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34 cell-node mimetic domain decomposition methods Ch. 2
2.5.2. A problem on a smooth grid. The second example considers the semilinear
parabolic problem (2.1) posed on the following space-time domain:
Ω × (0, T ] = {x ≡ (x, y) ∈ R 2 : 0 < x < 1, 0 < y < 1 + 1
2 sin(2πx)} × (0, 1].
Both tensor K(x) and function g(ψ) are chosen to be the same as in the previous
example. Likewise, data functions f(x, t) and ψ0(x) are such that (2.45) is the
exact solution of the problem.
The discretization process follows the steps described in Subsection 2.5.1.
Initially, we define a smooth logically rectangular grid Ωh ≡ {(xi,j, yi,j)} N i,j=1 ,
which discretizes the spatial domain Ω by using a collection of N nodes in each
direction (that is, Nx = Ny = N) and a spatial mesh size h = 1/(N − 1). The
coordinates of the (i, j)-node are then given by:
xi,j = (i − 1) h,
yi,j = (1 + 1
2 sin(2πxi,j)) (j − 1) h,
(2.54)
for i, j ∈ {1, 2, . . . , N}. Figure 2.10 shows an example of such a smooth grid, for
N = 17. More examples of sinusoidal grids can be found in Shashkov (1996).
Next, using the MFD method introduced in Section 2.3, we obtain a semidiscrete
scheme of the form (2.28).
As in the preceding test, the spatial domain Ω is decomposed into m = 4
overlapping subdomains {Ωk} m k=1 , each of which consists of mk = 4 disjoint
connected components {Ωkl} mk
l=1 , for k = 1, 2, 3, 4. In this case, we introduce the
following intervals:
I1 ≡ ( 0, 1
4 + ε) ∪ ( 1 3
2 − ε, 4 + ε) ,
I2 ≡ ( 1
4
− ε, 1
2 + ε) ∪ ( 3
4 − ε, 1) ,
I1,x ≡ ( 0, ( 1
4 + ε) γ(x) ) ∪ (( 1
2 − ε) γ(x), ( 3
4 + ε) γ(x) ) ,
I2,x ≡ (( 1
4 − ε) γ(x), ( 1
2 + ε) γ(x) ) ∪ (( 3
4 − ε) γ(x), γ(x) ) ,
where γ(x) = 1 + 1
1
2 sin(2πx) and ε = 16 . Now, the overlapping width in the
x-direction takes a constant value of ξx = 2ε = 1
8 , whereas that in the y-direction
depends on x as ξy(x) = 2εγ(x) = 1
8γ(x). Hence, the subdomains {Ωk} m k=1 are
defined as:
Ω1 ≡ I1 × I1,x, Ω2 ≡ I2 × I1,x,
Ω3 ≡ I1 × I2,x, Ω4 ≡ I2 × I2,x.
(2.55)
On the other hand, the partition of unity {ρk(x)} m k=1 is constructed upon the
aforementioned functions φ1(x) (cf. formula (2.49)) and φ2(x). Furthermore,