Relaxation Methods for Solving the Laplace Equation

The two graphics represent the progress of two different algorithms for solving the Laplace equation. They both calculate the electric potential in 2D space around a conducting ellipse with excess charge. The potential is constant on the ellipse and falls to zero as the distance from the ellipse increases.

Both algorithms use the method of relaxation in which grid cells are iteratively updated to equal the mean value of their neighbors. Grid cells inside the conductor remain at a constant potential, and the unknown potential values outside the conductor are initially zero.

The first algorithm, shown on the left, uses a high-resolution grid. The second algorithm, shown on the right, improves on the first algorithm by using a low-resolution grid that gradually increases in resolution. After several iterations of relaxation, each cell divides into four subcells, and the relaxation continues.

Compare the graphics to observe the significant improvement of the second algorithm. Using the same number of iterations as the first, the second algorithm is able to more accurately calculate the potential farther away from the conductor. The second algorithm is also faster because only a fraction of the total iterations is run on the high-resolution grid.

The Laplace algorithm is applied in many applications, including electrostatic and heat distributions.