An Implicit Interpolation Error-Based Error Estimation Strategy for HP-Adaptivity

*Hp*-adaptive finite element methods require error estimates of
the solution at the current order and one order higher. In [Moore,94]
it was proved that a *p*-refinement (hierarchical) error estimation
strategy was asymptotically exact for nonlinear parabolic equations.
An extension of this strategy was proposed for computing higher-order estimates
[Flaherty, Moore95]. Recently a new approach, interpolation error
based (IEB) error estimation, for constructing *a posteriori* error
estimates at both orders has been developed. I show that: i) IEB
error estimation can be applied to semilinear two-point boundary value
problems and parabolic equations in one space dimension; ii) the hierarchical
estimator is an implicit IEB method and thus, works for semilinear two-point
boundary value problems; iii) the hierarchical extension for computing
higher-order error estimates is asymptotically exact. Computational
results illustrating the theory and comparing the implicit (hierarchical)
strategy with the earlier explicit IEB methods are presented.