4,272 research outputs found
A mesh adaptivity scheme on the Landau-de Gennes functional minimization case in 3D, and its driving efficiency
This paper presents a 3D mesh adaptivity strategy on unstructured tetrahedral
meshes by a posteriori error estimates based on metrics, studied on the case of
a nonlinear finite element minimization scheme for the Landau-de Gennes free
energy functional of nematic liquid crystals. Newton's iteration for tensor
fields is employed with steepest descent method possibly stepping in.
Aspects relating the driving of mesh adaptivity within the nonlinear scheme
are considered. The algorithmic performance is found to depend on at least two
factors: when to trigger each single mesh adaptation, and the precision of the
correlated remeshing. Each factor is represented by a parameter, with its
values possibly varying for every new mesh adaptation. We empirically show that
the time of the overall algorithm convergence can vary considerably when
different sequences of parameters are used, thus posing a question about
optimality.
The extensive testings and debugging done within this work on the simulation
of systems of nematic colloids substantially contributed to the upgrade of an
open source finite element-oriented programming language to its 3D meshing
possibilities, as also to an outer 3D remeshing module
Convergence of Adaptive Finite Element Approximations for Nonlinear Eigenvalue Problems
In this paper, we study an adaptive finite element method for a class of a
nonlinear eigenvalue problems that may be of nonconvex energy functional and
consider its applications to quantum chemistry. We prove the convergence of
adaptive finite element approximations and present several numerical examples
of micro-structure of matter calculations that support our theory.Comment: 24 pages, 12 figure
Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations
The iterative diagonalization of a sequence of large ill-conditioned
generalized eigenvalue problems is a computational bottleneck in quantum
mechanical methods employing a nonorthogonal basis for {\em ab initio}
electronic structure calculations. We propose a hybrid preconditioning scheme
to effectively combine global and locally accelerated preconditioners for rapid
iterative diagonalization of such eigenvalue problems. In partition-of-unity
finite-element (PUFE) pseudopotential density-functional calculations,
employing a nonorthogonal basis, we show that the hybrid preconditioned block
steepest descent method is a cost-effective eigensolver, outperforming current
state-of-the-art global preconditioning schemes, and comparably efficient for
the ill-conditioned generalized eigenvalue problems produced by PUFE as the
locally optimal block preconditioned conjugate-gradient method for the
well-conditioned standard eigenvalue problems produced by planewave methods
Adaptive Finite Element Approximations for Kohn-Sham Models
The Kohn-Sham equation is a powerful, widely used approach for computation of
ground state electronic energies and densities in chemistry, materials science,
biology, and nanosciences. In this paper, we study the adaptive finite element
approximations for the Kohn-Sham model. Based on the residual type a posteriori
error estimators proposed in this paper, we introduce an adaptive finite
element algorithm with a quite general marking strategy and prove the
convergence of the adaptive finite element approximations. Using D{\" o}rfler's
marking strategy, we then get the convergence rate and quasi-optimal
complexity. We also carry out several typical numerical experiments that not
only support our theory,but also show the robustness and efficiency of the
adaptive finite element computations in electronic structure calculations.Comment: 38pages, 7figure
All-electron density functional theory and time-dependent density functional theory with high-order finite elements
We present for static density functional theory and time-dependent density
functional theory calculations an all-electron method which employs high-order
hierarchical finite element bases. Our mesh generation scheme, in which
structured atomic meshes are merged to an unstructured molecular mesh, allows a
highly nonuniform discretization of the space. Thus it is possible to represent
the core and valence states using the same discretization scheme, i.e., no
pseudopotentials or similar treatments are required. The nonuniform
discretization also allows the use of large simulation cells, and therefore
avoids any boundary effects.Comment: 11 pages, 9 figures; final (=published) versio
Towards chemical accuracy using a multi-mesh adaptive finite element method in all-electron density functional theory
Chemical accuracy serves as an important metric for assessing the
effectiveness of the numerical method in Kohn--Sham density functional theory.
It is found that to achieve chemical accuracy, not only the Kohn--Sham
wavefunctions but also the Hartree potential, should be approximated
accurately. Under the adaptive finite element framework, this can be
implemented by constructing the \emph{a posteriori} error indicator based on
approximations of the aforementioned two quantities. However, this way results
in a large amount of computational cost. To reduce the computational cost, we
propose a novel multi-mesh adaptive method, in which the Kohn--Sham equation
and the Poisson equation are solved in two different meshes on the same
computational domain, respectively. With the proposed method, chemical accuracy
can be achieved with less computational consumption compared with the adaptive
method on a single mesh, as demonstrated in a number of numerical experiments.Comment: 19pages, 17 figure
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