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Multi-Level Adaptive Solutions to Boundary-Value Problems

Achi Brandt
Mathematics of Computation
Vol. 31, No. 138 (Apr., 1977), pp. 333-390
DOI: 10.2307/2006422
Stable URL: http://www.jstor.org/stable/2006422
Page Count: 58
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Multi-Level Adaptive Solutions to Boundary-Value Problems
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Abstract

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 $O(n)$ operations ($40n$ 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 "$\infty$-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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