Optimality of a Standard Adaptive Finite Element Method

In this paper, an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever for some s > 0, the solution can be approximated to accuracy O(n−s) in energy norm by a continuous piecewise linear function on some partition with n triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate with piecewise constants, then the adaptive method produces approximations that converge with this rate, taking a number of operations that is of the order of the number of triangles in the output partition. The method is similar in spirit to that from [SINUM, 38 (2000), pp.466–488] by Morin, Nochetto, and Siebert, and so in particular it does not rely on a recurrent coarsening of the partitions. Although the Poisson equation in two dimensions with piecewise linear approximation is considered, it can be expected that the results generalize in several respects.