A new approach to energy-based sparse finite-element spaces

^{We show that the logarithmic factor in the standard error estimatefor sparse finite element (FE) spaces in arbitrary dimensiond is removable in the energy (H1) norm. Via a penalized sparsegrid condition, we then propose and analyse a new version ofthe energy-based sparse FE spaces introduced first in Bungartz(1992, Dünne Gitter und deren Anwendung bei der adaptivenLösung der dreidimensionalen Poisson-Gleichung. Dissertation.Munich, Germany: TU München) and known to satisfy an optimalapproximation property in the energy norm.}