Neumann and Dirichlet BCs in discrete Poisson EQ

In summary, it is possible to impose Dirichlet boundary conditions and specify a known derivative of the function in a certain region, but it is important to ensure compatibility to avoid an over constrained system.
  • #1
freechus9
9
0
Hello all.

I am working on a problem and I am getting a bit confused.

Suppose we have a poisson equation that we wish to solve subject to certain boundary conditions. Let's say we are in 1D (we can later extrapolate to more dimensions).

Is it possible to impose Dirichlet boundary conditions on the boundary, but also specify a known derivative of the function in a certain region?

In a physical context, I want to solve poisson's eq but want to specifiy the potential at the boundary and set the field to zero in a certain region. Will this overconstrain the system?

Thanks and please let me know if this is unclear.
 
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  • #2
Yes, you can impose Dirichlet boundary conditions on the boundary and also specify a known derivative of the function in a certain region. However, this will lead to an over constrained system unless the values you set for the boundary and the derivative are compatible. For example, if the derivative of the function is zero at the boundary then it will be possible to solve the system.
 

1. What are Neumann and Dirichlet boundary conditions in discrete Poisson equations?

Neumann and Dirichlet boundary conditions are two types of constraints that are imposed on the boundary of a domain for solving a discrete Poisson equation. Neumann boundary conditions specify the value of the derivative of the solution at the boundary, while Dirichlet boundary conditions specify the value of the solution itself at the boundary.

2. What is the difference between Neumann and Dirichlet boundary conditions?

The main difference between Neumann and Dirichlet boundary conditions is what they specify at the boundary of a domain. Neumann boundary conditions specify the derivative of the solution, while Dirichlet boundary conditions specify the value of the solution itself. This means that Neumann boundary conditions involve the normal vector to the boundary, while Dirichlet boundary conditions do not.

3. How do Neumann and Dirichlet boundary conditions affect the solution of a discrete Poisson equation?

Neumann and Dirichlet boundary conditions play a crucial role in determining the unique solution of a discrete Poisson equation. They provide essential information about the behavior of the solution at the boundary of the domain, which is necessary for solving the equation. Without these boundary conditions, the solution may not be well-defined or may have multiple solutions.

4. Can Neumann and Dirichlet boundary conditions be combined?

Yes, Neumann and Dirichlet boundary conditions can be combined in some cases. This is known as mixed boundary conditions and involves specifying both the value of the solution and its derivative at different parts of the boundary. Mixed boundary conditions are often used in more complex problems where different types of constraints are needed at different boundaries.

5. How are Neumann and Dirichlet boundary conditions applied in practice?

In practice, Neumann and Dirichlet boundary conditions are typically applied by modifying the equations that govern the behavior of the solution. This is done by incorporating the boundary conditions into the discretization process, resulting in a modified set of equations that can be solved numerically. The specific implementation of these boundary conditions may vary depending on the numerical method used to solve the discrete Poisson equation.

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