V V
where
z ( C …¤ ¤ B … & 4
Y
t
¦ ¨ ¢ £ &¥
7
@ ¤
z # C … ¤ ¤ B p… ¨ £ £ tY ¥
7 &
&
@ ¤
Thus, the diagonal elements of the matrix are nonnegative, while its offdiagonal elements
are nonpositive. In addition, the rowsum of the elements, i.e., the sum of all elements in
the same row, is equal to zero. This is because
z … ¤ … & ¤ … ¤ ¤ … & C … ¤ ¤ B … & — C … ¤ ¤ B … & …p … & — 4
¦ 7
&
@ @ @ @
¤ ¤ ¤ ¤
The matrices obtained have the same desirable property of weak diagonal dominance seen
in the onedimensional case. A disadvantage of upwind schemes, whether in the context of
irregular grids or in onedimensional equations, is the loss of accuracy due to the low order
of the schemes.
) " £) ¨ §
1 Derive Forward Difference formulas similar to (2.8), i.e., involving 'c §hf B 'c §hd'c 9
¥ 9¦f ¥
¥ 9¦
(
, which are of second and third order. Write down the discretization errors explicitly.
WW5 ¦hf B
55
d¥ © g¢ ¤© ©
n¥ u¡ ¡
§ £ ¥
©
2 Derive a Centered Difference formula for the ¬rst derivative, similar to (2.13), which is at least
of third order.
¡
3 Show that the Upwind Difference scheme described in 2.2.4, when and are constant, is stable 7
for the model problem (2.7).
4 Develop the two ninepoint formulas illustrated in Figure 2.4. Find the corresponding discretiza 5
tion errors. [Hint: Combine of the ¬vepoint formula (2.17) plus of the same formula based
0 EB 0 ¥'Whf B 3EB 80 'c hf B 0 IB 2'GDf B 3¦EB 2'Whf9H8 ¦ 'c
on the diagonal stencil ¦ ¥5c ¥ c ¦ ¥
¥¦
c¦ ¦
¦
¤f B
¢ £
to get one formula. Use the reverse combination , to get the other formula.] H8 8
5 Consider a (twodimensional) rectangular mesh which is discretized as in the ¬nite difference
approximation. Show that the ¬nite volume approximation to yields the same matrix as an 9 5¥ ¡
upwind scheme applied to the same problem. What would be the mesh of the equivalent upwind
¬nite difference approximation?
6 Show that the righthand side of equation (2.16) can also be written as
¤9E
&
7¢
E ¨© F §5 B
5
¦ ¦
3
7 Show that the formula (2.16) is indeed second order accurate for functions that are in .
6
8 Show that the functions ™s de¬ned by (2.31) form a basis of .
c
¨
9 Develop the equivalent of Green™s formula for the elliptic operator de¬ned in (2.6).
10 Write a short FORTRAN or C program to perform a matrixbyvector product when the matrix
is stored in unassembled form.
11 Consider the ¬nite element mesh of Example 2.1. Compare the number of operations required to
perform a matrixbyvector product when the matrix is in assembled and in unassembled form.
Compare also the storage required in each case. For a general ¬nite element matrix, what can
the ratio be between the two in the worst case (consider only linear approximations on triangular
elements) for arithmetic? Express the number of operations in terms of the number of nodes and
edges of the mesh. You may make the assumption that the maximum number of elements that
¤
are adjacent to a given node is (e.g., ). $ $
5
¦ Wp5 ¦ & P
¡
C C C C
12 Let be a polygon in with edges, and let , for , where is the 55
length of the th edge and is the unit outward normal at the th edge. Use the divergence
C
P P
) !
theorem to prove that . "% C
HC
"#
NOTES AND REFERENCES. The material in this chapter is based on several sources. For a basic
description of the ¬nite element method, the book by C. Johnson is a good reference [128]. Axelsson
and Barker [16] gives a treatment which includes various solution techniques emphasizing iterative
techniques. For ¬nite difference and ¬nite volume methods, we recommend C. Hirsch [121], which
also gives a good description of the equations and solution methods for ¬‚uid ¬‚ow problems.
handle very large problems that cannot be tackled by the usual “dense” solvers.
economical, both in terms of storage and computational effort. Sparse direct solvers can
trix technology, was to devise direct solution methods for linear systems. These had to be
for matrices with irregular structure. The main issue, and the ¬rst addressed by sparse ma
works in the 1960s were the ¬rst to exploit sparsity to solve general sparse linear systems
techniques are straightforward to develop. Electrical engineers dealing with electrical net
by engineers in various disciplines. In the simplest case involving banded matrices, special
The natural idea to take advantage of the zeros of a matrix and their location was initiated

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