Quasi-optimal convergence of AFEM based on separate marking, Part II

• Published in: Journal of numerical mathematics Vol. 23; no. 2; pp. 157 - 174
• Main Author: Rabus, Hella
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.06.2015; Walter de Gruyter GmbH
• Subjects: Adaptive algorithms; adaptive finite element method; AFEM; Algorithms; ANCFEM; Computational fluid dynamics; Convergence; Crouzeix-Raviart; Estimators; Finite element analysis; Finite element method; linear elasticity; Mathematical analysis; Mathematical models; nonconform FEM; Nonlinear programming; nonstandard FEM; optimal convergence; separate marking; Stokes equations
• ISSN: 1570-2820; 1569-3953
• DOI: 10.1515/jnma-2015-0011

Various applications in computational fluid dynamics and solid mechanics motivate the development of reliable and efficient adaptive algorithms for nonstandard finite element methods (FEMs). Standard adaptive finite element algorithms consist of the iterative loop of the basic steps Solve, Estimate, Mark, and Refine. For separate marking strategies, this standard scheme may be universalised. The (total) error estimator is split into a volume term and an error estimator term. Since the volume term is independent of the discrete solution, an appropriate data approximation may be realised by a high degree of local mesh refinement. This observation results in a natural adaptive algorithm based on separate marking. Its quasi-optimal convergence is proven in this second part for the pure displacement problem in linear elasticity and the Stokes equations and nonconforming Crouzeix-Raviart FEM. The proofs follow the same general methodology as for the Poisson model problem in the first part of this series. The numerical experiments confirm the optimal convergence rates and reveal its flexibility.