Numerical solutions to differential equations frequently make use of the weak formulation. That is, we take the inner product of each PDE with some test function. There are many different ways to evaluate inner products numerically; i.e. trapezoidal rule, midpoint rule, or higher-order approximations. A simple method for evaluating inner products on a numerical grid is to apply the midpoint rule; which is used by the discretize package.
Here, we demonstrate how to approximate various classes of inner products numerically. If this is known, the user will be capable of properly discretizing any term in a problem specific PDE.
Basic Inner Products