Assume that u satis es the essential boundary

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Unformatted text preview: al problem belong? 3.3 Overview of the Finite Element Method Let us conclude this chapter with a brief summary of the key steps in constructing a niteelement solution in two or three dimensions. Although not necessary, we will continue to focus on (3.1.1) as a model. 1. Construct a variational form of the problem. Generally, we will use Galerkin's method to construct a variational problem. As described, this involves multiplying the di erential equation be a suitable test function and using the divergence theorem to get 1 a symmetric formulation. The trial function u 2 HE and, hence, satis es any prescribed essential boundary conditions. The test function v 2 H01 and, hence, vanishes where essential boundary conditions are prescribed. Any prescribed Neumann or Robin boundary conditions are used to alter the variational problem as, e.g., with (3.1.6) or (3.1.8b), respectively. 1 Nontrivial essential boundary conditions introduce di erences in the spaces HE and N H01. Furthermore, the nite element subspace SE cannot satisfy non-polynomial boundary conditions. One way of overcoming this is to transform the di erential equation to one having trivial essential boundary conditions (cf. Problem 1 at the end of this section). This approach is di cult to use when the boundary data is discontinuous or when the problem is nonlinear. It is more important for theoretical than for practical reasons. 3.3. Overview of the Finite Element Method 17 The usual approach for handling nontrivial Dirichlet data is to interpolate it by the nite element trial function. Thus, consider approximations in the usual form U (x y) = N X j =1 cj j (x y) (3.3.1) however, we include basis functions k for mesh entities (vertices, edges) k that are on @ E . The coe cients ck associated with these nodes are not varied during the solution process but, rather, are selected to interpolate the boundary data. Thus, with a Lagrangian basis where k (xj yj ) = k j , we have U (xk yk ) = (xk yk ) = ck (xk yk ) 2 @ E : The interpolation is more di cult with hierarchical functions, but it is manageable (cf. Section 4.4). We will have to appraise the e ect of this interpolation on solution accuracy. N N Although the spaces SE and S0 di er, the sti ness and mass matrices can be made symmetric for self-adjoint linear problems (cf. Section 5.5). A third method of satisfying essential boundary conditions is given as Problem 2 at the end of this section. 2. Discretize the domain. Divide into nite elements having simple shapes, such as triangles or quadrilaterals in two dimensions and tetrahedra and hexahedra in three dimensions. This nontrivial task generally introduces errors near @ . Thus, the problem is typically solved on a polygonal region ~ de ned by the nite element mesh (Figure 3.3.1) rather than on . Such errors may be reduced by using nite elements with curved sides and/or faces near @ (cf. Chapter 4). The relative advantages of using fewer curved elements or a larger number of smaller straight-sided or planar-faced elements will have to be determined. 3. Generate the element sti ness and mass matrices and element load vector. PieceN N wise polynomial approximations U 2 SE of u and V 2 S0 of v are chosen. The approxN N 1 imating spaces SE and S0 are supposed to be...
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