1,911 research outputs found
A Posteriori Error Control for the Binary Mumford-Shah Model
The binary Mumford-Shah model is a widespread tool for image segmentation and
can be considered as a basic model in shape optimization with a broad range of
applications in computer vision, ranging from basic segmentation and labeling
to object reconstruction. This paper presents robust a posteriori error
estimates for a natural error quantity, namely the area of the non properly
segmented region. To this end, a suitable strictly convex and non-constrained
relaxation of the originally non-convex functional is investigated and Repin's
functional approach for a posteriori error estimation is used to control the
numerical error for the relaxed problem in the -norm. In combination with
a suitable cut out argument, a fully practical estimate for the area mismatch
is derived. This estimate is incorporated in an adaptive meshing strategy. Two
different adaptive primal-dual finite element schemes, and the most frequently
used finite difference discretization are investigated and compared. Numerical
experiments show qualitative and quantitative properties of the estimates and
demonstrate their usefulness in practical applications.Comment: 18 pages, 7 figures, 1 tabl
Dual weighted residual method for laser surface hardening of steel problem
Abstract. The main focus of this article is on the development of Adaptive Finite Element Method (AFEM) for the optimal control problem of laser surface hardening of steel governed by a dynamical system consisting of a semi-linear parabolic equation and an ordinary differential equation using Dual Weighted Residual Method (DWR). A posteriori error estimators using DWR method have been developed when a continuous piecewise linear discretization has been used for the finite element approximation of space variables and a discontinuous Galerkin method has been used for time and control discretizations. Further numerical results obtained are presented are compared with residual method numerical results. Key Words. Laser surface of steel problem, Adaptive finite element methods, Dual weighted residual methods, a posteriori error estimates. 1
An adaptive finite element method for laser surface hardening of steel problem
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Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems
In this paper we develop adaptive iterative coupling schemes for the Biot
system modeling coupled poromechanics problems. We particularly consider the
space-time formulation of the fixed-stress iterative scheme, in which we first
solve the problem of flow over the whole space-time interval, then exploiting
the space-time information for solving the mechanics. Two common
discretizations of this algorithm are then introduced based on two coupled
mixed finite element methods in-space and the backward Euler scheme in-time.
Therefrom, adaptive fixed-stress algorithms are build on conforming
reconstructions of the pressure and displacement together with equilibrated
flux and stresses reconstructions. These ingredients are used to derive a
posteriori error estimates for the fixed-stress algorithms, distinguishing the
different error components, namely the spatial discretization, the temporal
discretization, and the fixed-stress iteration components. Precisely, at the
iteration of the adaptive algorithm, we prove that our estimate gives
a guaranteed and fully computable upper bound on the energy-type error
measuring the difference between the exact and approximate pressure and
displacement. These error components are efficiently used to design adaptive
asynchronous time-stepping and adaptive stopping criteria for the fixed-stress
algorithms. Numerical experiments illustrate the efficiency of our estimates
and the performance of the adaptive iterative coupling algorithms
A Posteriori Error Estimation for the p-curl Problem
We derive a posteriori error estimates for a semi-discrete finite element
approximation of a nonlinear eddy current problem arising from applied
superconductivity, known as the -curl problem. In particular, we show the
reliability for non-conforming N\'{e}d\'{e}lec elements based on a residual
type argument and a Helmholtz-Weyl decomposition of
. As a consequence, we are also able to derive an a
posteriori error estimate for a quantity of interest called the AC loss. The
nonlinearity for this form of Maxwell's equation is an analogue of the one
found in the -Laplacian. It is handled without linearizing around the
approximate solution. The non-conformity is dealt by adapting error
decomposition techniques of Carstensen, Hu and Orlando. Geometric
non-conformities also appear because the continuous problem is defined over a
bounded domain while the discrete problem is formulated over a weaker
polyhedral domain. The semi-discrete formulation studied in this paper is often
encountered in commercial codes and is shown to be well-posed. The paper
concludes with numerical results confirming the reliability of the a posteriori
error estimate.Comment: 32 page
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