  This manuscript presents an adaptive high order discretization technique for
elliptic boundary value problems. The technique is applied to an updated
version of the Hierarchical Poincar\'e-Steklov (HPS) method. Roughly speaking,
the HPS method is based on local pseudospectral discretizations glued together
with Poincar\'e-Steklov operators. The new version uses a modified tensor
product basis which is more efficient and stable than previous versions. The
adaptive technique exploits the tensor product nature of the basis functions to
create a criterion for determining which parts of the domain require additional
refinement. The resulting discretization achieves the user prescribed accuracy
and comes with an efficient direct solver. The direct solver increases the
range of applicability to time dependent problems where the cost of solving
elliptic problems previously limited the use of implicit time stepping schemes.
