m . __ E! j_: 4 ._ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Computer methods in applied ELSEVIER Comput. Methods Appl. Mech. Engrg. 117 (1994) 225-230 mechanics and engineering On the limitations of bubble functions Leopold0 P. Franca* ’ * zyxwvutsrqponmlkjihgfedcbaZYXWVUTS Department of Mathematics, University of Colorado at Denver, P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364, USA Charbel Farhat Department of Aerospace Engineering Sciences, and Center for Aerospace Structures, University of Colorado at Boulder, Boulder, CO 803094429, USA zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR Abstract We present two examples that demonstrate no advantage in enriching a finite element subspace with bubble functions. 1. Quadratics or linears? Let us consider the problem of finding the scalar valued function u(x) defined on the unit interval and satisfying -u,,, =f on((Al) , (1) u(0) = u(l) = 0 . (2) Multiplying (1) by an arbitrary function u E H:(0), where H:(0) denotes the Hilbert space of functions satisfying (2) with square integrable value and derivative on the unit interval, and integrating on (0, 1) by parts, yields the variational formulation: Find u E H:(0) such that a(& v) =f(u) > u E HI@9 (3) with a(u7 v) = (u,,, u,J f(u) = (f?u) and (* , -) denotes the integral on (0,l). (4) (5) The standard Galerkin method is obtained by computing with the same variational formulation (3) on a subspace of H;(0) consisting of continuous functions that are piecewise polynomials on a partition of the unit interval. On each subinterval (or element) of the partition, we define basis functions in a fixed reference coordinate 5 that varies on (- 1,l). For piecewise linears, we have two basis functions on each element, those mapped to the 5 coordinate are given by &(S) = +(I - 5) 7 (6) * Corresponding author. ’ Visiting Associate Professor from LNCC, Rua, Lauro Miiller 455, 22290 Rio de Janeiro, Brazil. 004%7825/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0045-7825(93)E0219-X