Numerical Methods for Parabolic Equations

(1)  ut −∆u = f in Ω× (0, T ), u = 0 on ∂Ω× (0, T ), u(·, 0) = u0 in Ω. Here u = u(x, t) is a function of spatial variable x ∈ Ω and time variable t ∈ (0, T ). The Laplace differential operator ∆ is taking with respect to the spatial variable. For the simplicity of exposition, we consider only homogenous Dirichlet boundary condition and comment on the adaptation to Neumann and other type of boundary conditions. Comparing with elliptic equation, we also need to assign the value at time t = 0 which is called initial condition. The ending time T could be +∞. For parabolic equations, the boundary ∂Ω× (0, T )∪Ω×{t = 0} is called the parabolic boundary. Therefore the initial condition can be also thought as a boundary condition.