Adaptive refinement and convergence

Let nonnegative local squared indicators sum to eta_h squared. In the convention used here, Dörfler marking selects a set whose sum is at least theta times that total, with theta strictly between zero and one. Some references put theta squared in this formula, so the numerical parameter must be translated before comparisons. Sorting by indicator size produces a simple bulk set, although practical near-minimal algorithms need not sort every cell exactly.

Refinement must actually reduce the marked local scales while maintaining a conforming shape-regular mesh family, for example through an admissible bisection strategy and the necessary closure refinements. The marked cells are not necessarily the entire refined set: neighbors may also need refinement to remove hanging-node incompatibilities. Choosing a large indicator does not authorize a degenerate sliver or guarantee that the final cell count rises by the number of marks. Mesh closure and overlay properties enter later complexity estimates.

For nested conforming spaces and a symmetric exact bilinear form, energy orthogonality gives the old error squared equal to the new error squared plus the squared energy change between discrete solutions. This supplies a quantitative descent identity. Estimator reduction on refined cells usually includes a perturbation term involving that solution change, while estimator stability controls unrefined cells. A suitable combination of error and estimator can then contract under the hypotheses of a classical adaptive theorem.

The more general axiomatic framework records stability, reduction, quasi-orthogonality, and discrete reliability. Discrete reliability bounds the change between discrete solutions by indicators near the refined region; it is different from the global error reliability of the previous lesson. In nonsymmetric or nonlinear settings exact Pythagoras may be replaced by a suitable quasi-orthogonality statement. An arbitrary refinement loop is not covered merely because its estimator decreases on a few observed steps.

A contraction indexed by refinement steps proves decay of a combined quantity, but the number of new cells per step can vary substantially. An optimal algebraic rate instead compares estimator decay with the number of degrees of freedom relative to a best-admissible-mesh approximation class. Proving it needs further assumptions such as a suitable marking range, quasi-minimal marking, controlled mesh closure and an overlay property. The threshold on the bulk parameter in an optimality theorem can be stricter than that used for basic convergence.

Data oscillation also affects how an estimator-based approximation class corresponds to a best solution-error class. In the cited axiomatic theory, efficiency is not required merely for estimator convergence, while it helps relate classes to error plus data resolution. Inexact solves, nonzero boundary approximation, and changing polynomial degree require further compatible analysis. These distinctions let one make a precise claim: a verified adaptive method may converge without a proof of the best possible error per cell, and a mesh that looks refined near a corner is not itself evidence of either theorem.