Constrained first-order System least-squares for nonlinear problems

This thesis addresses two nonlinear problems: elastoplastic deformation and sea ice dynamics. For both problems, a posteriori error estimators in the form of Least-Squares functionals are derived and validated. For the first problem, the focus is on making the estimation stable with respect to the incompressible limit case. The elastoplastic deformation problem presents challenges due to its nature as a variational inequality, resulting in reduced regularity. Consequently, a classical Newton method is not suitable for solving this problem; therefore, the thesis discusses the use of a semismooth Newton method instead. In analyzing the sea ice problem, a general theory for error estimators in Banach spaces is first established and then applied. The Gauss-Newton method is used for the iterative solving of the sea ice problem. The well-posedness of this variational problem is also discussed. Both problems are supported by numerical examples, which not only confirm the theoretical findings, but also provide additional cases that extend beyond those that are theoretically covered.