An H1-Conforming Solenoidal Basis for Velocity Computation on Powell-Sabin Splits for the Stokes Problem

TL;DR

This paper introduces a new solenoidal basis for velocity computation in the Stokes problem, reducing problem size and improving efficiency by eliminating pressure variables and ensuring sparse, symmetric positive definite matrices.

Contribution

A novel H1-conforming solenoidal basis for velocity on Powell-Sabin splits is constructed, enabling efficient, pressure-free velocity computation with proven properties.

Findings

Reduces linear system size and complexity.

Ensures sparse, symmetric positive definite matrices.

Demonstrates computational efficiency gains.

Abstract

A solenoidal basis is constructed to compute velocities using a certain finite element method for the Stokes problem. The method is conforming, with piecewise linear velocity and piecewise constant pressure on the Powell-Sabin split of a triangulation. Inhomogeneous Dirichlet conditions are supported by constructing an interpolating operator into the solenoidal velocity space. The solenoidal basis reduces the problem size and eliminates the pressure variable from the linear system for the velocity. A basis of the pressure space is also constructed that can be used to compute the pressure after the velocity, if it is desired to compute the pressure. All basis functions have local support and lead to sparse linear systems. The basis construction is confirmed through rigorous analysis. Velocity and pressure system matrices are both symmetric, positive definite, which can be exploited to solve their corresponding linear systems. Significant efficiency gains over the usual saddle-point formulation are demonstrated computationally.