A locally calculable $P^3$-pressure in a decoupled method for incompressible Stokes equations

TL;DR

This paper introduces a low-cost, decoupled finite element method for incompressible Stokes equations that efficiently computes a $P^4$-velocity and a locally calculable $P^3$-pressure with optimal convergence, avoiding singularities.

Contribution

The paper presents a novel decoupled finite element approach that computes a $P^4$-velocity and a locally calculable $P^3$-pressure for Stokes equations, improving efficiency and handling singularities.

Findings

Locally calculable $P^3$-pressure with optimal convergence.

Reduced computational cost by decoupling velocity and pressure.

Overcomes issues with singular vertices or corners.

Abstract

This paper will suggest a new finite element method to find a $P^4$-velocity and a $P^3$-pressure solving incompressible Stokes equations at low cost. The method solves first the decoupled equation for a $P^4$-velocity. Then, using the calculated velocity, a locally calculable $P^3$-pressure will be defined component-wisely. The resulting $P^3$-pressure is analyzed to have the optimal order of convergence. Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the $P^4$-velocity. Besides, the method overcomes the problem of singular vertices or corners.