Solutions of the Finite Difference Discretized Laplace Equation

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

This Demonstration shows the dependence of the solution of the finite difference discretized Laplace equation on a square grid as a function of the given values at the discretization nodes.

[more]

On an square grid, the simplest finite difference approximation of the Laplace operator is . This means that generically values can be prescribed. In most textbooks, these values are on the boundary, but they can also be in the interior. Not all configurations of prescribed values are consistent. A consistent set of prescribed values leads to a unique solution.

Drag the sliders by the double arrows into any square (other than the four corners) to change the arrangement of prescribed sets of values. Use the sliders in the squares to change the values of the prescribed . (The color of the crosses indicates the values of the sliders.)

[less]

Contributed by: Michael Trott with permission of Springer (March 2011)
From: The Mathematica Guidebook for Programming, second edition by Michael Trott (© Springer, 2008).
Open content licensed under CC BY-NC-SA


Snapshots


Details

For various results on the constellations of determining sets for the Laplace equation, see

A. Rubinstein, J. Rubinstein and G. Wolansky, "Determining Sets for the Discrete Laplacian," SIAM Review, 49(2), 2007 pp. 315-324. DOI‐Link.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send