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G. Gantner, D. Praetorius:
"Adaptive BEM for elliptic PDE systems, Part I: Abstract framework for weakly-singular integral equations";
in: "ASC Report 10/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 36.

In the present work, we consider weakly-singular integral equations arising from linear second-order strongly-elliptic PDE systems with constant coefficients, including, e.g., linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract properties
for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee convergence at optimal algebraic rate of an adaptive algorithm driven by the weighted-residual error. These properties are satisfied, e.g., for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include local
inverse estimates for the (non-local) boundary integral operators associated to the PDE system.

boundary element method, a posteriori error estimates, adaptive algorithm, optimal convergence, inverse estimates. 1

http://www.asc.tuwien.ac.at/preprint/2020/asc10x2020.pdf