Mixed finite elements on sparse grids

Summary.

This paper generalizes the idea of approximation on sparse grids to discrete differential forms that include $\vec H({\rm div}; \Omega$)- and $\vec (H{\bf curl}; \Omega)$-conforming mixed finite element spaces as special cases. We elaborate on the construction of the spaces, introduce suitable nodal interpolation operators on sparse grids and establish their approximation properties. We discuss how nodal interpolation operators can be approximated. The stability of $\vec H({\rm div}; \Omega)$-conforming finite elements on sparse grids, when used to approximate second order elliptic problems in mixed formulation, is investigated both theoretically and in numerical experiments.