Skeleton-stabilized IsoGeometric Analysis: High-regularity Interior-Penalty methods for incompressible viscous flow problems
For researchers in computational fluid dynamics, this provides a practical isogeometric framework that simplifies discretization by using equal-order B-splines/NURBS while maintaining stability and efficiency.
The paper proposes a Skeleton-stabilized IsoGeometric Analysis (SIGA) method for incompressible viscous flow problems, achieving oscillation-free solutions and optimal convergence rates for Stokes and Navier-Stokes equations in 2D and 3D. The method uses identical spaces for velocity and pressure, with stabilization of high-order derivative jumps, and demonstrates smaller matrix bandwidth compared to Lagrange basis functions.
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 $C^α$-regularity ($0 \leq α< k$), only the derivative of order $α+1$ 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.