Computational analysis demonstrates that multi-level adaptive discretization solves boundary-value problems in O(n) operations, highlighting optimal efficiency even around singularities.
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.
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Achi Brandt (1977) studied this question.
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