How to apply divergence free (∇.v=0) in nodal finite element method?

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SUMMARY

The discussion focuses on applying the divergence-free condition (∇·v=0) in the context of the nodal finite element method (FEM) for Maxwell's and Stokes equations. The user expresses a preference for nodal base elements over edge elements, which are typically used to enforce this condition. For 2D problems, the use of a stream function is recommended, while for 3D problems, a toroidal-poloidal decomposition is suggested. The user seeks clarification on the potential formulation for curl equations in fluid dynamics, specifically regarding the Navier-Stokes equations.

PREREQUISITES
  • Nodal finite element method (FEM)
  • Maxwell's equations
  • Navier-Stokes equations
  • Stream function formulation
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  • Research the implementation of stream functions in 2D Navier-Stokes equations using nodal FEM.
  • Explore toroidal-poloidal decomposition techniques for 3D fluid dynamics problems.
  • Study the mathematical foundations of divergence-free vector fields in computational fluid dynamics.
  • Examine potential formulations for curl equations in the context of finite element analysis.
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Researchers, engineers, and developers working on computational fluid dynamics, particularly those focused on implementing nodal finite element methods for solving Maxwell's and Stokes equations.

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I know how to apply boundary condition like Dirichlet, Neumann and Robin but i have been struggling to apply divergence free condition for Maxwells or Stokes equations in nodal finite element method. to overcome this difficulties a special element was developed called as edge element but i don't want to use this element because my complete programming depends on the nodal base element.
 
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Divergence free is not usually a boundary condition; it is, certainly in both the examples you give, a condition which is true throughout the domain. The way to enforce it is to express your vector field in a manner which satisfies the condition automatically.

For 2D problems, use a stream function.

For 3D problems, you can use a toroidal-poloidal decomposition.
 
Can I use such potential formulation for curl, curl equation of Maxwell or Navier- Stokes equation in fluid flow?
 
Would you give me the example of NS equation using stream function in fem?
 

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