APPROXIMATION THEORY, II ACADEMIC PRfSS. INC. New York San Francioco London. J. T. aden

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1 Rprintd trom; APPROXIMATION THORY, II 1976 ACADMIC PRfSS. INC. Nw York San Francioco London \st. I IITBRID FINIT L}ffiNT }ffithods J. T. adn Som nw tchniqus for dtrmining rror stimats for so-calld hybrid finit-lmnt mthods for th approximation of solutions of linar lliptic boundary valu problms of scond-ordr ar dscribd. 1. Introduction I ), Th hybrid finit lmnt mthods, dvlopd largly by Pian and Tong and thir associats (.g. [1]) hav bn usd succssfully in th numrical solution of a varity of practical problms, and thr appar to b crtain classs of problms in which ths mthods hav advantap,s ovr convntional finit lmnt tchniqus (.g. shll problns, lasto-plasticity problms, problms of cracks and strss singularitis). Howvr, th intrinsic mathmatical proprtis of ths mthods hav only rcntly bgun to b invstigatd [2-4]. \~ dscrib hr a fairly gnral thory of mixd-hybrid mthods dvlopd in [4]. 2. A Hixd-Hybrid Variational Principl Considr as a modl problm, (2.1) - liu + u f in n u '" 0 on an 2 whr IIis th Laplacian oprator in m, f f L 2 (11), and 11 1s 2 a convx polygonal domain inm - - Lt P dnot a triangulation of n into triangls n, i.., 485

2 J. T. ODN (2.2) fl '" U n l W also dfin a collction of boundary pics of P (2.3) r= U r =l r = an - S rf = rn r f whr S is th st of vrtics of fl. W now introduc th following dfinitions and convntions; I. Spacs Dfind on th Partition P (2.4) 1 < < } (Z.5) (2.6) whr W (n = compltion of L 2 (n in II II W(n I t i (2.7) in which II II _k is a dual norm of l- H 2(an ) 11 2 (an ) II. Product Spac (Z.8) X = UI(P)x ~Z(P)x W(r) (2.9) II AII ~ X Z ( II u I IIII (P) +IIoII Z +IIl/i11 ]~ - ~Z(P) W(r) whr ~2(P) = LZ(P) x LZ(P) and ~ = (01'oZ)' III. Spcial Bilinar and Linar Forms 486

3 HYBRID FINIT LMNT MTHODS (2.10) B:XxX... lr, B(A,I) \' b (A,I) L - - l arbitrary tripls in X; A b (A I ) ~- (2.11) Vu + u u + (Vu - 0 ) 0 Jdx dx - - l 2 + f (l/i do ~ + '$ u )ds (2.12) F(~) = \' f (I ) L - l f (I ) - IV. Proprtis of th Forms B(',') and F(') tain proprtis which guarant that th variational problm In [7], th forms B(A,I) and F(I) ar shown to hav cr- of finding A ~ X such that (2.13) v I (X has a uniqu solution in X. Morovr, it can b shown that th mixd-hybrid variational problm (2.15) is quivalnt to th wak form of (2.1). 3. Finit lmnt Approximations I ~ Th variational principl dscribd in th prvious oction has bn dsignd so as to b naturally adaptabl to mixd-hybrid finit-lmnt approximations. As xpctd, th triangls 0 ar now viwd as finit lmnts. ll W introduc th spacs of polynomials: Qk(P) = {un (P), U. Pk(fl ), 1 2 \2 }, g~(p) = (~. ~2(P), ~ : ~/fl)' 1 < < }, and Q-2(r) = {'I': W(n, l/i f = l/il r (P (r f)' - - t t f 487

