A posteriori bound

The error bound (2.8) for the solution demonstrates that the error behaves like if   exists, is continuous, and is bounded. Generally, the function will be differentiable, and the bounds and can be calculated. However, the error bound so obtained many not be very good because the largest value of and will have to be chosen. If we have some knowledge of the solution, and assume that its second derivative is continuous and bounded by a known quantity, say we can get a better bound.

We first express by using a Taylor's series expansion at with remainder term to get

for

Therefore,

(2.9)

This is an A posteriori bound because it depends on knowledge of the second derivative of the solution.