TY - JOUR
T1 - Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation
AU - Pacheco, Douglas R.Q.
AU - Steinbach, Olaf
N1 - Publisher Copyright:
© 2023 Walter de Gruyter GmbH, Berlin/Boston 2023.
PY - 2023/11/21
Y1 - 2023/11/21
N2 - 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) {p\in L^{2}(\Omega)}. 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.
AB - 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) {p\in L^{2}(\Omega)}. 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.
KW - Least-Squares Formulation
KW - Pressure Poisson Equation
KW - Ultra-Weak Variational Formulation
UR - http://www.scopus.com/inward/record.url?scp=85178038160&partnerID=8YFLogxK
U2 - 10.1515/cmam-2021-0242
DO - 10.1515/cmam-2021-0242
M3 - Article
AN - SCOPUS:85178038160
SN - 1609-4840
JO - Computational Methods in Applied Mathematics
JF - Computational Methods in Applied Mathematics
ER -