Since the integrands are quadratic functions of x

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Unformatted text preview: Simpson's rule to yield ( Finally, ( j j Ut ) = hj (_j;1 + 2_j ) + hj6+1 (2_j + c_j+1): c c 6c f) Zx j xj ;1 j f (x)dx + Zx xj j +1 j f (x)dx: Although integration of order one would do, we'll, once again, use Simpson's rule to obtain ( j f ) hj (2fj;1=2 + fj ) + hj6+1 (fj + 2fj+1=2): 6 We could replace fj;1=2 by the average of fj;1 and fj to obtain a similar formula to the one obtained for ( j Ut ) thus, ( j f ) hj (fj;1 + 2fj ) + hj6+1 (2fj + fj+1): 6 Combining these results yields the discrete nite element system hj (_ c 6 j ;1 p p + 2_j ) + hj+1 (2_j + cj+1) + j;1=2 (cj ; cj;1) ; j+1=2 (cj+1 ; cj ) c c_ 6 hj hj + 1=2 6 Parabolic Problems = hj (fj;1 + 2fj ) + hj+1 (2fj + fj+1) j = 1 2 : : : N ; 1: 6 6 (We have dropped the and written the equation as an equality.) If p is constant and the mesh spacing h is uniform, we obtain h (_ c 6 j ;1 + 4_j + cj+1) ; p (cj;1 ; 2cj + cj+1) = h (fj;1 + 4fj + fj+1) c_ h 6 j = 1 2 : : : N ; 1: The discrete systems may be written in matrix form and, for simplicity, we'll do so for the constant coe cient, uniform mesh case to obtain _ Mc + Kc = l where 24 61 h6 M= 66 6 6 4 (9.2.4a) 3 1 7 41 7 ... ... ... 7 7 7 1 4 15 14 2 2 ;1 6 ;1 2 ;1 p6 K= h6 6 ... ... 6 4 ;1 3 7 7 7 7 5 ;1 7 ... 2 ;1 2 2f 6f l= h6 66 4 + 4f1 + f2 1 + 4f2 + f3 ... fN ;2 + 4fN ;1 + fN 0 c = c1 c2 : : : cN ;1]T : 3 7 7 7 5 (9.2.4b) (9.2.4c) (9.2.4d) (9.2.4e) The matrices M, K, and l are the global mass matrix, the global sti ness matrix, and the global load vector. Actually, M has little to do with mass and should more correctly be called a global dissipation matrix however, we'll stay with our prior terminology. In practical problems, element-by-element assembly should be used to construct global matrices and vectors and not the nodal approach used here. The discrete nite element system (9.2.4) is an implicit system of ordinary di erential _ equations for c. The mass matrix M can be \lumped" by a variety of tricks to yield an 9.2. Semi-Discrete Galerkin Problems 7 explicit ordinary di erential system. One such trick is to approximate ( the right-rectangular rule on each element to obtain ( j Ut) = Zx j j xj ;1 (_j;1 j;1 + cj j )dx + c _ Zx j +1 xj j (_j j + cj+1 c _ j +1 j Ut) by using )dx hcj : The resulting nite element system would be _ hIc + Kc = l: Recall (cf. Section 6.3), that a one-point quadrature rule is satisfactory for the convergence of a piecewise-linear polynomial nite element solution. N With the initial data determined by L2 projection onto SE , we have ( j j = 1 2 : : : N ; 1: U ( 0)) = ( j u0) Numerical integration will typically be needed to evaluate ( j u0) and we'll approximate it in the manner used for the loading term ( j f ). Thus, with uniform spacing, we have 2u 6u Mc(0) = u = h 6 66 4 3 + 4u0 + u0 1 2 + 4u0 + u0 7 2 3 7: 7 ... 5 0 0 0 uN ;2 + 4uN ;1 + uN 0 0 0 1 0 (9.2.4f) If the initial data is consistent with the trivial Dirichlet boundary data, i.e., if u0 2 H01 then the above system reduces to cj (0) = u0(xj ) j =...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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