## Chapter 2Differentiation

Assuming a polynomial approximation of the form:

where χ(ξ) is the mapping from the standard region ξ ∈ Ωs to the region containing x in the interval [a,b], we can differentiate u(x) using the chain rule to obtain

The differentiation of uδ(x) is therefore dependent on evaluating dϕp(ξ)∕dξ and . In this section we shall consider the case where ϕp(ξ) is the Lagrange polynomial hp(ξ) and discuss how to evaluate dϕp(ξ)∕dξ. If χ(ξ) is an isoparametric mapping this technique can also be used to evaluate = -1. Differentiation of this form is often referred to as differentiation in physical space or collocation differentiation.

If we assume that uδ(ξ) is a polynomial of order equal to or less than P [that is, uδ(ξ) ∈ PP ([-1,1])], then it can be exactly expressed in terms of Lagrange polynomials hi(ξ) through a set of q nodal points ξi (0 ≤ i ≤ q - 1) as

where q ≥ P + 1. Therefore the derivative of u(ξ) can be represented as

Typically, we only require the derivative at the nodal points ξi which is given by

where

An alternative representation of the Lagrange polynomial is

Taking the derivative of hi(ξ) we obtain

Finally, noting that because numerator and denominator of this expression are zero as ξ → ξi, and because Pqi) = 0 by definition,

so we can write dij as

 (2.1)

Equation (2.1) is the general representation of the derivative of the Lagrange polynomials evaluated at the nodal points ξi (0 ≤ i ≤ q - 1). To proceed further we need to know specific information about the nodal points ξi which will allow us to deduce alternative forms of g′qi) and g′′qi).

### 2.1 Legendre Formulae

The most common differentiation matrices dij are those corresponding to the Gauss-Legendre quadrature points. In this section we illustrate the final form of the differential matrices that correspond to the use of Gauss-Legendre, Gauss-Radau-Legendre, and Gauss-Lobatto-Legendre quadrature points. Denoting by ξi,P α,β the P zeros of the Jacobi polynomial PP α,β(ξ) such that

the derivative matrix dij used to evaluate at ξi, that is,

is defined as:

(1) Gauss-Legendre