TY - JOUR

T1 - High-order finite element methods for a pressure Poisson equation reformulation of the Navier–Stokes equations with electric boundary conditions

AU - Rosales, Rodolfo Ruben

AU - Seibold, Benjamin

AU - Shirokoff, David

AU - Zhou, Dong

N1 - Funding Information:
The authors wish to acknowledge support by the National Science Foundation, United States of America through the grants DMS-1719637 and DMS-1614043 (Rosales), DMS-1719640 (Seibold and Zhou), DMS-2012271 (Seibold), and DMS-1719693 and DMS-2012268 (Shirokoff).
Funding Information:
This research includes calculations carried out on HPC resources supported in part by the National Science Foundation, United States of America through major research instrumentation grant number 1625061 and by the US Army Research Laboratory under contract number W911NF-16-2-0189 .
Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - Pressure Poisson equation (PPE) reformulations of the incompressible Navier–Stokes equations (NSE) replace the incompressibility constraint by a Poisson equation for the pressure and a suitable choice of boundary conditions. This yields a time-evolution equation for the velocity field only, with the pressure gradient acting as a nonlocal operator. Thus, numerical methods based on PPE reformulations are representatives of a class of methods that have no principal limitations in achieving high order. In this paper, it is studied to what extent high-order methods for the NSE can be obtained from a specific PPE reformulation with electric boundary conditions (EBC). To that end, implicit–explicit (IMEX) time-stepping is used to decouple the pressure solve from the velocity update, while avoiding a parabolic time-step restriction; and mixed finite elements are used in space, to capture the structure imposed by the EBC. Via numerical examples, it is demonstrated that the methodology can yield at least third order accuracy in space and time.

AB - Pressure Poisson equation (PPE) reformulations of the incompressible Navier–Stokes equations (NSE) replace the incompressibility constraint by a Poisson equation for the pressure and a suitable choice of boundary conditions. This yields a time-evolution equation for the velocity field only, with the pressure gradient acting as a nonlocal operator. Thus, numerical methods based on PPE reformulations are representatives of a class of methods that have no principal limitations in achieving high order. In this paper, it is studied to what extent high-order methods for the NSE can be obtained from a specific PPE reformulation with electric boundary conditions (EBC). To that end, implicit–explicit (IMEX) time-stepping is used to decouple the pressure solve from the velocity update, while avoiding a parabolic time-step restriction; and mixed finite elements are used in space, to capture the structure imposed by the EBC. Via numerical examples, it is demonstrated that the methodology can yield at least third order accuracy in space and time.

KW - Electric boundary conditions

KW - IMEX schemes

KW - Incompressible Navier–Stokes

KW - Mixed finite elements

KW - Pressure Poisson equation

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U2 - 10.1016/j.cma.2020.113451

DO - 10.1016/j.cma.2020.113451

M3 - Article

AN - SCOPUS:85094322362

VL - 373

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

M1 - 113451

ER -