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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?

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  • $\begingroup$ I may be wrong but I expect that the usual analysis would carry through. $\endgroup$ – timur Nov 29 '13 at 7:32
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I didn't read this in detail, but I think it should have the gist of the kind of proof you're looking for.

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  • $\begingroup$ Thanks for the reference; however, they just demonstrate equivalence of the least squares error with the energy norm. There's an extra step in the proof of adaptive convergence which demonstrates decrease of error given some bulk-chasing adaptive marking strategy. isc.tamu.edu/publications-reports/technical_reports/0709.pdf has an example for an IPDG method. $\endgroup$ – Jesse Chan Sep 23 '13 at 0:15

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