Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver

04/27/2020
by   Alexander Haberl, et al.
0

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. We in particular 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 / linerization / 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/01/2022

Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs

We consider a general nonsymmetric second-order linear elliptic PDE in t...
research
01/27/2021

Goal-oriented adaptive finite element methods with optimal computational complexity

We consider a linear symmetric and elliptic PDE and a linear goal functi...
research
10/19/2022

Optimal computational costs of AFEM with optimal local hp-robust multigrid solver

In this work, an adaptive finite element algorithm for symmetric second-...
research
11/08/2022

Cost-optimal adaptive iterative linearized FEM for semilinear elliptic PDEs

We consider scalar semilinear elliptic PDEs where the nonlinearity is st...
research
02/27/2017

Landau Collision Integral Solver with Adaptive Mesh Refinement on Emerging Architectures

The Landau collision integral is an accurate model for the small-angle d...
research
09/25/2022

Scalable adaptive algorithms for next-generation multiphase simulations

The accuracy of multiphysics simulations is strongly contingent up on th...
research
05/18/2023

Stopping Criteria for the Conjugate Gradient Algorithm in High-Order Finite Element Methods

We introduce three new stopping criteria that balance algebraic and disc...

Please sign up or login with your details

Forgot password? Click here to reset