Two-side a posteriori error estimates for the DWR method
This work provides theoretical guarantees for error estimation in the DWR method, benefiting computational scientists using adaptive finite elements.
The authors derive two-sided a posteriori error estimates for the DWR method, providing lower bounds that confirm estimator efficiency for nonlinear PDEs and functionals. Numerical tests validate the theory.
In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we derive lower bounds yielding the efficiency of the error estimator. These results hold true for both nonlinear partial differential equations and nonlinear functionals of interest. Furthermore, the DWR method employed in this work accounts for balancing the discretization error with the nonlinear iteration error. We also perform careful studies of the remainder term that is usually neglected. Based on these theoretical investigations, several algorithms are designed. Our theoretical findings and algorithmic developments are substantiated with some numerical tests.