Adaptive BEM with inexact PCG solver yields almost optimal computational costs

Abstract

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 H −1/2 (Γ). The main results also hold for the hyper-singular integral equation with energy space H 1/2 (Γ).

Adaptive boundary element methods for the computation of the electrostatic capacity on complex polyhedra

“…Figure 7 displays the condition number of the arising linear systems (10). We emphasize that the MLAS preconditioning is optimal in the sense that the condition number of the arising systems remains bounded, independently of the number of elements; see [FHPS19]. This is confirmed for MLAS as well as empirically observed also for operator preconditioning with the hypersingular operator in Algorithm 1.…”

“…Remark 7. The recent work [FHPS19] proves that Theorem 6 remains valid, if (18) is solved inexactly by PCG with optimal additive Schwarz preconditioner. With the PCG iterates Φ ,k ≈ Φ , the iterative solver is stopped if ||| Φ ,k−1 − Φ ,k ||| ≤ λµ (Φ ,k ), where the residual error estimator is evaluated at Φ ,k instead of the exact Galerkin solution.…”

“…• Algorithm 1 with its built-in operator preconditioning vs. diagonal preconditioning vs. non-preconditioning; • Algorithm 5 with multilevel additive Schwarz preconditioning (MLAS) [FHPS19] vs. diagonal preconditioning [GM06] vs. non-preconditioning; We consider lowest-order BEM for different polyhedral domains Ω starting from simple shapes, e.g., the unit cube (Section 3.1), and moving to more complicated shapes, e.g., a star-shaped domain (Section 3.3). If not stated otherwise, we employ Algorithm 1 with the Dörfler parameter θ = 1/2 and operator preconditioning.…”

Adaptive BEM with optimal convergence rates for the Helmholtz equation

“…are solved by GMRES. Preconditioning can be done by diagonal or multilevel additive Schwarz preconditioners; see, e.g., [GM06,FHPS18] for BEM for the Laplace problem. To reduce the cost for storage of the system matrices, BEM++ supports H-matrices.…”

Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods

“…The resulting linear systems are solved by PCG. We refer to the recent work [FHPS18] for the interplay of PCG solver and optimal adaptivity. We compare the preconditioners to simple diagonal preconditioning.…”