Multilevel Isogeometric Projection Stabilization via Quasi-Interpolation for Advection-Dominated Problems

TL;DR

This paper introduces a multilevel isogeometric stabilization method for advection-dominated problems that reduces oscillations and parameter sensitivity, with proven theoretical support and effective numerical results.

Contribution

It develops a novel multilevel projection stabilization technique using B-spline quasi-interpolants, avoiding sensitive parameters and discontinuous auxiliary spaces.

Findings

Significantly reduces spurious oscillations in benchmarks.

Achieves comparable performance to nonlinear shock-capturing schemes.

Reduces parameter sensitivity in stabilization methods.

Abstract

This paper presents a novel multilevel projection-based stabilization method for advection-dominated convection--diffusion problems within the framework of Isogeometric Analysis. The proposed approach extracts and penalizes fine-scale fluctuations using continuous B-spline quasi-interpolants, avoiding both the highly sensitive parameters used in residual-based stabilization methods and the discontinuous auxiliary spaces required by classical Local Projection Stabilization. Stabilization is applied hierarchically across nested levels of the discrete space via explicit mesh-dependent weights. We establish the theoretical foundation of the method by deriving a priori error estimates, supplemented by a discrete inf-sup condition established for the one-dimensional setting with constant advection under a numerically validated stability hypothesis that ensures robust streamline derivative control. Numerical experiments on stringent benchmarks demonstrate the method's ability to significantly reduce spurious oscillations across a variety of regimes, including the limiting cases of pure advection and advection--reaction. Notably, despite being a fully linear formulation, the method achieves significant reduction of undershoots near sharp layers, delivering performance comparable to complex nonlinear shock-capturing schemes. Furthermore, by utilizing a robust global parameter scaling, the proposed approach significantly alleviates the parameter sensitivity that typically affects residual-based alternatives, reducing the strong dependence on problem-dependent tuning.