PDEs (partial differential equations) are ubiquitous in the modelling of real-world problems. The exact solutions of these equations can never be found in practice, so that numerical schemes (mathematical algorithms executed on a computer) are required for their approximation. The ultimate goal of the RANPDEs project is to design algorithms that allow to compute a numerical approximation with error below some desired tolerance at the expense of minimal computational cost. To this end, we first accurately quantify the overall error and identify its different components:

• discretization error from the finite element method,
• linearization error from the iterative linearization scheme (fixed-point, Newton,…),
• algebraic error from the iterative algebraic solver,
• …

We then apply adaptivity in a broad sense:

• mesh refinement,
• polynomial degree adjustment,
• stopping criteria for the iterative solvers,
• …

Robust a posteriori error estimators are our main tool. We focus on proofs of contraction and optimality with respect to the total cost of the numerical simulation, for several classes of important model nonlinear partial differential equations. Numerical experiments illustrate all the theoretical developments.