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Fig. 2.3
L
∞
(J
;
L
2
(G))
convergence rates for
θ
=
1
1
2
(
bottom
)
(
top
)and
θ
=
1
2
For
θ
(h)
and obtain, in terms of the mesh
width
h
, second order convergence for both FDM and FEM, i.e.
=
(Crank-Nicolson), we let
k
=
O
h
2
ε
m
(h
2
).
su
m
2
=
O
(2.28)
Both convergence rates are shown in Fig.
2.3
.
The convergence rates (
2.27
)-(
2.28
) have been shown for the finite difference
method in Theorem
2.3.8
. In the next chapter, we show that these also hold for the
finite element method.
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