Talk:Screened Poisson equation

Solutions: first para

I'm no expert in this particular branch of PDEs otherwise I would just edit it, but it looks like this:

"When λ is zero, the equation reduces to Poisson's equation. Therefore, when λ is very small, the solution approaches that of the unscreened :Poisson equation"

should instead say that the solution to the unscreened Poisson equation should approach that of the Poission equation for small λ. Tweet7 (talk) 13:35, 20 November 2012 (UTC)[reply]

Application

Might be of interest..

A. Pigazzini et al, have made a notable contribution to the study of the homogeneous case in the context of differential geometry. Their research, focusing particularly on Einstein warped product manifolds, explores cases where the warped function satisfies the homogeneous version of this equation. This work demonstrates that under specific conditions, the manifold size, Ricci curvature, and screening parameter are interconnected via a quadratic relationship, revealing deeper geometric implications.

https://arxiv.org/abs/2407.20381 (Preprint of manuscript accepted for publication in the Contemporary Mathematics series of the American Mathematical Society - Book entitled: "Recent Advances in Differential Geometry and Related Areas" to appear in 2025). Maurixxt (talk) 21:05, 27 October 2024 (UTC)[reply]

Filtering in Topology Optimization

Often referred to as "Helmholtz filter", a very common filtering technique in topology optimization is actually a screened Poisson equation. Solve this equation for the filtered density ${\tilde {\rho }}$

$-r^{2}\nabla ^{2}{\tilde {\rho }}+{\tilde {\rho }}=\rho ,$

where $r$ is the filter radius, and $\rho$ is the original density field. The filtering removes oscillatory features and enforces a minimum length scale related to $r$ .

https://onlinelibrary.wiley.com/doi/10.1002/nme.3072 128.15.198.123 (talk) 21:13, 3 April 2025 (UTC)[reply]