Optimality of a Standard Adaptive Finite Element Method |
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Authors: | Rob Stevenson |
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Institution: | (1) Department of Mathematics, Utrecht University, The Netherlands |
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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 O(ε-1/s) 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 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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Keywords: | |
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