C. Carstensen, D. Praetorius:

"Convergence of adaptive boundary element methods";

Journal of Integral Equations and Applications,24(2012), 1; 1 - 23.

In many applications, adaptive mesh-refinement is observed to be an

efficient tool for the numerical solution of partial differential equations

and integral equations. Convergence of adaptive schemes to the correct

solution, however, is so far only understood for certain kind of differential

equations. In general, it cannot be excluded that the adaptive algorithm

computes a convergent sequence of discrete approximations with a limit

which is not the correct solution. This work proposes a feedback loop

which guarantees the convergence of the computed discrete approximations

to the correct solution. Although stated for Symm's integral equation of

the first kind, the main part of this work is written for a general audience

in the context of weak forms as Riesz representations in Hilbert spaces.

Numerical examples illustrate the adaptive strategies.

http://dx.doi.org/10.1216/JIE-2012-24-1-1

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