University of Groningen
High-order finite-difference methods for Poisson's equation
van Linde, Hendrik Jan
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1971
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van Linde, H. J. (1971). High-order finite-difference methods for Poisson's equation Groningen: s.n.
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VII TWO GENERAL CONCLUDING REMARKS.
§ 1. Approximation of the derivatives.
Now that we have formulated the desired approximations for
the three boundary value problems we shall conclude by making
two more general remarks which hold for all three problems.
The first of these concerns the approximation of the derivatives of the solutions for these boundary value problems.
Bramble and Hubbard [23] have remarked that it may often be
of great importance to have, apart from information about the
discretization error as a function of the mesh width h, information about the behaviour of the difference quotients. This
information enables us to predict how well the derivatives of
the desired solution can be approximated. If, for instance,
one has shown for a certain finite-difference method that its
discretization error is
0(h 2 ), then it is clear that the first
difference quotients have an error which is not worse than
0(h). The question whether this is the best obtainable result
was answered in the negative by Bramble and Hubbard in the
above-mentioned paper; they have shown that under reasonable
conditions the errors in the difference quotients are also
0(h 2 ) in the whole field.
Examination of these results with the view to using them for
the approximation methods we have just derived, which yielded
0(h 4 ), 0(h 3 ) and 0(h 3 Jln hJ) errors for the Dirichlet, Robin
and Neumann problems respectively, shows that they do not hold
in all generality for our methods.
Bramble and Hubbard gave, however, for the type of approximations we use here, a seperate general proof which directly
86
leads us to the formulation of the following theorem, which
follows immediately from the above-mentioned paper:
Theorem 1 . Let R be a bounded, connected region, in which one
of the three boundary value problems for Poisson's equation
is approximated as described in one of the previous chapters.
Let further R h be the set of mesh points whose eight nearest
neighbours are also in R. Then Dnc(P), the n
th
difference quo-
tient of the discretization error E(P) in the point P has the
same order as (P), nrovided P e R.
§ 2. Sharpness of the estimates.
It is comparatively easy, following as so often in this work
Bramble and Hubbard's footsteps, to give examples illustrating
the contention that the error estimates given in the previous
chapters are sharp. We shall do this here for the mixed boundary value problem only:
Let R be the unit square with corners at (0,0), (0,7), (1,7)
and (7,0). We suppose u to be given on y = 0,7 and on x = 0.
On x = 7 we prescribe án + u. In a mesh point (7,y) on the
boundary we take (compare (5.75))
1
d n u(7,y) = h - {3u(7,y) - —10[u(7- h,y+h) +u(7- h,zy-h)]
(7. 7)
-
3 u(7,U +h) +u(7,U-h)] - 3u(1-h,U) }
From (7.1) we can see that
2
d
n
u
u
n
2
3
+-- Ax u -
24
F(u)+0(h4)
where
F(u) = u
+2u
+u
xxxx xxyy yyyy
We now take for u the function u(x,y) = 3x 2 y2, in 1?, and we
define the approximation U(P) of u(P) by
8,7
(9)
-A
2
U(P) = - Du - h
2 Mu
P <_ Rh
2
6 U(P)+U(P)= an + u- 2 Au + s A u
U(P)
= u (P)
(x-7)
P, C
h
PF
02h
(y=0, 7; x=0)
We now have for t (P) -- u (P) -U (P) :
A (9)
e(P) = 0
c(P)
D
P
=0
3
d n e(P)+E(P) _ - 24 F(u)+O(h 4 ) =
-1+0(h)] h
P
Using (4.7) we find
Ic(P)1
(7.2)
E Gh(P,Q)L1+0(h)]h3
=
Qec7h
In (4.10) we found
P
E Gh ( iQ) < 4k$iM
QEC1h
By similar arguments we can show that
E Gh (P,Q) > Wm
(7.3)
QEC7h
with
ip
a function satisfying similar conditions as q.
From (7.2) and (7.3) then follows
1c(P)
j
> kh3
which is what we wanted to show.
88
Rh
'2h
C^h