Traditional numerical methods like the Finite Difference Method or Finite Element Method solve PDEs by discretizing the domain into a grid or mesh. They approximate the solution \(u(x)\) by finding its values at these specific, discrete points.

For example, the Finite Difference method approximates the second derivative:

This transforms the differential equation into a system of algebraic equations for the values \(u_i\) at grid points \(x_i\). The result is a discrete representation of the solution.