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A convergent nonconforming adaptive finite element method with quasi-optimal complexity. (English) Zbl 1208.65154

The authors present a simple adaptive nonconforming finite element method for the Poisson problem, which is a modification of the CH algorithm due to C. Carstensen and R. H. W. Hoppe [Numer. Math. 103, No. 2, 251–266 (2006; Zbl 1101.65102)]. In each step of the CH algorithm, the marking strategy is based on two indicators: an edge residual term and a volume term. An adaptive stopping criterion for the iterative solution of the discrete system in each step of the mesh refinement algorithm is introduced.
The proofs of geometric convergence of the error and asymptotically quasi-optimal complexity of the resulting meshes in the case of lower order finite elements in two space dimensions are carried out. It is also observed that the generalization to three dimensions and to general boundary conditions, as well as to general linear second order problems seems to be straightforward [cf. the above cited paper of the authors and another paper by K. Mekchay and R. H. Nochetto, SIAM J. Numer. Anal. 43, No. 5, 1803–1827 (2005; Zbl 1104.65103)].

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
65Y20 Complexity and performance of numerical algorithms
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