[Paper Review] Finite Element Technique for Solving the Stream Function Form of a Linearized Navier-Stokes Equations Using Argyris Element

Faisal Fairag, Noah Al-Mulla|ArXiv.org|Jun 3, 2004

Advanced Numerical Methods in Computational Mathematics12 references3 citations

TL;DR

This paper presents a finite element method using Argyris elements to solve the stream function form of the linearized Navier-Stokes equations, focusing on minimizing matrix bandwidth through node reordering to improve computational efficiency. The approach achieves optimal convergence and stability, with numerical results showing accurate solutions for a manufactured exact solution on a unit square domain using Re=1 and adaptive quadrature rules.

ABSTRACT

The numerical implementation of finite element discretization method for the stream function formulation of a linearized Navier-Stokes equations is considered. Algorithm 1 is applied using Argyris element. Three global orderings of nodes are selected and registered in order to conclude the best banded structure of matrix and a fluid flow calculation is considered to test a problem which has a known solution. Visualization of global node orderings, matrix sparsity patterns and stream function contours are displayed showing the main features of the flow.

Motivation & Objective

• To develop a stable and efficient finite element discretization for the stream function formulation of the linearized Navier-Stokes equations.
• To evaluate the impact of different global node orderings on the bandwidth and sparsity structure of the resulting stiffness matrix.
• To implement and test the conjugate gradient stabilized (BiCGSTAB) method for solving the nonsymmetric, sparse linear systems arising from the finite element formulation.
• To validate the numerical scheme using a manufactured exact solution on a unit square domain with known analytical behavior.
• To analyze computational cost, convergence rates, and iterative solver performance across varying mesh refinements and quadrature rules.

Proposed method

• The stream function formulation eliminates pressure and enforces incompressibility automatically, reducing the problem to solving a fourth-order biharmonic-type PDE.
• The weak form of the linearized problem is derived using Sobolev spaces $ H^2_0(\Omega) $, with bilinear forms $ a(\psi,\varphi) $ and $ b(\psi^*;\psi,\varphi) $ representing the Laplacian and convective terms.
• Argyris elements are employed as conforming finite elements in $ H^2_0(\Omega) $, ensuring $ C^1 $ continuity necessary for fourth-order problems.
• Three global node reordering strategies are tested to minimize matrix bandwidth and improve the condition number of the stiffness matrix.
• The BiCGSTAB iterative solver is applied to the resulting sparse, nonsymmetric linear system, with convergence monitored via residual and iterate norm tolerances.
• Numerical integration uses four and six Gauss quadrature points to assess accuracy and computational cost in evaluating element matrices.

Experimental results

Research questions

• RQ1 How does node reordering affect the bandwidth and sparsity pattern of the finite element matrix in $ H^2 $-conforming discretizations?
• RQ2 What is the impact of quadrature point selection (4 vs. 6) on the accuracy and computational cost of the Argyris finite element solution?
• RQ3 How do the number of BiCGSTAB iterations and CPU time scale with mesh refinement ($ h = 1/3, 1/5, 1/9 $) for the linearized problem?
• RQ4 Can the Argyris finite element method achieve optimal convergence rates for the stream function formulation with a manufactured exact solution?
• RQ5 How does the choice of initial guess $ \psi^* $, derived from solving the biharmonic equation, affect the convergence of the iterative solver?

Key findings

• Node reordering 1 produced the narrowest matrix bandwidth and lowest CPU time (28.45 seconds), outperforming alternatives in computational efficiency.
• For $ h = 1/9 $, the method achieved a relative error of $ 1.423 \times 10^{-4} $ in the $ L^2 $-norm and $ 6.986 \times 10^{-4} $ in the $ H^1 $-norm, with 567.5 BiCGSTAB iterations.
• The number of arithmetic operations increased significantly with mesh refinement: from ~17.5 million at $ h=1/3 $ to ~3.5 billion at $ h=1/9 $, reflecting the $ O(h^{-4}) $ scaling of $ H^2 $-conforming elements.
• The use of six quadrature points slightly increased error and operation count compared to four points, but improved stability and convergence behavior.
• Streamline visualizations confirmed physically consistent flow patterns across all mesh sizes, validating the method’s ability to capture vortex structures.
• The method demonstrated robust convergence for all tested mesh sizes and quadrature rules, with residual and iterate norms falling below the tolerance $ 10^{-5} $.