Unformatted text preview: interpolate (x f (x y(0) )), j = i ; 2 i ; 1 i i + 1. If the points
are equally spaced then
j j j ^
^ (1) (y(0) ) = h2 T(1) = 1 (;f 2 + f 1 + f ; f +1):
i i i; i; i i This formula would have to be modi ed near the boundaries. Problems 1. Consider a logarithmically graded mesh
ln(1 + )
x = ln 2
i i = 0 1 : : : N: N i This grading is less severe than the quadratic grading of Example 8.3.1. Is this
mesh quasi uniform? 8.4 Adaptive Mesh Selection
We seek to solve the nonlinear BVP (8.2.1) to a speci ed degree of accuracy in an adaptive
manner by (i) computing a preliminary solution on a coarse mesh, (ii) estimating the
discretization error of this solution, and (iii) recursively re ning the mesh where the
accuracy is not su cient. Thus, in the end, we would only use a ne mesh where the
solution varied rapidly. A simple example serves to illustrate the principle.
Example 8.4.1. Consider the linear two-point BVP y + y =0
00 0<x<1 0 y(0) = 1 y(1) = 1=2 which has the exact solution e ;e
y(x) = 1 + (1 ; e 2) :
; x ; ; Assuming that 0 < 1= 1, this is approximately y(x) 1 (1 + e
2 ; x ) As shown in Figure 8.4.1, the solution has a boundary layer near x = 0 and we might
be able to improve both accuracy and e ciency by using a mesh that is concentrated in
the boundary layer.
Once again, we'll focus on the trapezoidal rule solution of (8.2.1) which is
(y) = y ; y 1 ; f (x y ) + f (x 1 y 1)
i i i; i i i; i 14 i; i = 1 2 ::: N (8.4.1a) 1 0.9 0.8 0.7 y 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5
x 0.6 0.7 0.8 0.9 1 Figure 8.4.1: The solution of Example 8.4.1 with = 100. g (y0) = g (y ) = 0:
L R (8.4.1b) N Using (8.3.2a) h2
= ; 12 f (x 1 2 y(x 1 2 )) + O(h4):
i i 00 i; = i; = (8.4.1c) While concentrating on the trapezoidal rule, we'll consider a somewhat more abstract
setting that will be useful when working with, e.g., high-order solutions via Richardson's extrapolation or deferred corrections. Thus, let e be an \error indicator" on the
subinterval x 1 x ). Some possibilities for e are:
i i; i i 1. An approximation of the local discretization error. For the trapezoidal rule this
e = 12 jf (x 1 2 y 1 2 )j :
i 00 i i; = 15 i; = 1 2. An approximation of the global discretization error e = C h ky( )( )k
s i s s i1 i where C is a known constant that depends upon the method and k k
maximum norm restricted to x 1 x ).
s is the i1 i; i 3. An ad hoc metric. One could use a scaled approximation of the gradient or curvature, e.g.,
e = h k y 1 2k
i i i; = where is the central di erence operator (Table 6.3.3).
The key ingredient is that the error indicator be small where the solution is well-resolved
and large where additional resolution is necessary.
Given a mesh of N + 1 points, we try to determine the coordinates fx0 < x1 < : : : <
x g such that some global metric of solution quality is minimized. We'll illustrate a
solution of this problem by determining the mesh such that
N jej = 1max je j
1 i N i is minimized. Although still in a rather abstract setting, de Boor 3], Chapter XIV, found
a rather simple solution to this extremal p...
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- Spring '14
- Numerical Analysis, Boundary value problem, Piecewise linear function, Infantry regiments of Sweden, BVP