Multi-level adaptive solutions to boundary-value problems

The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in O (n) O (n) operations (40 n additions and shifts for Poisson problems) ; (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc. ) to the evolving solution in a nearly optimal way, obtaining " ∞ -order" approximations and low n, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problems-confirm theoretical predictions. Similar techniques for initial-value problems are briefly discussed.