**Interpolation Error-Based A Posteriori Error Estimation in
Three Dimensions:**

Interpolation error-based a posteriori error estimation for elliptic
and parabolic equations requires finding an interpolant that is

asymptotically equivalent to the finite element solution. In
three dimensions such an interpolant is obtained by taking the tensor

product of the one-dimensional Lobatto interpolant. Formulas
for the error in L^2 and H^1 of this interpolant are derived. These
formulas involve high-order derivatives of the solution of the partial
differential equations. Approximations of these derivatives from
jumps in first and second derivatives of the interpolant across element
boundaries and from differences of high-order derivatives of the interpolant
are presented. Projections of the interpolant onto finite element
spaces constructed from hierarchical and modified-hierarchical bases are
examined and the effect of using these smaller bases on the error estimation
strategy is considered.