  This paper constitutes our initial effort in developing sparse grid
discontinuous Galerkin (DG) methods for high-dimensional partial differential
equations (PDEs). Over the past few decades, DG methods have gained popularity
in many applications due to their distinctive features. However, they are often
deemed too costly because of the large number of degrees of freedom of the
approximation space, which are the main bottleneck for simulations in high
dimensions. In this paper, we develop sparse grid DG methods for elliptic
equations with the aim of breaking the \emph{curse of dimensionality}. Using a
hierarchical basis representation, we construct a sparse finite element
approximation space, reducing the degrees of freedom from the standard
{$O(h^{-d})$ to $O(h^{-1}|\log_2 h|^{d-1})$} for $d$-dimensional problems,
where $h$ is the uniform mesh size in each dimension. Our method, based on the
interior penalty (IP) DG framework, can achieve accuracy of $O(h^{k}|\log_2
h|^{d-1})$ in the energy norm, where $k$ is the degree of polynomials used.
Error estimates are provided and confirmed by numerical tests in
multi-dimensions.
