Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Marcin Łoś, Tomasz Służalec, Maciej Paszyński, Eirik Valseth

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 Péclet numbers, while strongly advection-dominated problems are better handled with SUPG or Galerkin/least-squares.

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

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 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 -optimal test functions) outperforms the other stabilization methods for small Péclet numbers, while strongly advection-dominated problems are better handled with SUPG or Galerkin/least-squares.

Paper Structure

This paper contains 17 sections, 3 theorems, 47 equations, 14 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Introduction of the methods
  3. Model problems
  4. First model problem
  5. Second model problem
  6. Bubnov-Galerkin method for advection-difussion problems
  7. Least-squares finite element method for advection-diffusion model problem
  8. Bubnov-Galerkin method on a manually adapted grid
  9. Bubnov-Galerkin method on a uniform grid
  10. Petrov-Galerkin method with optimal test functions in $L_2$ setup on a uniform grid
  11. Galerkin/least-squares formulation
  12. The Streamline-Upwind Petrov-Galerkin method
  13. Streamline Upwind Petrov Galerkin method on uniform grid
  14. Comparison of the methods
  15. Stability Analysis
  16. ...and 2 more sections

Key Result

Lemma 1

There exists a constant $C>0$ independent of $h$ such that:

Figures (14)

  • Figure 1: The exact solutions for the first problem for $\epsilon \in \{0.1, 0.01, 0.003\}$
  • Figure 2: The exact solutions for the second problem for $\epsilon \in \{0.01, 0.001, 0.0001\}$
  • Figure 3: B-spline basis functions for non-uniform grid for the Eriksson-Johnsons problem.
  • Figure 4: The first model problem, for $\epsilon \in \{ 0.1, 0.01, 0.003 \}$. The solution of the Bubnov-Galerkin problem on a manually refined grid.
  • Figure 5: The second model problem, Eriksson-Johnson for $\epsilon \in \{0.01, 0.001, 0.0001 \}$. The solution of the Bubnov-Galerkin problem on a manually refined grid.
  • ...and 9 more figures

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof