36 research outputs found

    Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs

    Get PDF
    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 pp-Laplacian illustrate the tight overall error control and important computational savings achieved in our approach

    An hphp-Adaptive Newton-Galerkin Finite Element Procedure for Semilinear Boundary Value Problems

    Full text link
    In this paper we develop an hphp-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 hphp-version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully hphp-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

    Get PDF
    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

    Full text link
    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

    Full text link
    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
    corecore