  This paper constructs adaptive sparse grid collocation method onto arbitrary
order piecewise polynomial space. The sparse grid method is a popular technique
for high dimensional problems, and the associated collocation method has been
well studied in the literature. The contribution of this work is the
introduction of a systematic framework for collocation onto high-order
piecewise polynomial space that is allowed to be discontinuous. We consider
both Lagrange and Hermite interpolation methods on nested collocation points.
Our construction includes a wide range of function space, including those used
in sparse grid continuous finite element method. Error estimates are provided,
and the numerical results in function interpolation, integration and some
benchmark problems in uncertainty quantification are used to compare different
collocation schemes.
