My main focus is on the numerical solution of differential equations. A major focus has been the investigation of adaptive (moving mesh) methods for solving partial differential equations (PDEs). These methods are essential to numerically solve most complex physical problems, where both high speed parallel processors and sophisticated computer graphics play key roles as well. Recent emphasis has been on the study of adaptive methods for PDEs in higher spatial dimensions and the development of corresponding mathematical software. Another interest is in dynamical systems (where qualitative studies of solutions to differential equations is the main concern). This area of research, applicable many problems in science and engineers, relates to the study of highly nonlinear physical processes and necessitates numerical investigations.