Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation

Douglas R. Q. Pacheco; Olaf Steinbach

doi:10.1515/cmam-2021-0242

Article

Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation

Douglas R. Q. Pacheco and Olaf Steinbach

Published/Copyright: November 21, 2023

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From the journal Computational Methods in Applied Mathematics Volume 24 Issue 4

Abstract

Reconstructing the pressure from given flow velocities is a task arising in various applications, and the standard approach uses the Navier–Stokes equations to derive a Poisson problem for the pressure p. That method, however, artificially increases the regularity requirements on both solution and data. In this context, we propose and analyze two alternative techniques to determine p ∈ L 2 ⁢ ( Ω ) . The first is an ultra-weak variational formulation applying integration by parts to shift all derivatives to the test functions. We present conforming finite element discretizations and prove optimal convergence of the resulting Galerkin–Petrov method. The second approach is a least-squares method for the original gradient equation, reformulated and solved as an artificial Stokes system. To simplify the incorporation of the given velocity within the right-hand side, we assume in the derivations that the velocity field is solenoidal. Yet this assumption is not restrictive, as we can use non-divergence-free approximations and even compressible velocities. Numerical experiments confirm the optimal a priori error estimates for both methods considered.

Keywords: Pressure Poisson Equation; Ultra-Weak Variational Formulation; Least-Squares Formulation

MSC 2020: 65N30

Funding statement: The authors acknowledge Graz University of Technology for the financial support.

Acknowledgements

Part of the work was done when the first author was a PhD student at TU Graz within the lead-project: Mechanics, Modeling, and Simulation of Aortic Dissection.

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Received: 2021-12-21

Revised: 2023-10-13

Accepted: 2023-10-18

Published Online: 2023-11-21

Published in Print: 2024-10-01

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Articles in the same Issue

https://doi.org/10.1515/cmam-2021-0242

Keywords for this article

Pressure Poisson Equation; Ultra-Weak Variational Formulation; Least-Squares Formulation