Prove the augmented-boundary error estimate and recover fourth-order convergence

Establish a rigorous proof that the augmented boundary-variable error for the augmented high-order compact scheme satisfies the discrete estimate \(\|\mathbf{E}^q\|_2=O(h^{7/2})\), and determine whether a sharper treatment of the boundary contribution yields the optimal fourth-order error estimate suggested by the numerical results.

Background

The convergence theorem for the augmented high-order compact method is conditional on an O(h7/2)O(h^{7/2}) discrete L2L^2-error bound for the augmented boundary variable qq. The paper rigorously controls the remaining interior and stability terms under this assumption, obtaining an O(h7/2)O(h^{7/2}) estimate for the numerical solution.

The authors explicitly state that a proof of the required boundary estimate is unavailable. Numerical experiments instead indicate fourth-order convergence, while the proved estimate loses one-half order because the analysis does not sharply control the boundary-driven component. A rigorous boundary estimate and a sharper analysis are therefore needed to establish the observed optimal fourth-order rate.

References

The convergence analysis yields a conditional $O(h{7/2})$ error estimate, provided that the augmented boundary variables satisfy an $O(h{7/2})$ discrete $L2$-error bound. Although a rigorous proof of this boundary estimate is not yet available, it is consistently supported by the numerical results, under this assumption, the remaining convergence analysis is rigorous. The resulting estimate does not recover the full fourth-order rate observed numerically. This loss of half an order appears to arise from a non-sharp treatment of the boundary contribution. The numerical results suggest that the theoretical bound is not optimal and that the actual convergence rate is fourth order.

— A Matrix-free Augmented High Order Compact Solver for Variable-Coefficient Biharmonic Problems  (2609.16478 - Li et al., 15 Sep 2026) in Section 3.2, immediately after the two-dimensional convergence proof; see also the final paragraph of Section 3.3