Successive finite element methods for Stokes equations

TL;DR

This paper introduces a new finite element method for solving Stokes equations that efficiently computes velocity and pressure with optimal convergence, using a successive approach for pressure approximation.

Contribution

A novel finite element method for Stokes equations that sequentially computes velocity and pressure, achieving optimal convergence and reducing computational cost.

Findings

Achieves optimal order of convergence for pressure.

Superposes local pressures to improve accuracy.

Main computational cost is solving two linear systems.

Abstract

This paper will suggest a new finite element method to find a $P^4$-velocity and a $P^3$-pressure solving Stokes equations. The method solves first the decoupled equation for the $P^4$-velocity. Then, four kinds of local $P^3$-pressures and one $P^0$-pressure will be calculated in a successive way. If we superpose them, the resulting $P^3$-pressure shows the optimal order of convergence same as a $P^3$-projection. The chief time cost of the new method is on solving two linear systems for the $P^4$-velocity and $P^0$-pressure, respectively.