Métodos miméticos de pasos fraccionarios - E-Archivo - Universidad ...

§2.2 properties of the continuous operators 11
FSRKm+1 method and permits to reduce the nonlinear semidiscrete problem to
a collection of linear systems which can be efficiently parallelized.
The rest of the chapter is organized as follows. Section 2.2 describes some
relevant properties of the first- and second-order differential operators arising in
(2.1a) and (2.3). Based on these properties, the semidiscrete scheme is formulated
on general non-orthogonal meshes by using a second-order MFD method
(cf. Section 2.3). In Section 2.4, we set the basis for the domain decomposition
splitting technique and introduce a family of linearly implicit FSRKm+1 methods
of classical order 1. To conclude, we report on some numerical experiments
in Section 2.5.
2.2. Properties of the continuous operators
Let us denote Hs as the space of scalar functions endowed with the inner
product:
∫
(ψ, ϕ)Hs = ψϕ dx + ψϕ ds, ψ, ϕ ∈ Hs. (2.4)
∂Ω
Ω
The solution of problem (2.1) belongs to a subspace H0 s ⊆ Hs involving those
functions of Hs which are equal to zero on ∂Ω, i.e., ψ : [0, T ] → H0 s. Note that
the previous inner product reduces to the first integral in (2.4) if any of the
considered functions belongs to H0 s.
Next, we can define the second-order operator A : D(A) ⊆ Hs → Hs such
that Aψ = −div (Kgrad ψ). When applied to functions ψ ∈ H0 s, A can be proved
to be symmetric and positive definite in the inner product (2.4). To this end,
let us recall Green’s first identity:
∫
∫
φ div w dx + ⟨w, grad φ⟩ dx = φ ⟨w, n⟩ ds (2.5)
Ω
Ω
where ⟨·, ·⟩ denotes the standard dot product, φ and w are arbitrary smooth
scalar and vector functions, respectively, and n is the outward unit vector normal
to ∂Ω. Obviously, (2.5) reduces to:
∫
∫
φ div w dx + ⟨w, grad φ⟩ dx = 0, (2.6)
Ω
Ω
for those scalar functions φ ∈ H0 s. Hence, using (2.4) and (2.6), it holds that:
∫
∫
(Aψ, ϕ)Hs = − ϕ div (Kgrad ψ) dx = ⟨Kgrad ψ, grad ϕ⟩ dx, (2.7)
provided that ϕ ∈ H 0 s. Therefore:
Ω
Ω
(Aψ, ϕ)Hs = (ψ, Aϕ)Hs, (Aψ, ψ)Hs > 0, (2.8)
for all ψ, ϕ ∈ H 0 s, since K is assumed to be symmetric and positive definite.
∂Ω