The damped Kuramoto-Sivashinsky equation has emerged as a fundamental tool for the understanding of the onset and evolution of secondary instabilities in a wide range of physical phenomena. Most existing studies about this equation deal with its asymptotic states on one-dimensional settings or on periodic square domains. We utilize a large-scale numerical simulation to investigate the asymptotic states of the damped Kuramoto-Sivashinsky equation on annular two-dimensional geometries and three-dimensional domains. To this end, we propose an accurate, efficient, and robust algorithm based on a recently introduced numerical methodology, namely, isogeometric analysis. We compared our two-dimensional results with several experiments of directed percolation on square and annular geometries, and found qualitative agreement.