Commentationes Mathematicae Universitatis Carolinae

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1 Commentationes Mathematicae Universitatis Carolinae Ivo Babuška A remark to the finite element method Commentationes Mathematicae Universitatis Carolinae, Vol. 12 (1971), No. 2, Persistent URL: Terms of use: Charles University in Prague, Faculty of Mathematics and Physics, 1971 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Commentstiones Mathematicae Universitatis Carolinae 12,2 (1971) A REMARK TO THE FINITE ELEMENT METHOD Ivo BABUSKA, College Park I. Introduction. The finite element method, its different versions has become very important today in the theory and practice. See e.g. El] - L17J and others. One important form of the method is such that the approximate solution AA,j^ (X) on the domain JM has the following form (1.1) AJUJ^CX) ** X e (Jh } M,) g> ( j^ - **> ) where the function <p (X) has a compact support and the sum is over all multiintegers Jfe m (Jk f... t *%J ^i integers) such that Many times the function g? (x ), x 6 R.^ is taken in a special form (1.2) <p(x) -JT ytx 4 ), *- 1, 2,...,m,, JCsYoc,,.,., x^>. For some domains e.g. a square, for cp (X) smooth enough so called "inverse theorem" is valid for some sequences *% t + ~ *> 1 f"f M,^-* 0. AMS, Primary 65N30 Ref , Secondary

3 where W, (A) ia the usual Sobolav apace and C(Z) doe a not depend on i, Many vary important thaorama of the theory of the finite elementa method are baaed on the validity of the mentioned w inver8a theorem". Under thia aaaumption e.g. the computation of derivatives (aee 1143) could be made much easier as in tl4l» simply by derivating the approximate aolution. Without thia "inveree theorem" proofa of many thaorama will break down. The purpoaa of thia note ia to show that for smooth domains the inverae theorem ia not true for any choice of A.. To simplify the arguments wa ahall deal only with the two dimanaional case i.e. il c R. The inverse theorem ia in fact equivalent with a geometric property of the domain SI. Let us explain thia property. Let -0- c JL^ be a bounded domain. For every Jh, > 0 let ua define the mesh? a tijhjk,, K Z )} with k,, I intagara. Further let 0*^ ba the aat of the all aquarea s t ^ l s * ** * C x 4» x 2 *» ^"^ *-- ^ < >u (At +4), h i <c x^ <* Jh, (Jl + 4)J and < *«& *» <*" A* 0 1, P*«HJk,A), B% t e (&}. Let ua define now the function y^ (H, I) on f^ by the following way % - ib8 -

4 (1.4) ү к <.Ь,t) - /WlдM? Jh z wheгe <*tl л л ì meana the mea Ч...Ł г\ Jl. Furthar let (1.5) ^ (<h) m mum vf^(*,,l ), (h 7 l) e? K. The crucial question ia whether we may chooae a sequence fa^ > 0, ^.-sr 4,2 ;., auch that (1.6) Ibrrv Jim* v (Jh,. ) > oc. certainly if il ia a square then auch a sequence 4- clearly exists. In the next chapter we shall show that such a sequence does not exist in the case that the boundary has bounded curvature. For the simplicity we shall prove the statement for a circular domain only. By the same idea it is obviously possible to prove the general statement for domains with bounded curvature in %^. 11 Theorem. As we said we shall study the case of the circular domain It in R^ only* Let (2.11 X (*) s <L C*,^) j **> ty* * x, 1 } be a circle with the radius <** Further let us denote > ( x, <y-) the distance of the point (*x,/y,) to the boundary of the circle X. (tv). Let us prove the following theorem. 369

5 Theorem. Let ua denote (2.2) at (to) as /rrum, <* (M, f I) Jk, Jt integer a. (M,,l)e KM * Then for every sequence H,^ > <x> wa hav«* (2.3) tjumu Am* *e C*,.) -* CO + Thie theorem is obviously equivalent with the statement which we introduced in the firat chapter in the case that -Q- ia a circle. Proof. 1) Let ua denote by \l(fc) the number of all C*,, Z) in K (*>), (2.4) UU)» Obviously UC/c) ia a not decreasing atap function. Let i i f-i.j.4 > ft- *** *** aequence of all pointa of the diecontinuity of the function IL(fo) * Denote further (2.51 A U ) = Mux, 18, - 8,, I. Our theorem will be proved if we shall ahow that (2.6) &*%, AU) m 0. *,-+ CO In fact (2.7) aec^) -6 X(^). 2) Let ua aaaume on the contrary that there exist a a sequence &i f & m 4, 2,. #,, ty > K^, /c^ rco auch that 370 -

6 (2 8). JLvm, sunf $e(k,,) > 3 A > 0 * y CO + Then for all /_,.? ^ > }f we have (2.9) Xin,. ) * 2A. So we may construct the subsequence _ y such that (2.10) anđ Let us denote u t - tў. * - h - ** (2.11) <D. = : ЄІ. - з where [ex] means integral part of _x. We have (2.12) G^ **.,, a>^ -i-.*. Because a> and a)* -e -i is certainly the point of the discontinuity of the function IL (fv ) we have (2.12) ^ =4 ^_, * ^ + A4 g^._* ^ + 4 and (2.13) &, * cfy+0 where 0 ^ «, A. _r Let us define now the numbers oc^ rasp. y3' such that (2.14) <aj**j - f^-c^->^>*, (2.15) o> 2 $ + / * _ > : % -*.-«_,-><_.*.*>' 371

