Math 7731 { Mathematical Problems in Industry Assignment 3 (2017) 1. Consider the one-dimensional steady state heat equation (k(x)u0)0 = −f(x) (1) on the interval 0 ≤ x ≤ 1 with the boundary conditions u = 0 at x = 0 and − ku0 = h(u − 1) at x = 1: This corresponds to internal heat production of f per unit length per unit time, with a fixed temperature boundary condition at x = 0 and a cooling boundary condition at x = 1 (with external temperature being taken as 1. ) For simplicity take k = 1, f = 1 and h = 1. (So, in particular, k and f do not depend on x.) (a) Multiply (1) by a suitable test function and intergrate by parts to obtain the weak formulation of this problem. Be careful to specify any boundary conditions that might apply to the spaces of trial and test functions. What is the bilinear form a(: ; :) and the functional f(:) in this case? (b) Let W0 be a finite dimensonal subspace of the full space in (a). What is the approximate weak formulation corresponding to W0? (c) Consider the one-dimensional finite element mesh on [0; 1] shown below. x0 = 0 x1 xi-1 xi xi+1 x n = 1 xn-1 hi = xi - xi-1, i= 1,...,n Figure 1: Finite element mesh on the interval 0 ≤ x ≤ 1. hi is the length of the element (xi−1; xi). Let W be the finite element subspace of continuous, piecewise linear functions defined for this mesh, and let φi; i = 0; :::; n be the corresponding linear Lagrange basis functions of W . x0 = 0 x1 xi-1 xi xi+1 x n = 1 xn-1 hi = xi - xi-1, i= 1,...,n 1 f0 fi fn Figure 2: Basis functions associated with the above finite element mesh on the interval 0 ≤ x ≤ 1. In this case W0 = fw 2 W : w = 0 at x = 0g. Write down a set of basis functions for W0 in terms of the φi. 1(d) Write down the approximate version of the weak formulation in terms of these basis functions. (Hint: Express the approximate solution as a sum of basis functions with unknown scalar coefficients. Then use each of the basis functions in turn as the test function in (b).) (e) Explicitly evaluate all of the stiffness matrix elements a(φi; φj) and the load vector components f(φj) that arise in (d). Your answer should be expressed in terms of the quantities hi defined in Figure 1. (Hint: It is convenient to consider the cases where the test function φj corresponds to a node xj that lies in the interior of the interval (j = 1; :::; n − 1) separately from the cases where the test function φj corresponds to a node that is on the boundary of the interval (j = 0 and j = n.) (f) If the boundary condition u = 0 at x = 0 is changed to u = 1, for example, how would your answers to the above questions change? (Hint: Write the unknown solution w as w = u + U ~, where u = 0 at x = 0 and U ~ is some (arbitrary) extension of the boundary condition u = 1 at x = 0 to the entire interval [0;1]. We could take U ~ = 1 everywhere as that extension; however, for this problem it is convenient to instead take U ~ = φ0.) 2