Higher-order spectral element method for the stationary Stokes interface problem in two dimensions

TL;DR

This paper introduces a higher-order spectral element method for the 2D Stokes interface problem with piecewise constant viscosity, achieving exponential accuracy and validated through numerical examples.

Contribution

It develops a novel spectral element approach with blending functions for interface resolution, providing stability and exponential error decay for velocity and pressure.

Findings

Method is exponentially accurate for velocity and pressure.

Stability and error estimates are rigorously proven.

Numerical examples confirm theoretical results.

Abstract

This article presents a higher-order spectral element method for the two-dimensional Stokes interface problem involving a piecewise constant viscosity coefficient. The proposed numerical formulation is based on least-squares formulation. The mesh is aligned with the interface, and the interface is completely resolved using blending element functions. The higher-order spectral element functions are nonconforming, and the same-order approximation is used for both velocity and pressure variables. The interface conditions are added to the minimizing functional in appropriate Sobolev norms. Stability and error estimates are proven. The proposed method is shown to be exponentially accurate in both velocity and pressure variables. The theoretical estimates are validated through various numerical examples.