Domain decomposition methods - Basic notions

All engineering problems today are solved in parallel on computers with hundreds or even thousands of compute nodes. This article describes domain decomposition methods that can be applied to these new tools. Emile Picard teaches us in that to understand a theory, it's good to have a model problem in mind.

The approximation methods we use are theoretically applicable to any equation, but they only become really interesting for studying the properties of functions defined by differential equations if we go beyond generalities and consider certain classes of equations.

Throughout this presentation, we'll choose a common thread: the heat equation.

representing the variations in time and space of the temperature of a body filling the domain Ω, subjected to a heat source f (which will be called the second member), with a given initial temperature throughout the domain, and boundary conditions on the edge of the domain ∂Ω, e.g. Dirichlet (the temperature is fixed), i.e. u = g. ∂ _t u is the time derivative of u, Δ is the Laplace operator, Δu = ∂ ₁₁ u + ∂ ₂₂ u + ∂ ₃₃ u. To calculate an approximate solution to this equation on a computer, we can start with a semi-discretization in time. The simplest scheme is the implicit Euler scheme (see [AF 1 220] ). Let's divide the time interval [0, T] into sub-intervals [t _n , t _n+1 ] of length Δt. Let's denote u _n (x) the approximation of u at time t _n at point x, calculated by the recurrence formula