Finite element methods are not always optimal

Abstract

Consider & regularly-elliptic 2m-th order bound&ry problem LtI. = I '4'ith I e Hr{O), r ~-m. In previous work. we showed tht the 6nite-element method (FE~) using piecewise polynomi&!s of degree It is &!Ymptotic&!ly optim&l when It ~ 2m-1 + r. In this paper. we show th&t the FE~ is not a.symptotic&lly optim&l when this inequ&lity is violated. However, there exists &n &lgorithm. the Traub-Wa.silkowski-Wotni&kowski spline Iilgorithm. which is &lways optim&l. Moreover, the error of the FE~ can be arbitrvily larger th&n the erT"Or of the spline &lgorithm. We &!so obtain 8. nec~ary &nd sufficient condition for & G&lerkin method (or & gener&Jized G&lerkin method) to be• & spline Iilgorithm.

Galerkin method and optimal error algorithms in hilbert spaces

“…We have Hence, the error of the interpolatory algorithm is about 12% smaller than the error of the Galerkin algorithm. We now compare the errors for specific f E W. Let u be the solution of(6.7) for a fixed f and let i u,, u~ be the solutions of the interpolatory and Galerkin algorithm In this example we showed that for M=m, the error of the interpolatory algorithm may be considerably smaller than the error of the Galerkin algorithm for some functions f. This is true not only for this example but also for other problems of the form (2.3) (see for instance [6,7,8] From the triangle inequality, (6.9), (6.10) and (5.1) we find that II.-..II <s Ilstl. < 2_ cltsll,,,.…”

“…Thus, the Galerkin algorithm is optimal to within a constant for any parameters r and k. This should be contrasted with the result of Werschulz, see [6,7,8], who has considered linear information dependent only on f, i.e., 91,~ (f)= P'f, where P* is the orthogonal projector onto (v.Ie) *. He has proved that d(91.…”

“…Thus, the Galerkin method for the linear information 9l* is optimal to within a constant iff k => r + 1, whereas the Galerkin method for the nonlinear information 91re is optimal to within a constant independently whether k __> r + 1 or not. This also shows that the nonlinear information 91[e is significantly worse than the linear information 91" if r+l >k. [] It is known that dl,(L) is the diameter of linear information 9~,*(f)=P*f (see [6,7,8]). From Corollary 6.1 it follows that in the case M=m we can apply results obtained in [6,7,8] for d~ (L).…”

“…Werschulz has proved in [6,7,8] that the finite element method is not always optimal algorithm to solve (1.1) with a regularly-elliptic 2m-th order operator L, V=(Hm(Q)+essential boundary conditions), W= Hr(O), r>-m, Q c IR N. More precisely, the finite element method using a space V, of piecewise polynomials of degree k __> m is asymptotically (n-, oo) optimal iff k >=2m-1 +r. That is, its (asymptotical) optimality depends on the choice of the spaces V~ and W.…”

Optimal approximation of elliptic problems by linear and nonlinear mappings I

“…provided that no l,m is equal to k. This means that the finite number of singularity functions that is needed depends on the scale of spaces we are interested in, i.e., on the smoothness parameter k. According to (42), we have to estimate the Besov regularity of both, u S and u R , in the specific scale…”