Convergence and Optimality of Adaptive Least Squares Finite Element Methods

The first-order div least squares finite element methods (LSFEMs) allow for an immediate a posteriori error control by the computable residual of the least squares functional. This paper establishes an adaptive refinement strategy based on some equivalent refinement indicators. Since the first-order div LSFEM measures the flux errors in $H$(div), the data resolution error measures the $L^2$ norm of the right-hand side $f$ minus the piecewise polynomial approximation $\Pi f$ without a mesh-size factor. Hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment, e.g., by the thresholding second algorithm due to Binev and DeVore. The resulting novel adaptive LSFEM with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class.

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