alt="p plus 1"/> 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.
1.3.4 Computation of the right hand side vector
Computation of the right hand side vector involves evaluation of the functional , usually by numerical means. In particular, we write:
(1.73)
The element‐level integral is computed from the definition of vn on Ik:
(1.74)
where
(1.75)
which is computed from the given data and the shape functions.
Example 1.5 Let us assume that is a linear function on Ik. In this case can be written as
Using the Legendre shape functions we have:
Exercise 1.11 Assume that is a linear function on Ik. Using the Legendre shape functions compute and show that for . Hint: Make use of eq.
