An $H^1$-Conforming Solenoidal Basis for Velocity Computation on Powell-Sabin Splits for the Stokes Problem
Year: 2024
Author: Jeffrey M. Connors, Michael Gaiewski
International Journal of Numerical Analysis and Modeling, Vol. 21 (2024), Iss. 2 : pp. 181–200
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2024-1007
International Journal of Numerical Analysis and Modeling, Vol. 21 (2024), Iss. 2 : pp. 181–200
Published online: 2024-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 20
Keywords: Divergence free Powell-Sabin Stokes finite element saddle-point.