4 J. T. ODN 1 2, f 2, > f} whr Pk(ll) is th spac of polynomials of dgr < k on n,tc. Ths spacs hav th following in- - l 2 trpolation proprtis: givn u : H (n ), : (H q (11», m-3/2 - l/i. fi (n,an) (= compltion of th spac of normal dri vativs avian L 2 (an ) in th norm I Il/il I, -3/2 = H m (an) a I inf{ Ilvll ; m ~ 1, -!- = lji}) thr xists constants C l, Hm(n ) on an C, C > 0, indpndnt of th partition P, and lmnts and l/i in th rspctiv spacs in (3.1), such that U,, (3.1) Ila-Ii 2c2hvlloll -, L 2 (Q) - Hq(Q) /1 - 'I'll -k 2 C3hl 11/11 1Am-3/2 H 2(an ) II (an) whr h = dia(11 ) and (with 0 < s < l, m > 1), (3.2) µ = min(k+l-s, l-s); v = min(r+1,q); = min(t+t,m-l). Lt A = (U,,'!'). Qkl(P) x QO(P) x Q-\r) = X c X r t -n Thn th mixd-hybrid finit lmnt mthod consists of sking th ~ \ such that (3.3) B(A,A) F(!.) whr B(',-) and F(') ar th forms dfind prviously. Lt n l and no dnot orthogonal projctions of Hl(n ) and l L (n ) onto Qk (n ) and QO (n ), rspctivly. \~ introduc th r spcial stability paramtrs 488

5 HYBRID FINIT LMNT MTHODS lj v (3.4) Y B(P) Y Y min {µ - -2,v - -Z }, 0 < lj,v,y < 1 l<< - whr z Hl(n ) is th solution of an auxiliary problm, az - Vz z + z = 0 in ~ and ~ = 'I' on dn, ~ Q -~ (an), and 2 an t II z III n = II 'l' II_k2 an' W thn hav, ' Thorm 3.l. If B(P) > 0, thr xists a uniqu finit lmnt solution ~o ~ i (3.4) and th following stimat holds: (3.5) whr c = II II 2 Ch Cl II u 0 I; l ' X H (P) = (u o _ U o, Vu o _ O, - - l > Z o _ ~ _ '1'0) dn Cl = min{k,r+l,t+-f' -l}. W rmark hr that on asy way to guarant that B(P) > 0 is to choos th polynomial spacs such that k-l ~ r and k ~ t+l (t vn), k > t+z (t odd) ovr a triangular lmnt n. - A ncssary condition can also b givn. Thorm 3.2. In ordr that (3.3) has a uniqu solution, it is ncssary that lj > 0, In th spcial cas in which QO(P)C V(Qkl(P», thn y = 0, r v = l, and B(P) = min lj = lj. Thn lj > 0 is a ncssary and sufficint condition for th xistnc of a uniqu solution. 489

6 J. T. ODN A ncssary and sufficint condition that th paramtrs U b positiv is furnishd by th rank condition. - THORM 3.3. Th pa ramtr u in (3.4).i. > 0 if and - k only if th following condition holds: for any 'I' Q-2(r), t (3.6) f 'I' an U ds = 0 V U (Qk1(n) ~ 'I' = 0 [4]. Additional proprtis of such approximations ar givn in RFRNCS 1. Pian, T. II. H., "lmnt Stiffnss Matrics for Boundary Compatibility and for Prscribd Boundary Strsss," Procdings of th First Confrnc on Matrix Mthods in Structural Mchanics, Wright-Pattrson Air Forc Bas, 1965, AFFDL-TR Brzzi, F., "Sur la Mthod ds lmnts Finis liybrids Pour I Problm Biharmoniqu," Numr. Math. 24, 1975, l03-l3l. 3. Raviart, P. A., "Hybrid Finit lmnt Mthods for Solving 2nd Ordr lliptic quations," Confrnc on Numrical Analysis, Royal Irish Acadmy, Dublin, Babuska, I., Odn, J. T., and L, J. K., "Mixd Hybrid Finit lmnts for Scond-Ordr lliptic Boundary- Valu Problms," (to appar). Author: J. T. Odn Dpt. of Arospac ngr. and ngr. Mchanics nginring Lab Bldg. 305 Th Univrsity of Txas at Austin Austin, Txas Acknowldgmnt: Th author's work on this projct was supportd through AFOSR Grant No