I want to solve the Laplace equation on a grayscale image, say of size 1024 by 1024. The pixel values are imposed on a closed curve in the middle (say surrounding a quarter of the image area) and free elsewhere. The solution is only required outside the curve and if necessary, we can impose a constant value (such as the average of the outline) on the image edges.
Accuracy of the solution is not required, but speed is. Is there a known way to solve this efficiently, for instance via a multiscale approach ?