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.