391 research outputs found
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
Error estimation and adjoint-based adaptation in aerodynamics
In this article we give an overview of recent developments
in error estimation and in residual-based and goal-oriented
(adjoint-based) adaptation for Discontinuous Galerkin discretizations
of sub- and supersonic viscous compressible flows. We also give an
outlook on the planned continuation of this research in the EU project
ADIGMA
Goal-oriented error analysis of iterative Galerkin discretizations for nonlinear problems including linearization and algebraic errors
We consider the goal-oriented error estimates for a linearized iterative
solver for nonlinear partial differential equations. For the adjoint problem
and iterative solver we consider, instead of the differentiation of the primal
problem, a suitable linearization which guarantees the adjoint consistency of
the numerical scheme. We derive error estimates and develop an efficient
adaptive algorithm which balances the errors arising from the discretization
and use of iterative solvers. Several numerical examples demonstrate the
efficiency of this algorithm.Comment: submitte
Semilinear mixed problems on Hilbert complexes and their numerical approximation
Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010),
281-354] that linear, mixed variational problems, and their numerical
approximation by mixed finite element methods, can be studied using the
powerful, abstract language of Hilbert complexes. In another recent article
[arXiv:1005.4455], we extended the Arnold-Falk-Winther framework by analyzing
variational crimes (a la Strang) on Hilbert complexes. In particular, this gave
a treatment of finite element exterior calculus on manifolds, generalizing
techniques from surface finite element methods and recovering earlier a priori
estimates for the Laplace-Beltrami operator on 2- and 3-surfaces, due to Dziuk
[Lecture Notes in Math., vol. 1357 (1988), 142-155] and later Demlow [SIAM J.
Numer. Anal., 47 (2009), 805-827], as special cases. In the present article, we
extend the Hilbert complex framework in a second distinct direction: to the
study of semilinear mixed problems. We do this, first, by introducing an
operator-theoretic reformulation of the linear mixed problem, so that the
semilinear problem can be expressed as an abstract Hammerstein equation. This
allows us to obtain, for semilinear problems, a priori solution estimates and
error estimates that reduce to the Arnold-Falk-Winther results in the linear
case. We also consider the impact of variational crimes, extending the results
of our previous article to these semilinear problems. As an immediate
application, this new framework allows for mixed finite element methods to be
applied to semilinear problems on surfaces.Comment: 22 pages; v2: major revision, particularly sharpening of error
estimates in Section
Cost-optimal adaptive iterative linearized FEM for semilinear elliptic PDEs
We consider scalar semilinear elliptic PDEs where the nonlinearity is
strongly monotone, but only locally Lipschitz continuous. We formulate an
adaptive iterative linearized finite element method (AILFEM) which steers the
local mesh refinement as well as the iterative linearization of the arising
nonlinear discrete equations. To this end, we employ a damped Zarantonello
iteration so that, in each step of the algorithm, only a linear Poisson-type
equation has to be solved. We prove that the proposed AILFEM strategy
guarantees convergence with optimal rates, where rates are understood with
respect to the overall computational complexity (i.e., the computational time).
Moreover, we formulate and test an adaptive algorithm where also the damping
parameter of the Zarantonello iteration is adaptively adjusted. Numerical
experiments underline the theoretical findings
- …