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2
Cell-Node Mimetic Domain Decomposition
Methods
This chapter is devoted to the numerical approximation of semilinear timedependent
reaction–diffusion problems. Via the method of lines approach, we
combine a mimetic finite difference method with a family of fractional step time
integrators which consider domain decomposition operator splittings. Although
both discretization techniques have been separately used for solving evolutionary
problems, a proper combination of them has been lacking so far. The acronyms
MFD and FSRKm will be used throughout this chapter to denote mimetic finite
difference and m-part fractional step Runge–Kutta methods, respectively.
2.1. Introduction
Let us consider the following semilinear parabolic initial-boundary value problem:
Find ψ : Ω × [0, T ] → R such that
ψt − div (K(x) grad ψ) = g(ψ, t) + f(x, t), (x, t) ∈ Ω × (0, T ], (2.1a)
ψ(x, 0) = ψ0(x), x ∈ Ω, (2.1b)
ψ(x, t) = 0, (x, t) ∈ ∂Ω × (0, T ]. (2.1c)
The spatial domain Ω ⊂ R 2 is assumed to be a bounded polygonal open set with
boundary ∂Ω and K ≡ K(x) is a symmetric and positive definite tensor defined
to be:
K =
( K XX (x) K XY (x)
K XY (x) K Y Y (x)
)
, (2.2)
On the other hand, g(·) ≡ g(·, t) denotes a nonlinear function assumed to be
globally Lipschitz in the first variable and f ≡ f(x, t) is a sufficiently smooth
source/sink term. Here and henceforth, x ≡ (x, y).