Exact solutions of partial differential equations are usually impossible to find. One then employs numerical (computer) simulations to obtain approximate solutions. Some arising questions of primordial interest are:
• How large is the overall error between the unkown exact solution and the available numerical approximation?
• Where does it come from? Insufficient spatial mesh resolution? Time stepping? Iterative linearization? Iterative algebraic solver? And where is it localized? In which part of the computational domain? On which time step?
• How to make the numerical simulation efficient? I.e., how to obtain the best possible result for the smallest possible price (error with respect to algorithmic complexity, computing time, or memory usage)?
I will explain some basic principles of the theory of a posteriori error estimation and adaptivity which allows to indicate answers to the above questions. I will in particular treat:
• The Laplace equation, with focus on the control of the discretization error and on mesh adaptivity.
• The nonlinear Laplace equation, with focus on the overall error control (iterative linearization, iterative algebraic solver, discretization) and on adaptive steering which balances these arising error components.
• The reaction–diffusion equation, with a focus on robustness (uniform behavior) with respect to the reaction and diffusion parameters.
• The heat equation, with a focus on robustness with respect to the final time and on space–time error localization.
• Multiphase multicompositional flows in underground porous media (with phase appearance and disappearance), as an environmental application, with focus on practical implementations in an industrial code.