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This report gives a brief discussion of the different problem areas under which research papers have been published during the term of the grant as well as a short description of the main results. In this research finite element methods were applied to a simple model of shear band formation. They were also used for elliptic and parabolic problems that arise in optimal control and for equations that simulate phase transitions. These finite element schemes were analyzed and implemented. Also, studies of space-time schemes where the Galerkin technique is used to discretize time as well as space were the focus of several papers. Finally, a last paper contains a thorough analysis of an adaptive time discretization technique.