Applications of Lobatto Polynomials to an Adaptive Finite Element Method:
Estimating Solution Derivatives and Grid-to-Grid Interpolation

Hp-adaptive finite element methods require algorithms for estimating the error in the solution for different discretizations and for interpolating solutions between grids.  The first often involves estimating high-order derivatives of the solution.  The second typically leads to the solution of large linear systems.  The Lobatto interpolant, which possesses a variety of superconvergence properties for two-point boundary value problems and parabolic equations provides one approach to developing these algorithms.  I derive a ``Taylor-like'' series for the pointwise error in the Lobatto interpolant.  These estimates are extended to the finite element solution using the weak form of the equations.  Explicit formulas for the inverse of the Lobatto interpolation matrices are given.  Computational results illustrate the theory.