clough - Tocher Triangle

We study the unisolvence and interpolation properties of the reduced Hsieh-Clough-Tocher triangle. This finite element of class C , which has only nine degrees of freedom, can be used in the numerical approximation of plate problems. Introduction; Main Notation. The space R" is equipped with the Euclidean norm | • | and inner-product <•,•>. If B is a subset of R", we let hB = diameter of B, pB = sup {ft ß; 8 is a ball contained in B], meas(7?) = fB dx (assuming B to be measurable), v\B = restriction of the function v to the set B, Pk(B)={p\B;pEPk(Rn)}, fcGN, where Pk(R") denotes the space formed by all polynomials of degree < k in n variables. Given a multi-index a = (a,, a2, . . . , a„) E N" with length |a| = 2"=1a/, we use the usual notation davia) for the partial derivatives of order |a| of a function v at a point a E R", while the Fréchet derivatives are denoted Dmvia), or simply Dvia) if m = 1. We write for brevity lDmvia)i-m if|1=|2=... = |m=|) Dmv(aKxA2,...Am)=\ ( DmviaXr^ Am) if £l = É2 = * * • = Im-X = %■ Following [3], [4], let us recall some general definitions pertaining to finite elements: A finite element in R" is a triple (K, P, 2), where: o (i) K is a subset of R" with a nonempty interior K and a Lipschitz-continuous boundary in the sense of Necas [7], (ii) P is a vector space of finite dimension TV, whose elements are real-valued functions defined over the set K, (iii) 2 is a set of TV linear forms 0(-, 1 < / < TV, defined over the space P, and the set 2 is P-unisolvent in the following sense: Given arbitrary real numbers a,., 1 < i < TV, there exists one and only one function p E P which satisfies 4>,(p) = a„ Ki/• /■=! Of course, the above definition makes sense only if the function v is smooth enough so that the degrees of freedom 0(u) are well defined. For finite elements for which all the degrees of freedom are of the form davia) (as is the case in this paper), we shall require, for definiteness, that the function v be s times continuously differentiable over the set K, where s is the maximal order of partial derivatives found in the set 2. In other words, for such finite elements, we can define a P-interpolation operator XI: dorn II = C\K) —* P. A reduced Hsieh-Qough-Tocher triangle is a triple (7Í, P, 2) where the data K, P and 2 are defined as follows. (i) The set K is a triangle, with vertices ax, a2, a3. (if) Let a be any point in the interior of the triangle K. Denoting (cf. Figure 1) by K¡ the triangle with vertices a, ai+,, ai+ 2 (the indices are counted modulo 3 Figure 1 whenever necessary), and by K¡ the side opposite to the vertex a¡, the space P is given by (1) P={pE C\K); p\Ki S P3iKt), 9„p|*;. EPxiK\), 1 < i < 3}, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE REDUCED HSIEH-CLOUGH-TOCHER TRIANGLE 337 where dvp\K: is the (outer) normal derivative of the function p\xalong the side K¡. (iii) The set of degrees of freedom is (2) 2= {bapiat), Ki<3, |a|-unisolvent, with P and 2 given as in (1) and (2), we turn to the main object of this paper, which is to estimate the interpolation errors \v Xlv\m K, where the standard notation \TM\m,A = (\ £ \^\2dX\ , \JA\a\ = m / is used, and where, according to the general definition given above, the TMnterpolant Liu is uniquely determined by the conditions (3) XlvEP and 9a(IIu)(a,) = 3au(>.)> 1< / < 3, |a|< 1. We are then able to show that (Theorem 2), given a family of reduced Hsieh-CloughTocher triangles which is regular in a sense to be defined below, and given a function v E 7/3(7C) CC\K) = dorn II, one has (4) lu Xlv\m¡K = Oih3K~m), m = 0, 1, 2, i.e., the order of convergence is the same as one would expect from the inclusion P2iK) C PK for an affine family in the sense of [5] (this is thus another instance of an almost-affine family of finite elements, according to the terminology of [4, ChapLicense or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 338 PHILIPPE G. CIARLET ter 6] ). It appeared, however, that the standard techniques for getting interpolation error estimates for finite elements of class C1 (cf. Bramble and Zlámal [1], the author [3], [4], Raviart [9], Zenisek [10], Zlámal [13]) did not directly apply to this element, and this observation led to the present paper. Let us then assume that we are using this finite element for solving a fourthorder problem (such as a plate problem) posed over some open set Í2 C R2. If we let u and uh denote, respectively, the exact and approximate solution, then we get from (4), (5) HM-"iillff2(n)n. where h denotes the greatest diameters of the triangles found in the finite element space where the discrete solution uh is found. Notice that the above error estimate requires that the solution u be in the space 7/3(£2), but this is a mild regularity assumption, satisfied if £2 is a convex polygon for a plate problem. The reader interested in finite element methods for fourth-order problems in general may consult Zienkiewicz [12, Chapter 10] for a discussion from an engineering viewpoint, while a fairly complete description and a study of their convergence properties are given in [3, Sections 13 and 14] and [4, Chapters 6 and 7]. Unisolvence. Theorem 1. The set 2 of (2) is P-unisolvent, the space P being defined as in (1). Proof. Since the number of degrees of freedom is equal to the dimension of the space P, it suffices to prove that a function p which satisfies pEP and bap(at) = 0, 1|jr>/+1) = dvp\K¡(ai+2) = 0, obtained from the definition of the space P, implies that Kp\k'í = 0, 1 < i < 3. The unisolvence is then a consequence of the unisolvence established in [2] (see also Percell [8] for another proof) for the Hsieh-Clough-Tocher triangle: This finite element is a triple (K, P*, 2*), with ÍK = a triangle subdivided as in Figure 1, P* = {p E C\K), p\K¡ E P3iKt), 1 < i < 3}, 2* = {dapiai), \a\ < 1, b.pibi), 1 < i < 3}, where dvpib¡) denotes the normal derivative at the midpoint b¡ of the side K¡. D Interpolation Error Estimates. Our first objective is to appropriately describe a family of reduced Hsieh-Clough-Tocher triangles. Let K be a fixed triangle with vertices a¡. Given an arbitrary triangle K with vertices aiK (cf. Figure 2), we let FK License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE REDUCED HSIEH-CLOUGH-TOCHER TRIANGLE 339