Adaptive finite element methods with convergence rates.

*(English)*Zbl 1063.65120The authors present estimates of the convergence rate for an adaptive finite element method applied to the two-dimensional Poisson equation on a two-dimensional polygonal domain. The adaptive method is based on that of P. Morin, R. Nochetto and K. Siebert [SIAM J. Numer. Anal. 38, 466–488 (2000; Zbl 0970.65113)] using piecewise linear elements and newest vertex bisection. The current work introduces a coarsening step that is important for limiting growth in the number of elements.

It is proved that the mesh refinement required to acheive accuracy \(\epsilon\) is constructed from available quantities and that the number of nodes in the refined mesh is bounded by an expression involving the number of nodes in the original mesh, norms of the forcing term and the solution, and a fractional power of \(\epsilon\). Further, a preconditioned conjugate gradient method is shown to require a bounded number of iterations independent of mesh refinement so that the solution process requires total time proportional to the number of nodes in the refined mesh.

The proofs rely heavily on the deterministic nature of newest vertex bisection, the existence of a local error estimator whose sum over elements bounds the global energy norm of the error, and results of P. Binev and R. DeVore [“Fast computation in tree approximation”, Numer. Math. 97, No. 2, 193–217 (2004; Zbl 1075.65158)] on adaptive nonlinear interpolation.

It is proved that the mesh refinement required to acheive accuracy \(\epsilon\) is constructed from available quantities and that the number of nodes in the refined mesh is bounded by an expression involving the number of nodes in the original mesh, norms of the forcing term and the solution, and a fractional power of \(\epsilon\). Further, a preconditioned conjugate gradient method is shown to require a bounded number of iterations independent of mesh refinement so that the solution process requires total time proportional to the number of nodes in the refined mesh.

The proofs rely heavily on the deterministic nature of newest vertex bisection, the existence of a local error estimator whose sum over elements bounds the global energy norm of the error, and results of P. Binev and R. DeVore [“Fast computation in tree approximation”, Numer. Math. 97, No. 2, 193–217 (2004; Zbl 1075.65158)] on adaptive nonlinear interpolation.

Reviewer: Myron Sussman (Bethel Park)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65Y20 | Complexity and performance of numerical algorithms |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |