Least-squares formulations for Stokes equations with non-standard boundary conditions -- A unified approach

TL;DR

This paper introduces a unified non-conforming least-squares spectral element method for solving Stokes equations with diverse non-standard boundary conditions, avoiding complex regularity estimates and achieving exponential accuracy.

Contribution

A novel unified approach that handles various boundary conditions for Stokes equations without relying on ADN theory-based regularity estimates.

Findings

Achieved exponential accuracy in numerical experiments

Developed stability and error estimates for the method

Validated effectiveness in 2D and 3D cases

Abstract

In this paper, we propose a unified non-conforming least-squares spectral element approach for solving Stokes equations with various non-standard boundary conditions. Existing least-squares formulations mostly deal with Dirichlet boundary conditions are formulated using ADN theory-based regularity estimates. However, changing boundary conditions lead to a search for parameters satisfying supplementing and complimenting conditions [4] which is not easy always. Here we have avoided ADN theory-based regularity estimates and proposed a unified approach for dealing with various boundary conditions. Stability estimates and error estimates have been discussed. Numerical results displaying exponential accuracy have been presented for both two and three-dimensional cases with various boundary conditions.