Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions
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,...
| Main Authors: | , , , , |
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| Format: | Article |
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Springer
2018
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| Online Access: | http://hdl.handle.net/1721.1/116017 https://orcid.org/0000-0002-8828-5930 |
| _version_ | 1811095226522009600 |
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| author | Zhou, Dong Seibold, Benjamin Shirokoff, David Chidyagwai, Prince Rosales, Rodolfo |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Zhou, Dong Seibold, Benjamin Shirokoff, David Chidyagwai, Prince Rosales, Rodolfo |
| author_sort | Zhou, Dong |
| collection | MIT |
| description | 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 Babuška 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: Meshfree Finite-differences; Navier-Stokes; Incompressible; Vector Poisson equation; Pressure Poisson equation; Reformulation; Manufactured solution; High-order |
| first_indexed | 2024-09-23T16:13:06Z |
| format | Article |
| id | mit-1721.1/116017 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T16:13:06Z |
| publishDate | 2018 |
| publisher | Springer |
| record_format | dspace |
| spelling | mit-1721.1/1160172022-09-29T18:58:53Z Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions Zhou, Dong Seibold, Benjamin Shirokoff, David Chidyagwai, Prince Rosales, Rodolfo Massachusetts Institute of Technology. Department of Mathematics Rosales, Rodolfo 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 Babuška 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: Meshfree Finite-differences; Navier-Stokes; Incompressible; Vector Poisson equation; Pressure Poisson equation; Reformulation; Manufactured solution; High-order National Science Foundation (U.S.) (Grant DMS–1318942) 2018-05-31T14:06:26Z 2018-05-31T14:06:26Z 2015 2018-05-30T13:53:11Z Article http://purl.org/eprint/type/ConferencePaper 978-3-319-06897-8 978-3-319-06898-5 1439-7358 2197-7100 http://hdl.handle.net/1721.1/116017 Zhou, Dong et al. “Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions.” Meshfree Methods for Partial Differential Equations VII (November 2014): 223–246 © 2015 Springer International Publishing Switzerland https://orcid.org/0000-0002-8828-5930 http://dx.doi.org/10.1007/978-3-319-06898-5_12 Meshfree Methods for Partial Differential Equations VII Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Springer arXiv |
| spellingShingle | Zhou, Dong Seibold, Benjamin Shirokoff, David Chidyagwai, Prince Rosales, Rodolfo Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions |
| title | Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions |
| title_full | Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions |
| title_fullStr | Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions |
| title_full_unstemmed | Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions |
| title_short | Meshfree Finite Differences for Vector Poisson and Pressure Poisson Equations with Electric Boundary Conditions |
| title_sort | meshfree finite differences for vector poisson and pressure poisson equations with electric boundary conditions |
| url | http://hdl.handle.net/1721.1/116017 https://orcid.org/0000-0002-8828-5930 |
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