In numerical analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

, a superconvergent method is one which converges faster than generally expected. For example in the Finite Element Method

Finite element method

The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...

 approximation to Poisson's equation

Poisson's equation

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics...

 in two dimensions, using piecewise linear elements, the average error in the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

 is first order. However under certain conditions it's possible to recover the gradient at certain locations within each element to second order.