Abstract
We consider the preconditioned conjugate gradient method (PCG) with optimal preconditioner in the frame of the boundary element method 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}(Gamma ). The main results also hold for the hyper-singular integral equation with energy space H^{1/2}(Gamma ).
Highlights
1.1 Model problemLet ⊂ Rd with d = 2, 3 be a bounded Lipschitz domain with polyhedral boundary ∂
That the quasi-error is linearly convergent in each step of the adaptive algorithm
We monitor the condition numbers of the arising boundary element method (BEM) systems for diagonal preconditioning [7], the proposed additive Schwarz preconditioning from Sect. 3.1, and no preconditioning
Summary
Let ⊂ Rd with d = 2, 3 be a bounded Lipschitz domain with polyhedral boundary ∂. Let ⊆ ∂ be a (relatively) open and connected subset. TN }, consider the standard basis χ, j : j = 1, . Tj where the matrix A ∈ RN×N is positive definite and symmetric. For a given initial triangulation T0, we consider an adaptive mesh-refinement strategy of the type solve −→ estimate −→ mar k −→ r e f ine (6). We note that the condition number of the Galerkin matrix A from (5) depends on the number of elements of T , as well as the minimal and maximal diameter. The step solve requires an efficient preconditioner as well as an appropriate iterative solver. Adaptive BEM with inexact PCG solver yields almost optimal
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