Abstract
We consider a second-order elliptic boundary value problem with strongly monotone and Lipschitz-continuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the Banach–Picard linearization, and a contractive linear algebraic solver. In particular, we identify stopping criteria for the algebraic solver that on the one hand do not request an overly tight tolerance but on the other hand are sufficient for the inexact (perturbed) Banach–Picard linearization to remain contractive. Similarly, we identify suitable stopping criteria for the Banach–Picard iteration that leave an amount of linearization error that is not harmful for the residual a posteriori error estimate to steer reliably the adaptive mesh-refinement. For the resulting algorithm, we prove a contraction of the (doubly) inexact iterates after some amount of steps of mesh-refinement/linearization/algebraic solver, leading to its linear convergence. Moreover, for usual mesh-refinement rules, we also prove that the overall error decays at the optimal rate with respect to the number of elements (degrees of freedom) added with respect to the initial mesh. Finally, we prove that our fully adaptive algorithm drives the overall error down with the same optimal rate also with respect to the overall algorithmic cost expressed as the cumulated sum of the number of mesh elements over all mesh-refinement, linearization, and algebraic solver steps. Numerical experiments support these theoretical findings and illustrate the optimal overall algorithmic cost of the fully adaptive algorithm on several test cases.
Highlights
Let ⊂ Rd with d ≥ 1 be a bounded Lipschitz domain with polytopal boundary
We propose an adaptive algorithm of the type estimate total error and its components advance algebra/advance linearization/mark and refine mesh elements which monitors and adequately stops the iterative linearization and the linear algebraic solver as well as steers the local mesh-refinement
Under this so-called realistic assumption on the algebraic solver, we have proved in [23] that the overall strategy leads to optimal convergence rates with respect to the number of degrees of freedom as well as to almost optimal convergence rates with respect to the overall computational cost
Summary
Let ⊂ Rd with d ≥ 1 be a bounded Lipschitz domain with polytopal boundary. Given f ∈ L2( ) and a nonlinear operator A : Rd → Rd , we aim to numerically. Approximate the weak solution u ∈ H01( ) of the nonlinear boundary value problem −div A(∇u ) = f in , (1) u = 0 on ∂ To this end, we propose an adaptive algorithm of the type estimate total error and its components. Advance algebra/advance linearization/mark and refine mesh elements which monitors and adequately stops the iterative linearization and the linear algebraic solver as well as steers the local mesh-refinement. The goal of this contribution is to perform a first rigorous mathematical analysis of this algorithm in terms of convergence and quasi-optimal computational cost
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