@inproceedings{735757ed3f6b45dfa22733e5d0b92508,

title = "Meshfree finite differences for vector Poisson and pressure Poisson equations with electric boundary conditions",

abstract = "We demonstrate how meshfree finite difference methods can be applied to solve vector Poisson problems with electric boundary conditions. In these, the tangential velocity and the incompressibility of the vector field are prescribed at the boundary. Even on irregular domains with only convex corners, canonical nodalbased finite elements may converge to the wrong solution due to a version of the Babu{\v s}ka paradox. In turn, straightforward meshfree finite differences converge to the true solution, and even high-order accuracy can be achieved in a simple fashion. The methodology is then extended to a specific pressure Poisson equation reformulation of the Navier-Stokes equations that possesses the same type of boundary conditions. The resulting numerical approach is second order accurate and allows for a simple switching between an explicit and implicit treatment of the viscosity terms.",

keywords = "Finite-differences, High-orde, Incompressible, Manufactured solution, Meshfree, Navier-Stokes, Pressure Poisson equation, Reformulation, Vector Poisson equation",

author = "Dong Zhou and Benjamin Seibold and David Shirokoff and Prince Chidyagwai and Rosales, {Rodolfo Ruben}",

note = "Funding Information: P. Chidyagwai, R. R. Rosales, B. Seibold, and D. Zhou wish to acknowledge support by the National Science Foundation. This work was supported through grants DMS–1115269 and DMS-1115278. Furthermore, R. R. Rosales would like to acknowledge partial support by NSF through grant DMS–1318942, and B. Seibold would like to acknowledge partial support by NSF through grants DMS–1318641 and DMS–1318709. D. Shirokoff acknowledges partial funding by NSERC. Publisher Copyright: {\textcopyright} Springer International Publishing Switzerland 2015.; 7th International Workshop on Meshfree Methods for Partial Differential Equations, 2013 ; Conference date: 09-09-2013 Through 11-09-2013",

year = "2015",

doi = "10.1007/978-3-319-06898-5_12",

language = "English (US)",

series = "Lecture Notes in Computational Science and Engineering",

publisher = "Springer Verlag",

pages = "223--246",

editor = "Michael Griebel and Schweitzer, {Marc Alexander}",

booktitle = "Meshfree Methods for Partial Differential Equations VII",

address = "Germany",

}