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32 cell-node mimetic domain decomposition methods Ch. 2
Subordinate to such a decomposition, we introduce the splittings Ah ∑
=
m
k=1 Ah k and F h (t) = ∑m k=1 F h k (t), for the discrete diffusion operator and the
semidiscrete source/sink term, respectively. Both Ah k and F h k (t) are defined in
the spirit of Subsection 2.4.1. These splittings are then combined with the linearly
implicit variant of Yanenko’s method in order to obtain a totally discrete
scheme of the form (2.42). In this case, the linear system to solve at the kth
internal stage is reduced to a set of four uncoupled subsystems associated
to the corresponding disjoint components {Ωkl} 4 l=1 , for k = 1, 2, 3, 4. These
subsystems can be readily solved in parallel, thus decreasing the number of unknowns
from (N − 1) × (N − 1) to a number between n1 × n1 and n2 × n2, being
n1 = (N − 1) ( 1
4 + ε) and n2 = (N − 1) ( 1
4 + 2ε) .
Remark 2.6 A sharp question concerning the previous technique is whether the
overlapping size affects the numerical behaviour of the method. Theoretically,
the scheme under study is likely to be second-order convergent in space and firstorder
convergent in time, which means that the global error is expected to be
bounded by C(h2 + τ) in a suitable discrete norm. The constant C > 0 in this
expression is independent of h and τ, but depends on negative powers of the
overlapping size ξ; that is to say, such a constant deteriorates as ξ → 0 + (cf.
Vabishchevich (1994); Mathew et al. (1998); Samarskiĭ et al. (2002)). Thus, it
is convenient to set the value of ξ to a fixed quantity in order to compute the
numerical orders of convergence. Typically, for a given fixed ξ, the computed
orders will become lower than expected if the number of mesh cells lying inside
the overlapping region, nξ, is not large enough. This is due to the fact that such
a region is the only part of the domain in which the partition of unity functions
{ρk(x)} m k=1 smoothly vary from 0 to 1. In other words, a small value of nξ will
prevent the totally discrete scheme (2.42) from capturing the smoothness of each
function ρk(x). On the other hand, a decreasing value of ξ reduces the amount
of CPU time used by the algorithm. The compromise between the size of the
error constant and the required CPU time determines the optimal value for the
overlapping size. For this example, the adopted value of ξ = 1
8 is a good choice,
according to the mesh sizes to be considered in the sequel.
In our first experiment, we compare each component of the numerical solution
Ψ h n to the evaluation of the exact solution ψ(x, t) at the corresponding cell
center and time level. If we consider a sufficiently small fixed time step τ, this
quantity measures the global error in space and can be used to test the order of
convergence in space of the proposed method. Such an error, denoted by E h τ , is
computed in the discrete L 2 -norm in space and the discrete maximum norm in
time, i.e.:
Eh ⎛
N−1 ∑
τ 2 = max ⎝
n∈{1,2,...,NT +1}
i=1
N−1 ∑
j=1
|Ω i+1/2,j+1/2|
(
ψ(x i+1/2,j+1/2, tn)
− (Ψ h n) i+1/2,j+1/2
)2 ) 1/2
.
(2.51)