A Developed Numerical Algorithm for Solving Stokes Equations Using Adaptive Elements

Abstract

The Stokes model is an idealized, canonical model of incompressible viscous fluid flow and appears in many types of applications, from microfluidics to geophysical simulations.  AFEM holds best promise for optimal efficiency but demands also reliable error estimators and proven convergence. More recent substitutes, i.e., locally adaptive penalty methods, relax the incompressibility constraint to ease treatment but lose physical fidelity by allowing non-zero velocity divergence, constituting an essential deterrent to mass-sensitive applications. We present an original adaptive FEM algorithm based on an inexact but residual-based a posteriori error estimator based on Dörfler marking and recent vertex bisection refinement. We describe a new method guaranteeing inf-sup stability for any adaptive mesh, based on Taylor-Hood (P₂–P₁) element pair. We give strong guarantees of linear contraction and quasi-optimal complexity showing that the algorithm converges at the optimal algebraic rate in terms of degrees of freedom. We tested the method on two canonical benchmarks, a smooth manufactured solution over the unit square, and the singular L-shaped domain. We evaluated the error in energy norm, order of convergence, effectivity index, and number of degrees of freedom, and contrasted these against uniform refinement and latest developments in adaptive techniques like the penalty technique by Fang. The algorithm exhibited (optimal) second-order convergence on the smooth problem, and optimal first-order convergence (rate 1.0) on the L-shaped domain, in which uniform refinement saturated (with rate 0.5<0.5). The adaptive scheme resulted in 50% error reduction of the entire error compared to uniform refinement at 2,000 DOFs and dominated the penalty scheme [14] of Fang (error 1.12e1. vs 1, respectively). 30e-1). The error estimator was in all instances evidently reliable (effectivity index Θ ≈ 2.0). Importantly, our technique enforces the constraint ∇·uh = 0 in an exact sense and thereby preserves mass — a desirable property over penalty schemes. This paper presents an effective and largely inexpensive adaptive finite element method (AFEM) of the Stokes equations at optimal accuracy at very little computational expense while under the condition of minimizing inequalities. We demonstrate this interplay between engineering and mathematics through an integration of shown error estimation, stable refinement, and formal convergence analysis. The adaptive penalty methods can't impose strict incompressibility, and our algorithm achieves an entirely new level of fidelity for adaptive flow simulations. It becomes evident and potential to extend to 3D and time-dependent Navier-Stokes problems.