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The second term of the bilinear form is also computed as a sum of integrals over the elements:

(1.67)

We will be concerned with evaluation of the integral

Defining:

(1.68)

the following expression is obtained:

(1.69)

where , and

The terms of the coefficient matrix are computable from the mapping, the definition of the shape functions and the function . The matrix is called the element‐level Gram matrix¹³ or the element‐level mass matrix. Observe that is symmetric. In the important special case where is constant on Ik it is possible to compute once and for all. This is illustrated by the following example.

Example 1.4 When is constant on Ik and the Legendre shape functions are used then the element‐level Gram matrix is strongly diagonal. For example, for the Gram matrix is:

(1.70)

Remark 1.5 For a simple closed form expression can be obtained for the diagonal terms and the off‐diagonal terms. Using eq. (1.55) it can be shown that:

(1.71)

and all off‐diagonal terms are zero for , with the exceptions:

(1.72)

Remark 1.6 It has been proposed to make the Gram matrix perfectly diagonal by using Lagrange shape functions of degree p with the node points coincident with the Lobatto points. Therefore where is the Kronecker delta¹⁴. Then, using Lobatto points, we get:

where wi is the weight of the ith Lobatto point. There is an integration error associated with this term because the integrand is a polynomial of degree . To evaluate this integral exactly Lobatto points would be required (see Appendix E), whereas only Lobatto points are used. Throughout this book we will be concerned with errors of approximation that can be controlled by the design of mesh and the assignment of polynomial degrees. We will assume that the errors of integration and errors in mapping are negligibly small in comparison with the errors of discretization.

Exercise 1.9 Assume that is constant on Ik. Using the Lagrange shape functions of degree , with the nodes located in the Lobatto points, compute numerically using 4 Lobatto points. Determine the relative error of the numerically integrated term. Refer to Remark 1.6 and Appendix E.

Exercise 1.10 Assume that is constant on Ik. Using the Lagrange shape functions of degree , compute and in terms of ck and ℓk.