, Volume 97, Issue 2, pp 219-268

Adaptive Finite Element Methods with convergence rates

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Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages of adaptive strategies versus non-adaptive strategies. The present paper modifies the adaptive method of Morin, Nochetto, and Siebert [17] for solving the Laplace equation with piecewise linear elements on domains in ℝ2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm (which is equivalent to the H 1 norm). Namely, it is shown that whenever s>0 and the solution u is such that for each n≥1, it can be approximated to accuracy O(n −s ) in the energy norm by a continuous, piecewise linear function on a triangulation with n cells (using complete knowledge of u), then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process. Moreover, the number of arithmetic computations in the proposed method is also of order O(n) for each n≥1. The construction and analysis of this adaptive method relies on the theory of nonlinear approximation.

Mathematics Subject Clasification (2000): 65N30, 65Y20, 65N12, 65N50, 68W40, 68W25.
This work has been supported by the Office of Naval Research Contract Nr. N00014-03-10051, the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation Grants DMS 0221642, DMS 9872890 the Deutsche Forschungsgemeinschaft grant SFB 401, the European Community’s Human Potential Programme under Contract HPRN-CT-2002-00286, ‘‘Breaking Complexity’’.