A posteriori error estimators for the first-order least-squares finite element method
The aim of the authors is to construct an a posteriori error estimator based on the computation of adequate norms of the residual for the model problem written in a first-order least-squares finite element setting. Their analysis is based on the argument given in the previously obtained and published study on the weak coupling of solutions of least-square methods. In the first-order least-squares method, the built-in functional provides a natural error estimator. However, when someone is interested in certain quantities of interest, then the built-in energy error estimator may not be optimal and may overestimate the actual error. This phenomenon is observed in the accomplished numerical experiments. The authors are constructing a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method for second-order elliptic problems. In the beginning of the paper, they describe the first-order least-squares method and state known a priori error estimates. Then, they are deriving an a posteriori error estimator based on the residuals of the least-squares finite element solution. It is proven that an a posteriori error estimator yields the global upper bounds and the local lower bounds on the error of interest. Finally, some numerical results are presented to show the actual performance of the obtained estimator. The effectivity indices computed from numerical examples show that constructed error estimator is asymptotically exact. According to the author's knowledge, this is the first result providing an a posteriori error estimator different from the usual built-in energy error indicators.
scientific article; zbMATH DE number 5804885
scientific article