1,243 research outputs found
Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes
This paper is concerned with the design, analysis and implementation of
preconditioning concepts for spectral Discontinuous Galerkin discretizations of
elliptic boundary value problems. While presently known techniques realize a
growth of the condition numbers that is logarithmic in the polynomial degrees
when all degrees are equal and quadratic otherwise, our main objective is to
realize full robustness with respect to arbitrarily large locally varying
polynomial degrees degrees, i.e., under mild grading constraints condition
numbers stay uniformly bounded with respect to the mesh size and variable
degrees. The conceptual foundation of the envisaged preconditioners is the
auxiliary space method. The main conceptual ingredients that will be shown in
this framework to yield "optimal" preconditioners in the above sense are
Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic
nested dyadic grids as well as specially adapted wavelet preconditioners for
the resulting low order auxiliary problems. Moreover, the preconditioners have
a modular form that facilitates somewhat simplified partial realizations. One
of the components can, for instance, be conveniently combined with domain
decomposition, at the expense though of a logarithmic growth of condition
numbers. Our analysis is complemented by quantitative experimental studies of
the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents
for better readability, part on wavelet preconditioner adde
Three real-space discretization techniques in electronic structure calculations
A characteristic feature of the state-of-the-art of real-space methods in
electronic structure calculations is the diversity of the techniques used in
the discretization of the relevant partial differential equations. In this
context, the main approaches include finite-difference methods, various types
of finite-elements and wavelets. This paper reports on the results of several
code development projects that approach problems related to the electronic
structure using these three different discretization methods. We review the
ideas behind these methods, give examples of their applications, and discuss
their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status
solidi (b) - basic solid state physics" devoted to the CECAM workshop "State
of the art developments and perspectives of real-space electronic structure
techniques in condensed matter and molecular physics". v2: Minor stylistic
and typographical changes, partly inspired by referee comment
On the Hierarchical Preconditioning of the Combined Field Integral Equation
This paper analyzes how hierarchical bases preconditioners constructed for
the Electric Field Integral Equation (EFIE) can be effectively applied to the
Combined Field Integral Equation (CFIE). For the case where no hierarchical
solenoidal basis is available (e.g., on unstructured meshes), a new scheme is
proposed: the CFIE is implicitly preconditioned on the solenoidal Helmholtz
subspace by using a Helmholtz projector, while a hierarchical non-solenoidal
basis is used for the non-solenoidal Helmholtz subspace. This results in a
well-conditioned system. Numerical results corroborate the presented theory
Adaptive boundary element methods with convergence rates
This paper presents adaptive boundary element methods for positive, negative,
as well as zero order operator equations, together with proofs that they
converge at certain rates. The convergence rates are quasi-optimal in a certain
sense under mild assumptions that are analogous to what is typically assumed in
the theory of adaptive finite element methods. In particular, no
saturation-type assumption is used. The main ingredients of the proof that
constitute new findings are some results on a posteriori error estimates for
boundary element methods, and an inverse-type inequality involving boundary
integral operators on locally refined finite element spaces.Comment: 48 pages. A journal version. The previous version (v3) is a bit
lengthie
Locally supported, piecewise polynomial biorthogona wavelets on non-uniform meshes
In this paper, biorthogonal wavelets are constructed on non-uniform meshes. Both primal and dual wavelets are explicitly given locally supported, continuous piecewise polynomials. The wavelets generate Riesz bases for the Sobolev spaces H s
for j s j < 3 2 . The wavelets at the primal side span standard Lagrange nite element spaces
The wavelet-based theory of spatial naturally curved and twisted linear beams
The paper presents the wavelet-based discretization of the linearized finite-strain beam theory which assumes small displacements, rotations and strains but is capable of considering an arbitrary initial geometry and material behaviour. In the numerical solution algorithm, we base our derivations on the vector of strain measures as the only unknown functions in a finite element. In such a way the determination of the beam quantities does not require the differentiation. This is an important advantage which allows a wider range of shape functions. In the present paper, the classical polynomial interpolation is compared to scaling and wavelet function interpolations. The computational efficiency of the method is demonstrated by analyzing initially curved and twisted beams
An optimal adaptive wavelet method for First Order System Least Squares
In this paper, it is shown that any well-posed 2nd order PDE can be
reformulated as a well-posed first order least squares system. This system will
be solved by an adaptive wavelet solver in optimal computational complexity.
The applications that are considered are second order elliptic PDEs with
general inhomogeneous boundary conditions, and the stationary Navier-Stokes
equations.Comment: 40 page
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