289 research outputs found
An a posteriori error analysis of a mixed finite element Galerkin approximation to second order linear parabolic problems
In this article, a posteriori error estimates are derived for a mixed finite element Galerkin approximation to second order linear parabolic initial and boundary value problems. Using mixed elliptic reconstruction method, a posteriori error estimates in and -norms with optimal order of convergence for the solution as well as its flux are proved for the semidiscrete scheme. Finally, based on backward Euler method, a completely discrete scheme is analyzed and a posteriori bounds are derived, which improves earlier results on a posteriori estimates for mixed parabolic problems
A Posteriori error control & adaptivity for Crank-Nicolson finite element approximations for the linear Schrodinger equation
We derive optimal order a posteriori error estimates for fully discrete
approximations of linear Schr\"odinger-type equations, in the
norm. For the discretization in time we use the Crank-Nicolson
method, while for the space discretization we use finite element spaces that
are allowed to change in time. The derivation of the estimators is based on a
novel elliptic reconstruction that leads to estimates which reflect the
physical properties of Schr\"odinger equations. The final estimates are
obtained using energy techniques and residual-type estimators. Various
numerical experiments for the one-dimensional linear Schr\"odinger equation in
the semiclassical regime, verify and complement our theoretical results. The
numerical implementations are performed with both uniform partitions and
adaptivity in time and space. For adaptivity, we further develop and analyze an
existing time-space adaptive algorithm to the cases of Schr\"odinger equations.
The adaptive algorithm reduces the computational cost substantially and
provides efficient error control for the solution and the observables of the
problem, especially for small values of the Planck constant
Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations
We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed for an abstract nonlinear parabolic PDE and subsequently applied to linear heat and nonlinear Cahn-Hilliard equations. The error due to finite element approximations is estimated with a residual weighted approximate-dual solution which is computed with two primal approximations at nested levels. We prove that the exact error is estimated by our estimator up to higher-order remainder terms. Numerical experiments confirm the theory regarding consistency of the dual-based two-level estimator. We also present a novel space-time adaptive strategy to control errors based on the new estimator
Adaptive computational methods for aerothermal heating analysis
The development of adaptive gridding techniques for finite-element analysis of fluid dynamics equations is described. The developmental work was done with the Euler equations with concentration on shock and inviscid flow field capturing. Ultimately this methodology is to be applied to a viscous analysis for the purpose of predicting accurate aerothermal loads on complex shapes subjected to high speed flow environments. The development of local error estimate strategies as a basis for refinement strategies is discussed, as well as the refinement strategies themselves. The application of the strategies to triangular elements and a finite-element flux-corrected-transport numerical scheme are presented. The implementation of these strategies in the GIM/PAGE code for 2-D and 3-D applications is documented and demonstrated
An adaptive space-time Newton–Galerkin approach for semilinear singularly perturbed parabolic evolution equations
Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)In this article, we develop an adaptive procedure for the numerical solution of semilinear parabolic problems with possible singular perturbations. Our approach combines a linearization technique using Newton’s method with an adaptive discretization – which is based on a spatial finite element method and the backward Euler time-stepping scheme – of the resulting sequence of linear problems. Upon deriving a robust a posteriori error analysis, we design a fully adaptive Newton-Galerkin time-stepping algorithm. Numerical experiments underline the robustness and reliability of the proposed approach for various examples
An adaptive space-time Newton–Galerkin approach for semilinear singularly perturbed parabolic evolution equations
Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)In this article, we develop an adaptive procedure for the numerical solution of semilinear parabolic problems with possible singular perturbations. Our approach combines a linearization technique using Newton’s method with an adaptive discretization – which is based on a spatial finite element method and the backward Euler time-stepping scheme – of the resulting sequence of linear problems. Upon deriving a robust a posteriori error analysis, we design a fully adaptive Newton-Galerkin time-stepping algorithm. Numerical experiments underline the robustness and reliability of the proposed approach for various examples
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