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Beginner
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## Does anyone know if the MKL Fast Poisson Solver can be used for the nonlinear Poisson eqn?

Hello,

Is it possible to modify the Intel MKL Fast Poisson Solver for the problem of type:

Δ .[K(u). Δ(u) ] = f

where Δ is the gradient symbol (I didn't find the reverse triangle in the special characters). K(u) is a positive differentiable function dependent on the position. Check the equation here:

The difference between the above equation and the demonstrated Poisson eqn. at MKL Poisson solver page is the term K(u).

5 Replies
Black Belt
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If the boundary values of u are known, you can reduce the problem to the linear Poisson problem by employing the Kirchoff transformation φ =  \int K(u) du. What types of boundary conditions are you given?

Beginner
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The boundary values are known.

I have a 2D domain where the right and left boundaries are period and the top and bottom are Neumann BCs. However, in my case, I dont have an exact function for K(u) to integrate. K values are given over the domain. Hence, K(x,y) is a very in-homogenous distribution. Let me rewrite the equation as:

Δ .[ K(x,y). Δ( u(x,y) ) ] = f

where the BCs are:

Neumann @ top/bottom

Period @ left/right

Black Belt
80 Views

Originally you wrote K = K(u), now you say K = K(x,y). The two are not equivalent, and Kirchoff's transformation does not help if K is not known in terms of u.

Beginner
80 Views

I was also confused when I first generated this topic. I saw the 1st equation form in that paper and I asked my question. Nevertheless, is there any fast method to solve the below equation other than Successive over relation (SOR) method?

Δ .[ K(x,y). Δ( u(x,y) ) ] = f

Black Belt
80 Views

If you know K(x,y) and the source/sink function f is not dependent on u, the problem is linear. The title of the thread could be misleading.

In contrast to the case where K is a constant, the coefficients in the difference equations vary over the grid. You may use any sparse linear equation solver for your problem, such as Pardiso.