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.