Stabilization of isogeometric finite element method with optimal test functions computed from $L_2$ norm residual minimization

TL;DR

This paper compares stabilization techniques in isogeometric analysis, deriving an optimal test function method from $L_2$ residual minimization, and evaluates its performance against other methods for advection-diffusion problems.

Contribution

It introduces a least-squares finite element method with optimal test functions for isogeometric analysis, ensuring stability and comparing it with existing stabilization techniques.

Findings

Least-squares method outperforms others at low Péclet numbers.

SUPG and Galerkin/least-squares better handle strongly advection-dominated problems.

Derived coercivity bounds for B-spline basis functions.

Abstract

We compare several stabilization methods in the context of isogeometric analysis and B-spline basis functions, using an advection-dominated advection\revision{-}diffusion as a model problem. We derive (1) the least-squares finite element method formulation using the framework of Petrov-Galerkin method with optimal test functions in the $L_2$ norm, which guarantee automatic preservation of the \emph{inf-sup} condition of the continuous formulation. We also combine it with the standard Galerkin method to recover (2) the Galerkin/least-squares formulation, and derive coercivity constant bounds valid for B-spline basis functions. The resulting stabilization method are compared with the least-squares and (3) the Streamline-Upwind Petrov-Galerkin (SUPG)method using again the Eriksson-Johnson model problem. The results indicate that least-squares (equivalent to Petrov-Galerkin with $L_2$-optimal test functions) outperforms the other stabilization methods for small P\'eclet numbers, while strongly advection-dominated problems are better handled with SUPG or Galerkin/least-squares.