Adaptive BEM for elliptic PDE systems, part I: abstract framework, for weakly-singular integral equations

Abstract

In the present work, we consider weakly-singular integral equations arising from linear second-order 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 conditions for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee that the weighted-residual error estimator is reliable and converges at optimal algebraic rate if used within an adaptive algorithm. These conditions are satisfied, e.g. for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include the localization of (non-local) fractional Sobolev norms and local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.

Keywords:

• Boundary element method
• a posteriori error estimates
• adaptive algorithm
• optimal convergence
• inverse estimates

2010 Mathematics Subject Classifications:

• 65N15
• 65N30
• 65N38
• 65N50

Acknowledgments

The authors acknowledge support through the Austrian Science Fund (FWF) under grant J4379-N,P29096,W1245.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 For $\hat{\omega }\subseteq {\mathbb{R}}^{d-1}$ and $\omega \subseteq {\mathbb{R}}^d$, a mapping $\gamma :\hat{\omega }\to \omega$ is bi-Lipschitz if it is bijective and γ as well as its inverse ${\gamma }^{-1}$ are Lipschitz continuous.

2 A compact Lipschitz domain is the closure of a bounded Lipschitz domain. For d = 2, it is the finite union of compact intervals with non-empty interior.

3 We use the convention $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(T,\mathrm{\varnothing }):=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}\left(\mathrm{\Gamma }\right)$.

Additional information

Funding

The authors acknowledge support through the Austrian Science Fund (FWF) under grant J4379-N,P29096,W1245.