Interpolation Error-Based *A Posteriori* Error Estimation
for Two-Point Boundary Value Problems and Parabolic Equations in One Space
Dimension

I derive *a posteriori* error estimates for two-point boundary
value problems and parabolic equations in one dimension based on interpolation
error estimates. The interpolation error estimates are obtained from
an extension of the error formula for the Lagrange interpolating polynomial
in the case of symmetrically-spaced interpolation points. From this
formula pointwise and H^1 seminorm *a priori* estimates of the interpolation
error are derived. The interpolant in conjunction with the *a priori*
estimates is used to obtain asymptotically exact *a posteriori* error
estimates of the interpolation error. These *a posteriori* error
estimates are extended to linear two-point boundary value problems and
parabolic equations. Computational results demonstrate the convergence
of *a posteriori* error estimates and their effectiveness when combined
with an *hp*-adaptive code for solving parabolic systems.