Adaptive BEM with inexact PCG solver yields almost optimal computational costs
This work provides a theoretically grounded adaptive algorithm for BEM that ensures both optimal convergence rates and near-optimal computational cost, addressing efficiency in solving integral equations.
The paper presents an adaptive boundary element method with an inexact PCG solver that achieves convergence with optimal algebraic rates and almost optimal computational complexity for elliptic first-kind integral equations.
We consider the preconditioned conjugate gradient method (PCG) with optimal preconditioner in the frame of the boundary element method (BEM) for elliptic first-kind integral equations. Our adaptive algorithm steers the termination of PCG as well as the local mesh-refinement. Besides convergence with optimal algebraic rates, we also prove almost optimal computational complexity. In particular, we provide an additive Schwarz preconditioner which can be computed in linear complexity and which is optimal in the sense that the condition numbers of the preconditioned systems are uniformly bounded. As model problem serves the 2D or 3D Laplace operator and the associated weakly-singular integral equation with energy space $\widetilde{H}^{-1/2}(Γ)$. The main results also hold for the hyper-singular integral equation with energy space $H^{1/2}(Γ)$.