Transport Problems
In this chapter we study the important transport equation that models transport of various physical quantities, such as density, momentum, and energy, for instance. In particular, the transportation of heat through convection is modeled by this equation.
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Transport Problems
Abstract In this chapter we study the important transport equation that models transport of various physical quantities, such as density, momentum, and energy, for instance. In particular, the transportation of heat through convection is modeled by this equation. That is, the transfer of heat by some external physical process, such as air blown by a fan, or a moving fluid, for instance. Often, high convection takes place alongside low diffusion (i.e., uniform spreading) of heat, leading to large temperature gradients. As we shall see, this may cause numerical instabilities unless special care is taken. To do so, we introduce the Galerkin Least Squares (GLS) method, which is more robust than the standard Galerkin method. We illustrate with numerical examples.
10.1 The Transport Equation The transport equation is given by u C b ru D f; u D 0;
in ˝
(10.1a)
on @˝
(10.1b)
where > 0 is a (small) parameter, b a given vector field, and f is a given function. For this problem to be well-posed we must assume that r b D 0. For simplicity, we also assume homogeneous Dirichlet boundary conditions. However, other types of boundary conditions are, of course, possible. Indeed, for the numerical experiments we shall use both Neumann and Robin boundary conditions. In (10.1) each of the two operators , and b r play a specific role in determining what the solution u will look like, and can be given simple interpretations. Loosely speaking, the first one smears u proportionally to , while the second one transports u in the direction of the vector b. Therefore, we say that these operators
M.G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Texts in Computational Science and Engineering 10, DOI 10.1007/978-3-642-33287-6__10, © Springer-Verlag Berlin Heidelberg 2013
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10 Transport Problems
model the physical processes of diffusion, and convection, respectively. In fact, the transport equation is sometimes referred to as the Convection-Diffusion equation.
10.1.1 Weak Form The weak form of (10.1) reads: find u 2 V D H01 .˝/ such that a.u; v/ D l.v/;
8v 2 V
(10.2)
where the bilinear and linear forms a.; / and l./ are given by a.u; v/ D .ru; rv/ C .b ru; v/ l.v/ D .f; v/
(10.3) (10.4)
The existence and uniqueness of u 2 V follows from the Lax-Milgram lemma, since a.; / is continuous and coercive on V , and l./ is continuous on V .
10.1.2 Standard Galerkin Finite Element Approximation Let Vh V be the usual space of continuous piecewise linears. The standard socalled Galerkin finite element approximation of (10.2) reads: find uh 2 Vh such that a.uh ; v/ D l.v/;
8v 2 Vh
(10.5)
i Now, let f'i gni D1 be the usual hat function basis for VP h , with ni the number of interior nodes. Expanding the finite element ansatz uh D nj iD1 j 'j , and choosing v D 'i , i D 1; 2; : : : ; ni , in (10.5) we obtain the linear system for the unknown nodal values j of uh
.A C C / D b
(10.6)
where the matrix and vector entries are given by Aij D
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