7 Therefore (2.16) oc- _ I/Co, + # )-,! > e.-" C2 + ff. OJ.-'*)^ (2.17) /J. _ a) (0.+A)b (2 + C«T. + _.>_>:" )--. a f & v %> But Therefore a>^ > ^ - A, 0 _& % < A. (2.18) oc. _ a>2 <** 2* (A +, ) a and hence (2.19) A- oc >: 2*cu* [(% + _})! *(< +! ) * j. But -t 1 4 (2.20) mim, t(x +A) 2 '-x 2 C1i-, ) 3 = Therefore for,*, big enough i.e. g, -> K we have (2.21) &-<*<_. A. - oc_ * > 2* (*. ( «_. -- 4)' A )* C with Cz* Q and hence for -t - jtf" >- N we have /3_ - oc s 2 ^ 2 i ' # Jr and therefore there exists an integer JtV- auch that % ^ *_J ^ **# for a11 * ^ ^a _L _t 4r and hence $._-(<_). *.Je. )* ia a point of discontinuity of the function VL(tt) But obviously

8 and this is a contradiction and the theorem ia proved. A closer analysis of the proof shows that obviously by the same idea the general statement introduced in 1 may be proved i.e. for m, -dimensional domain with bounded curvature of the boundary. R e f e r e n c e s [1] 0. FIX and G. STRANG: Fourier analysis of the finite element method in Ritz-Galerkin theory. Studies in Applied Mathematics 48(1969), C21 G. STRANG and G. FIX: A Fourier analysis of the finite element variational method. To appear. [3J J. P. AUBIN: Behavior of the error of the approximate solution of boundary value problems for linear elliptic operators by Galerkin's and finite differences methods, Annali della Scuola Normale di Pisa 21(1967), } J.P* AUBIN: Some variational methods for non-homogeneous boundary value problems, Calcolo, [5] J.H. BRAMBLE and A.H. SCHATZ: Raleigh, Ritz-Galerkin methods for Dirichlet's problem using subspaces without boundary conditions, Comm.Pure Appl.Math. (to appear). [6] J. NITSCHE: Ober ein Variationsprinzip fur Losung von Dirichlet Problemen bei Verwendung von Teilraumen, die keinen Randbedingungen unterworfen sind (to appear)

9 [7] J«H* BRAMBLE and A.H. SCHATZ: Least squares methods for 2nd order elliptic boundary value problems. To appear. [8] L.A. O0ANESJAN, L.A. RUCHOVEC: A study of rates of convergence of some variational difference schemes for elliptic equations of second order in a two dimensional domain with smooth boundary (Russian). Z.VySislit.Mat.i Mat. Fiz.9(1969), [9] I. BABUSKA: Error bounds for finite element method, Tech.Note BN-630(1969),University of Maryland, Institute for Fluid Dynamics and Applied Mathematics. Numerdsche M&them&tik 16, 322 ~ 333,1972. flo] I. BABUSKA: The rate of convergence for the finite element method, Tech.Note BN-646(1970), University of Maryland,Institute for Fluid Dynamics and Applied Mathematics. To appear in SIAM Journal,Num.Math. [Ill I* BABUSKA: The finite element method for elliptic equations with discontinuous coefficients, Tech.Note BN-631(1969),University of Maryland, Institute for Fluid.Dynamics and Applied Mathematics, Computing 5 f C1970), [12] I. BABUSKA: Finite element method for domains with corners, Techn.Note BN-636(1970),University of Maryland,Institute for Fluid Dynamics and Applied Mathematics, to appear in Computing

10 [13] I. BABUSKA: The finite element method for elliptic differential equations, Tech.Note BN-653, (1970),University of Maryland,Institute for Fluid Itynamics and Applied Mathematics. Numerical Solution of Partial Differential Equations II,SYNSPADE 1970.Edited by B.Hubbard, Academic Press 1971,New York-London,69-107* [14] I. BABUSKA: Computation of derivatives in the finit? element method, Tech.Note BN-650(1970),University of Maryland,Institute for Fluid Dynamics and Applied Mathematics, Comment. Math.Univ.Carolinae 11,3(1970), [15] I. BABUSKA: Approximation by Hill functions, Tech. Note BN-648(1970),University of Maryland, Institute for Fluid Dynamice and Applied Mathematics, Comment.Math-Univ.Carolinae ii f AKi9ro), rar-au * [16] I. BABUSKA: The finite element method for infinite domains I, Tech.Note BN-670(1970),University of Maryland, Inst.for Fluid Dynamics and Applied Mathematics. [17] I. BABUSKA and J. SEGETHOVX and K. SEGETH: Numerical experiments with finite element method I, Tech.Note BN-669(1970),University of Maryland, Inst.for Fluid Dynamics and Applied Mathematics* - ЭГ5-

11 Inatituta for Fluid Dynamica and Applied Mathematics Univaraity of Maryland Collage Park, Maryland U.S.A. (Oblatua )