An optimal adaptive finite element method. (English) Zbl 1081.65112

Stevenson, Rob

SIAM J. Numer. Anal. 42, No. 5, 2188-2217 (2005).

The Dirichlet problem in a polygonal domain \(\Omega\) in \({\mathbb R}^2\) for a second-order elliptic equation with disconinuous coefficients is considered. The finite element methods on triangulations of \(\bar\Omega \) are investigated and the main purpose of the paper is to generalize some published results dealing with convergence analysis of adaptive variants of the methods. “Nonconforming” partitions of triangles and special “wavelet” bases are in the center of the attention.
The author formulates (in several places) the next strange result “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 \(O(n^{-s})\), then given an \(\varepsilon>0 \), the adaptive method produces an approximation subordinate to a partition with \(O(\varepsilon ^{-1/s}) \) triangles, in only \(O(\varepsilon ^{-1/s}) \) operations.”
But it is a very known fact that the accuracy of the method depends on \(n\) only if the triangulations are quasi-uniform. Besides, the author allows the right-hand side to be in \(H^{-1}(\Omega)\); that gives no additional smoothness of the solution. It should be noted that in the theory of asymptotically optimal algorithms for classes of problems usually the estimates of \(N(\varepsilon)\)-widths (in the sense of Kolmogorov) for compact sets are used.

Reviewer: Evgenij D’yakonov (Moskva)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
65Y20	Complexity and performance of numerical algorithms
65T60	Numerical methods for wavelets
35J25	Boundary value problems for second-order elliptic equations
35R05	PDEs with low regular coefficients and/or low regular data
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs

Keywords:

adaptive finite element methods; quasi-optimization; quasi-wavelet basis; optimal computational complexity; a posteriori error estimator; discontinuous coefficients; Dirichlet problem; second-order elliptic equation; disconinuous coefficients; convergence