# Proving convergence of adaptive finite elements - min res FEM?

There's a body of work out there dealing with the discrete convergence of adaptive finite element methods using error estimators. Most deal with proving the property

$\|u-u_{k+1}\|_U \leq (1-\alpha) \|u-u_{k}\|_U, \quad 0<\alpha < 1$

by relying on Galerkin orthogonality, monotonicity of error, error estimator information, and some other tricks to bound $\| u_{k+1}-u_k\|_U \geq \alpha\|u-u_k\|_U$.

I'm curious about minimum residual (least squares) finite element methods - there's Galerkin orthogonality, minimization/monotonicity of error, and a built in error "estimator" (the residual in the proper norm), but I haven't found any work proving convergence of adaptive minimum residual finite element methods. Am I overlooking some papers, or is there a roadblock to proving adaptive convergence for minimum residual FEM?

• I may be wrong but I expect that the usual analysis would carry through. – timur Nov 29 '13 at 7:32