36 research outputs found
Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs
International audienceWe consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations of diffusion type. To solve these systems, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. We derive adaptive stopping criteria for both iterative solvers. Our criteria are based on an a posteriori error estimate which distinguishes the different error components, namely the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error does no longer affect the overall error significantly. Our estimates also yield a guaranteed upper bound on the overall error at each step of the nonlinear and linear solvers. We prove the (local) efficiency and robustness of the estimates with respect to the size of the nonlinearity owing, in particular, to the error measure involving the dual norm of the residual. Our developments hinge on equilibrated flux reconstructions and yield a general framework. We show how to apply this framework to various discretization schemes like finite elements, nonconforming finite elements, discontinuous Galerkin, finite volumes, and mixed finite elements; to different linearizations like fixed point and Newton; and to arbitrary iterative linear solvers. Numerical experiments for the -Laplacian illustrate the tight overall error control and important computational savings achieved in our approach
An -Adaptive Newton-Galerkin Finite Element Procedure for Semilinear Boundary Value Problems
In this paper we develop an -adaptive procedure for the numerical
solution of general, semilinear elliptic boundary value problems in 1d, with
possible singular perturbations. Our approach combines both a prediction-type
adaptive Newton method and an -version adaptive finite element
discretization (based on a robust a posteriori residual analysis), thereby
leading to a fully -adaptive Newton-Galerkin scheme. Numerical experiments
underline the robustness and reliability of the proposed approach for various
examples.Comment: arXiv admin note: text overlap with arXiv:1408.522
On the Convergence of Adaptive Iterative Linearized Galerkin Methods
A wide variety of different (fixed-point) iterative methods for the solution
of nonlinear equations exists. In this work we will revisit a unified iteration
scheme in Hilbert spaces from our previous work that covers some prominent
procedures (including the Zarantonello, Ka\v{c}anov and Newton iteration
methods). In combination with appropriate discretization methods so-called
(adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main
purpose of this paper is the derivation of an abstract convergence theory for
the unified ILG approach (based on general adaptive Galerkin discretization
methods) proposed in our previous work. The theoretical results will be tested
and compared for the aforementioned three iterative linearization schemes in
the context of adaptive finite element discretizations of strongly monotone
stationary conservation laws
Fully Adaptive Newton-Galerkin Methods for Semilinear Elliptic Partial Differential Equations
In this paper we develop an adaptive procedure for the numerical solution of
general, semilinear elliptic problems with possible singular perturbations. Our
approach combines both a prediction-type adaptive Newton method and an adaptive
finite element discretization (based on a robust a posteriori error analysis),
thereby leading to a fully adaptive Newton-Galerkin scheme. Numerical
experiments underline the robustness and reliability of the proposed approach
for different examples
Gradient Flow Finite Element Discretizations with Energy-Based Adaptivity for the Gross-Pitaevskii Equation
We present an effective adaptive procedure for the numerical approximation of
the steady-state Gross-Pitaevskii equation. Our approach is solely based on
energy minimization, and consists of a combination of gradient flow iterations
and adaptive finite element mesh refinements. Numerical tests show that this
strategy is able to provide highly accurate results, with optimal convergence
rates with respect to the number of freedom