Optimality of a Standard Adaptive Finite Element M

Authors: Rob Stevenson

Publish Date: 2006/07/05

Volume: 7, Issue: 2, Pages: 245-269

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Abstract

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 within a tolerance ε 0 in energy norm by a continuous piecewise linear function on some partition with Oε1/s triangles and one knows how to approximate the righthand 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 466488 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 the results generalize in several respects

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