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Unformatted text preview: a square. Problems
1. Consider a problem for Laplace's equation uxx + uyy = 0 (x y) 2
where is the region between a 4 4 square and a concentric 2 2 square (Figure
9.2.6). The Dirichlet boundary conditions are u = 1 on the outside of the 4 4
square and u = 0 on the edge of the 2 2 square. Due to symmetry, this problem
need only be solved on the one octant shown on the right of Figure 9.2.6. The
subscript n denotes di erentiation in the outer normal direction.
1.1. Construct a discrete approximation of the above problem on the region shown
on the right of Figure 9.2.6. Use the ve point di erence approximation for
Laplace's equation and, for simplicity, assume that the mesh spacing in the x
and y directions is the same, say, x = y = 1=N . Take appropriate steps
to ensure that the nite di erence approximations at the symmetry boundary
and the interior have O(1=N 2) accuracy.
1.2. For the special discretization when x = 1=2 the discrete problem has only
four unknowns. Write down an SOR procedure for determining these unknowns. Calculate the Jacobi iteration matrix MJ . Find an expression for 30 Solution Techniques for Elliptic Problems
the spectral radius (MSOR) of the SOR matrix. Plot (MSOR) and determine
the optimal relaxation parameter !. (This problem should be done symbolically.)
1 2 1 1 u=0 2 1 1 ux = 0 111111
000000
111111
000000
111111
000000
111111
000000 un 0
=
u=1
1 1 Figure 9.2.6: Domain for Problem 1 (left). Due to symmetry, the problem need only be
solved on the octant shown on the right. 9.3 Conjugate Gradient Methods
The xedpoint iterative methods of the previous section deteriorate in performance as
the dimension N of the linear system increases. The faster SOR and ADI techniques
depend on acceleration parameters that may be di cult to estimate. We seek to overcome these de ciencies without raising the storage requirements to the level of a direct
method. Solving the linear system (9.1.1) when A is symmetric positive de nite matrix
is equivalent to minimizing the quadratic functional...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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