Skeleton-stabilized IsoGeometric Analysis: High-regularity Interior-Penalty methods for incompressible viscous flow problems

TL;DR

This paper introduces a Skeleton-stabilized IsoGeometric Analysis method for incompressible viscous flows that achieves high-regularity approximation, stable pressure-velocity coupling, and optimal convergence, with computational efficiency advantages over traditional methods.

Contribution

It develops a high-regularity stabilization technique for isogeometric analysis that allows using identical spaces for pressure and velocity, improving stability and efficiency.

Findings

Achieves oscillation-free solutions for Stokes and Navier-Stokes equations.

Demonstrates optimal convergence rates in 2D and 3D.

Shows smaller bandwidth in algebraic systems with B-splines, enhancing computational efficiency.

Abstract

A Skeleton-stabilized IsoGeometric Analysis (SIGA) technique is proposed for incompressible viscous flow problems with moderate Reynolds number. The proposed method allows utilizing identical finite dimensional spaces (with arbitrary B-splines/NURBS order and regularity) for the approximation of the pressure and velocity components. The key idea is to stabilize the jumps of high-order derivatives of variables over the skeleton of the mesh. For B-splines/NURBS basis functions of degree $k$ with $C^{\alpha}$-regularity ($0 \leq \alpha < k$), only the derivative of order $\alpha +1$ has to be controlled. This stabilization technique thus can be viewed as a high-regularity generalization of the (Continuous) Interior-Penalty Finite Element Method. Numerical experiments are performed for the Stokes and Navier-Stokes equations in two and three dimensions. Oscillation-free solutions and optimal convergence rates are obtained. In terms of the sparsity pattern of the algebraic system, we demonstrate that the block matrix associated with the stabilization term has a considerably smaller bandwidth when using B-splines than when using Lagrange basis functions, even in the case of $C^0$-continuity. This important property makes the proposed isogeometric framework practical from a computational effort point of view.