An Optimal Adaptive Finite Element Method

• Authors: Rob Stevenson
• Venue: SIAM Journal on Numerical Analysis
• Year: 2004

Although existing adaptive finite element methods for solving second order elliptic equations often perform better in practical computations than nonadaptive ones, usually they are not even proven to converge. Only recently in the work of Dörfler [SIAM J. Numer. Anal., 33 (1996), pp. 1106--1124] and that of Morin, Nochetto, and Siebert [SIAM J. Numer. Anal., 38 (2000), pp. 466--488], adaptive methods were constructed for which convergence could be demonstrated. However, convergence alone does not imply that the method is more efficient than its nonadaptive counterpart. In [ Numer. Math.}, 97 (2004), pp. 219--268], Binev, Dahmen, and DeVore added a coarsening step to the routine of Morin, Nochetto, and Siebert, and proved that the resulting method is quasi-optimal in the following sense: If the solution is such that for some s 0, the error in energy norm of the best continuous piecewise linear approximations subordinate to any partition with n triangles is $\mathcal{O}(n^{-s})$, then given an $\eps0$, the adaptive method produces an approximation with an error less than $\eps$ subordinate to a partition with $\mathcal{O}(\eps^{-1/s})$ triangles, in only $\mathcal{O}(\eps^{-1/s})$ operations.In this paper, employing a different type of adaptive partition, we develop an adaptive method with properties similar to those of Binev, Dahmen, and DeVore's method, but unlike their method, our coarsening routine will be based on a transformation to a wavelet basis, and we expect it to have better quantitative properties. Furthermore, all our results are valid uniformly in the size of possible jumps of the diffusion coefficients. Since the algorithm uses solely approximations of the right-hand side, we can even allow right-hand sides in $H^{-1}(\Omega)$ that lie outside $L_2(\Omega)$, at least when they can be sufficiently well approximated by piecewise constants. In our final adaptive algorithm, all tolerances depend on an a posteriori estimate of the current error instead of an a priori one; this can be expected to provide quantitative advantages.