Interpolation Error-Based A Posteriori Error Estimation in Three Dimensions:
A First Step

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.