Differentiation

Assuming a polynomial approximation of the form:

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

The differentiation of u^{δ}(x) is therefore dependent on evaluating dϕ_{p}(ξ)∕dξ and . In this
section we shall consider the case where ϕ_{p}(ξ) is the Lagrange polynomial h_{p}(ξ) and discuss
how to evaluate dϕ_{p}(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 u^{δ}(ξ) is a polynomial of order equal to or less than P [that is,
u^{δ}(ξ) ∈ P_{P }([-1,1])], then it can be exactly expressed in terms of Lagrange polynomials h_{i}(ξ)
through a set of q nodal points ξ_{i} (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 ξ_{i} which is given by

where

An alternative representation of the Lagrange polynomial is

Taking the derivative of h_{i}(ξ) we obtain

Finally, noting that because numerator and denominator of this expression are zero as ξ → ξ_{i},
and because P_{q}(ξ_{i}) = 0 by definition,

so we can write d_{ij} as

| (2.1) |

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

The most common differentiation matrices d_{ij} 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 ξ_{i,P }^{α,β} the P zeros of the Jacobi polynomial P_{P }^{α,β}(ξ) such
that

the derivative matrix d_{ij} used to evaluate at ξ_{i}, that is,

is defined as:

(1) Gauss-Legendre

(2) Gauss-Radau-Legendre

(3) Gauss-Lobatto-Legendre

In a similar way to the quadrature formulae the construction of the differen- tiation matrices require the quadrature zeros to be determined numerically. Having determined the zeros, the components of the differentiation matrix can be evaluated directly from the above formulae by generating the Legendre polynomial from the recursion relationship.