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Unformatted text preview: roblem. He assumed the error indicator to be
a linear function of the mesh spacing, i.e., e =hw
i i (8.4.2) i where w is a weighting that is assumed to be independent of the mesh spacing. Weighting
could be obtained by taking roots of other more traditional measures. For example, taking
the s th root of a global discretization error on x 1 x ) measure might produce
i i; i e = h C ky( )( )k ]1 :
s i i s =s i1 The solution of the minimization problem is to determine the mesh such that e is the
same on every subinterval x 1 x ), i = 1 2 : : : N . With this solution, the minimization
problem is often called an \equidistribution" or \equilibration" problem. With
i i; i e=
i i = 1 2 ::: N
16 (8.4.3a) the equidistributing mesh is calculated from (8.4.2) as h =w i = 1 2 ::: N i i (8.4.3b) where h = x ; x 1:
i i (8.4.3c) i; It remains to calculate , and this can be done by summing (8.4.3b)
X
X1
h =b;a=
=1
=1 w
N N i i i i thus, = Pb ; a :
=1 1=w
Let us generalize the result and the procedure to continuous functions.
N (8.4.3d) i i De nition 8.4.1. A function w(x y(x)) is a monitor function if it is integrable and
w(x y(x)) , x 2 (a b), for some > 0.
De nition 8.4.2. A mesh fa = x0 < x1 < : : : < x = bg is equidistributing with respect
to a monitor function w(x y(x)) if there exists a constant > 0 such that
N Z i x i;1 x w(x y(x))dx = i = 1 2 : : : N: (8.4.4) If the monitor function w(xy(x)) is known, then it is as easy to nd the equidistributing mesh in the continuous case as it was in the discrete case. Let us de ne
Z
I (x) = w( y( ))d :
x a We observe that I (x) is a monotonically increasing function of x since w(x y(x)) > 0.
Now, the equidistribution problem (8.4.4) can be written as I (x ) = i
i i = 1 2 : : : N: We rst determine by setting i = N in (8.4.5) to obtain
I (x ) = Z w(x y(x))dx:
=N
b N a 17 (8.4.5) (8.4.6) I(x)
I(x N) I(x 1)
0 x
x0 x1 x2 xN Figure 8.4.2: Determining an equidistributing mesh from a monitor function.
With known, the mesh is calculated by nding the roots of (8.4.5). The situation is
shown in Figure 8.4.2. The roots of (8.4.5) can be determined sequentially without need
of solving a vector system of nonlinear equations.
Unfortunately w(x y(x)) is typically a function of the solution of the BVP and,
hence, it may only be known discretely with respect to a mesh. We'll view this mesh and
solution as being preliminary and use them to calculate an equidistributing mesh that
will subsequently be used to determine an improved solution. Thus, suppose that w ,
i = 1 2 : : : N , has been determined by using an input mesh fx0 < x0 < : : : < x0 g. Let
0
1
us calculate a continuous approximation w(x y(x)) to w , i = 1 2 : : : N , by polynomial
interpolation. We need not do this particularly carefully since precise solutions of the
equidistribution problem are not necessary. Assuming, for example, that kek is a
smooth function of x , i = 0 1 : : : N , then, near a minimum, an O(h) error in the
placement of the mesh would only result in an O(h2) change in the value of kek .
Using these arguments, it would su ce to construct a piecewise constant approximation of w , i = 1 2 : : : N ,...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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