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

• mdn
In summary, the speaker is knowledgeable about applying boundary conditions in finite element methods, but struggles with implementing a divergence free condition for Maxwell's or Stokes equations. They mention a special element called an edge element that can be used, but they prefer to use a nodal base element. The speaker also explains that divergence free is not a boundary condition, but can be enforced by expressing the vector field in a certain way. They suggest using a stream function for 2D problems and a toroidal-poloidal decomposition for 3D problems. They also ask for an example of using a stream function for the Navier-Stokes equation in fluid flow.
mdn
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.

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.

mdn
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?

## 1. How do I ensure divergence-free solutions in nodal finite element method?

In order to ensure divergence-free solutions in nodal finite element method, you need to apply the divergence-free condition (∇.v=0) to the displacement field. This can be achieved by using a suitable basis function for the displacement field that satisfies the divergence-free condition.

## 2. What is the significance of the divergence-free condition in nodal finite element method?

The divergence-free condition is important in nodal finite element method because it ensures that the solution is physically meaningful and satisfies the conservation of mass. It also helps to avoid numerical instabilities and inaccuracies in the solution.

## 3. Can the divergence-free condition be applied to any type of element in nodal finite element method?

Yes, the divergence-free condition can be applied to any type of element in nodal finite element method. However, for some elements, it may be more challenging to satisfy the condition due to the complexity of their shape functions.

## 4. How can I check if my solution satisfies the divergence-free condition in nodal finite element method?

You can check if your solution satisfies the divergence-free condition by calculating the divergence of the displacement field at each node and ensuring that it is equal to zero. This can be done using numerical methods or by using analytical solutions for simpler problems.

## 5. Are there any limitations to using the divergence-free condition in nodal finite element method?

One limitation of using the divergence-free condition in nodal finite element method is that it may increase the computational cost and complexity of the solution. This is because it requires the use of specialized basis functions and may also require additional constraints to be imposed on the solution.

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