350 research outputs found
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
Convergence of an adaptive hp finite element strategy in higher space-dimensions
We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients
Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows
In this article we develop both the a priori and a posteriori error analysis of hp–version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain = 2,3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm
Interpolation in Jacobi-weighted spaces and its application to a posteriori error estimations of the p-version of the finite element method
The goal of this work is to introduce a local and a global interpolator in
Jacobi-weighted spaces, with optimal order of approximation in the context of
the -version of finite element methods. Then, an a posteriori error
indicator of the residual type is proposed for a model problem in two
dimensions and, in the mathematical framework of the Jacobi-weighted spaces,
the equivalence between the estimator and the error is obtained on appropriate
weighted norm
Adaptive Discontinuous Galerkin Methods for Variational Inequalities with Applications to Phase Field Models
Solutions of variational inequalities often have limited regularity. In particular, the nonsmooth parts are local, while other parts of the solution have higher regularity. To overcome this limitation, we apply hp-adaptivity, which uses a combination of locally finer meshes and varying polynomial degrees to separate the different features of the the solution. For this, we employ Discontinuous Galerkin (DG) methods and show some novel error estimates for the obstacle problem which emphasize the use in hp-adaptive algorithms.
Besides this analysis, we present how to efficiently compute numerical solutions using error estimators, fast algebraic solvers which can also be employed in a parallel setup, and discuss implementation details.
Finally, some numerical examples and applications to phase field models are presented
Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows
In this article we develop both the a priori and a posteriori error analysis of hp– version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain Ω ⊂ R^d, d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm
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