Greedy Local Refinement for Analysis-Suitable ${\rm T}$-Splines with Linear Complexity

Abstract

Achieving linear complexity is crucial for demonstrating optimal convergence rates in adaptive refinement. It has been shown that the existing linear complexity local refinement algorithm for ${\rm T}$-splines generally produces more degrees of freedom than the existing greedy refinement, which lacks linear complexity. This paper introduces a novel greedy local refinement algorithm for analysis-suitable ${\rm T}$-splines, which achieves linear complexity and requires fewer control points than existing algorithms with linear complexity. Our approach is based on the observation that confining refinements around each ${\rm T}$-junction to a pre-established feasible region ensures the algorithm’s linear complexity. Building on this constraint, we propose a greedy optimization local refinement algorithm that upholds linear complexity while significantly reducing the degrees of freedom relative to previous linear complexity local refinement methods.