Convergence of adaptive finite element methods in computational mechanics
Applied Numerical Mathematics
Convergence analysis of an adaptive edge element method for Maxwell's equations
Applied Numerical Mathematics
Convergence of a standard adaptive nonconforming finite element method with optimal complexity
Applied Numerical Mathematics
Convergence of an Adaptive Finite Element Method for Controlling Local Energy Errors
SIAM Journal on Numerical Analysis
Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method
SIAM Journal on Numerical Analysis
An Adaptive Finite Element Approximation of a Variational Model of Brittle Fracture
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
A Posteriori Error Control for Discontinuous Galerkin Methods for Parabolic Problems
SIAM Journal on Numerical Analysis
Convergence of an Adaptive Mixed Finite Element Method for Kirchhoff Plate Bending Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Estimator reduction and convergence of adaptive BEM
Applied Numerical Mathematics
Advances in Engineering Software
Journal of Scientific Computing
Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity
Computational Mechanics
Convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
On Adaptive Eulerian---Lagrangian Method for Linear Convection---Diffusion Problems
Journal of Scientific Computing
New error estimates of the Morley element for the plate bending problems
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications
Computers & Mathematics with Applications
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We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive the optimal cardinality of the AFEM. We show that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.