510c where ex ux u x 9510d equation 959b

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Unformatted text preview: ction V is also an element of S N however, G( x) cannot be approximated to the same precision as u because it may be less smooth. To elaborate further, consider kG( ) ; V k = 2 1 where kuk = 2 1j N X j =1 Zx j xj ;1 kG( ) ; V k2 j 1 (u0)2 + u2]dx: If 2 (xk;1 xk ), k = 1 2 : : : N , then the discontinuity in Gx( x) occurs on some interval and G( x) cannot be approximated to high order by V . If, on the other hand, = xk , k = 0 1 : : : N , then the discontinuity in Gx( x) is con ned to the mesh and G( x) is smooth on every subinterval. Thus, in this case, the Green's function can be approximated to O(hp) by the test function V and, using (9.5.11c), we have u(xk ) = Ch2p k = 0 1 : : : N: (9.5.12) The solution at the vertices is converging to a much higher order than it is globally. Equation (9.5.11c) suggests that there are two ways of minimizing the pointwise error. The rst is to have U be a good approximation of u and the second is to have V be a good approximation of G( x). If the problem is not singularly perturbed, then the two conditions are the same. However, when 1, the behavior of the Green's function is hardly polynomial. Let us consider two simple examples. Example 9.5.2 5]. Consider (9.5.5) in the case when ! (x) > 0, x 2 0 1]. Balancing the rst two terms in (9.5.5a) implies that there is a boundary layer near x = 1 thus, at points other than the right endpoint, the small second derivative terms in (9.5.5) may be neglected and the solution is approximately !u0R + quR = f 0<x<1 uR(0) = 0 9.5. Convection-Di usion Systems 33 where uR is called the reduced solution. Near x = 1 the reduced solution must be corrected by a boundary layer that brings it from its limiting value of uR(1) to zero. Thus, for 0 < 1, the solution of (9.5.5) is approximately u(x) uR(x) ; uR(1)e;(1;x)!(1)= : Similarly, the Green's function (9.5.8) has boundary layers at x = 0 and x = ;. At points other than these, the second derivative terms in (9.5.8a) may be neglected and the Green's function satis es the reduced problem ;(!GR )0 + qGR = 0 x 2 (0 ) ( 1) GR( x) 2 C (0 1) GR ( 1) = 0: Boundary layer jumps correct the reduced solution at x = 0 and x = and determine an asymptotic approximation of G( x) as ;!(0)x= if x ( G( x) c( ) GRx; )x)( ; GR( 0)e e;( ! )= if x > : The function c( ) is given in Flaherty and Mathon 5]. Knowing the Green's function, we can construct test functions that approximate it accurately. To be speci c, let us write it as G( x) = N X j =1 G( xj ) j (x) (9.5.13) where j (x), j = 0 1 : : : N , is a basis. Let us consider (9.5.5) and (9.5.8) with ! > 0, x 2 0 1]. Approximating the Green's function for arbitrary is di cult, so we'll restrict to xk , k = 0 1 : : : N , and establish the goal of minimizing the pointwise error of the solution. Mapping each subinterval to a canonical element, the basis j (x), x 2 (xj;1 xj+1) is j where (x) = ^( x ; xj ) h 8 ;e; < e; ^(s) = e; ;;;e; : 0 ;e 1 (1+s) 1 1 where s if ; 1 s < 0 if 0 s < 1 otherwise = h! (9.5....
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