(BQ) Part 2 book "The finite element method" has contents: Further topics in the finite element method, the boundary element method, convergence of the finite element method, computational aspects. | 5 Further topics in the finite element method So far, elliptic problems only have been considered, and we shall see in Section how, with reference to Poisson problems, the variational approach is equivalent to Galerkin’s method. Of course, for many problems of practical interest, such variational principles may not exist, or where they do, a suitable functional may not be known. In this chapter we shall consider procedures for a wide variety of problems, including parabolic, hyperbolic and non-linear problems. It is not intended to be any more than an introduction, and the ideas are presented by way of particular problems. The reader with a specific interest in any one subject area will find the references useful for further detail. The variational approach We seek a finite element solution of the problem given by eqns ()–(), viz. −div(k grad u) = f (x, y) () in D, with the Dirichlet boundary condition () u = g(s) on C1 and the Robin boundary condition () k(s) ∂u + σ(s)u = h(s) ∂n on C2 . The functional for this problem is found from eqn () as I[u] = k D ∂u ∂x 2 +k ∂u ∂y 2 − 2uf (σu2 − 2uh) ds dx dy + C2 and the solution, u, of eqns ()–() is that function, u0 , which minimizes I[u] subject to the essential boundary condition u0 = g(s) on C1 . We follow exactly the finite element philosophy of Section , writing () ue (x, y) ˜ u(x, y) = ˜ e 172 The Finite Element Method with ue (x, y) = Ne (x, y)Ue . ˜ () Then I[˜] = u ⎧ ⎨ k D⎩ 2 ∂ ∂x ⎧ ⎨ + σ C2 ⎩ ue ˜ ∂ ∂y +k e 2 −2 ue ˜ e ue h ˜ e 2 −2 ue ˜ e ue f ˜ e ⎫ ⎬ ⎭ dx dy ⎫ ⎬ ⎭ ds. Now, since ue is zero outside element [e], the only non-zero contribution to I[˜] ˜ u from ue comes from integration over the element itself. Thus ˜ k I[˜] = u [e] e 2 +k ∂ ue ˜ ∂y 2 − 2˜e f u dx dy 2 u σ (˜e ) − 2˜e h ds u + e I e, = ∂ ue ˜ ∂x C2 say. e The second term applies only if the element has a boundary coincident with C2 ; see .
