Journal ArticleDOI
Convergence and optimal complexity of adaptive finite element eigenvalue computations
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Both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved.Abstract:
In this paper, an adaptive finite element method for elliptic eigenvalue problems is studied. Both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved. The analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper.read more
Citations
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Convergence of Adaptive Finite Element Methods
TL;DR: This work constructs a simple and efficient adaptive FEM for elliptic partial differential equations and proves that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants.
Journal ArticleDOI
Axioms of adaptivity
TL;DR: In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant.
Journal ArticleDOI
Guaranteed lower bounds for eigenvalues
TL;DR: This paper introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the corresponding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing.
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Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods
Jun Hu,Yunqing Huang,Qun Lin +2 more
TL;DR: A new systematic method that can produce lower bounds for eigenvalues is introduced and the saturation condition for most nonconforming elements is proved, which provides a guidance to modify known non Conforming elements in literature and to propose new nonconform elements.
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DFT-FE – A massively parallel adaptive finite-element code for large-scale density functional theory calculations
TL;DR: In this article, the authors present an accurate, efficient and massively parallel finite-element code, DFT-FE, for large-scale ab-initio calculations (reaching ∼ 100, 000 electrons) using Kohn-Sham density functional theory.
References
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Book
A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques
TL;DR: Introduction.
Journal ArticleDOI
Error Estimates for Adaptive Finite Element Computations
I. Babuvška,W. C. Rheinboldt +1 more
TL;DR: The main theorem gives an error estimate in terms of localized quantities which can be computed approximately, and the estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same.
Journal ArticleDOI
An optimal control approach to a posteriori error estimation in finite element methods
Roland Becker,Rolf Rannacher +1 more
TL;DR: The ‘dual-weighted-residual method’ is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems.
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A convergent adaptive algorithm for Poisson's equation
TL;DR: In this paper, a converging adaptive algorithm for linear elements applied to Poisson's equation in two space dimensions is presented, and it is proved that the error, measured in the energy norm, decreases at a constant rate in each step until a prescribed error bound is reached.
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Iterative methods by space decomposition and subspace correction
TL;DR: A unified theory for a diverse group of iterative algorithms, such as Jacobi and Gauss–Seidel iterations, diagonal preconditioning, domain decomposition methods, multigrid methods,Multilevel nodal basis preconditionsers and hierarchical basis methods, is presented by using the notions of space decomposition and subspace correction.