A comparison of adaptive refinement techniques for elliptic problems
ACM Transactions on Mathematical Software (TOMS)
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Error estimators for nonconforming finite element approximations of the Stokes problem
Mathematics of Computation
Local bisection refinement for N-simplicial grids generated by reflection
SIAM Journal on Scientific Computing
A recursive approach to local mesh refinement in two and three dimensions
Journal of Computational and Applied Mathematics
A convergent adaptive algorithm for Poisson's equation
SIAM Journal on Numerical Analysis
A Posteriori Error Estimators for the Raviart--Thomas Element
SIAM Journal on Numerical Analysis
A posteriori error estimate for the mixed finite element method
Mathematics of Computation
Canonical construction of finite elements
Mathematics of Computation
Data Oscillation and Convergence of Adaptive FEM
SIAM Journal on Numerical Analysis
Convergence of Adaptive Finite Element Methods
SIAM Review
Adaptive Finite Element Methods with convergence rates
Numerische Mathematik
C0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains
Journal of Scientific Computing
Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation
SIAM Journal on Numerical Analysis
Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs
SIAM Journal on Numerical Analysis
The Morley element for fourth order elliptic equations in any dimensions
Numerische Mathematik
A posteriori error estimates for the Morley plate bending element
Numerische Mathematik
Optimality of a Standard Adaptive Finite Element Method
Foundations of Computational Mathematics
A unifying theory of a posteriori error control for nonconforming finite element methods
Numerische Mathematik
Framework for the A Posteriori Error Analysis of Nonconforming Finite Elements
SIAM Journal on Numerical Analysis
An Optimal Adaptive Finite Element Method for the Stokes Problem
SIAM Journal on Numerical Analysis
Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method
SIAM Journal on Numerical Analysis
An optimally convergent adaptive mixed finite element method
Numerische Mathematik
A new a posteriori error estimate for the Morley element
Numerische Mathematik
A Convergent Nonconforming Adaptive Finite Element Method with Quasi-Optimal Complexity
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
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Some reliable and efficient a posteriori error estimators are produced for a mixed finite element method (the Hellan-Herrmann-Johnson (H-H-J) method) for Kirchhoff plate bending problems (cf. [K. Hellan, Acta Polytech. Scand. Civil Engrg. Ser., 46 (1967), pp. 1-28], [L. Herrmann, J. Eng. Mech. Div. ASCE, 93 (1967), pp. 13-26], [C. Johnson, Numer. Math., 21 (1973), pp. 43-62]). Based on these results with $k=0,1$, where $k$ denotes the polynomial order of the discrete moment-field space, an adaptive mixed finite element method (AMFEM) is set up, and its convergence and complexity are studied thoroughly. The key points of the theoretical analysis include achieving a discrete Helmholtz decomposition and a discrete inf-sup condition, which serve as the main tools in deducing the quasi-orthogonality of the moment field and the discrete reliability of the estimator. It is shown that the AMFEM is a contraction for the sum of the moment-field error in an energy norm and the scaled error estimator between two consecutive adaptive loops. Moreover, an estimate for the AMFEM's complexity via the number of elements is developed.