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. ConvectionDi 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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 Spring '14
 JosephE.Flaherty
 The Land, Tn, Boundary value problem, Numerical ordinary differential equations, nite element, parabolic problems

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