37 research outputs found
An analysis of a class of variational multiscale methods based on subspace decomposition
Numerical homogenization tries to approximate the solutions of elliptic
partial differential equations with strongly oscillating coefficients by
functions from modified finite element spaces. We present in this paper a class
of such methods that are very closely related to the method of M{\aa}lqvist and
Peterseim [Math. Comp. 83, 2014]. Like the method of M{\aa}lqvist and
Peterseim, these methods do not make explicit or implicit use of a scale
separation. Their compared to that in the work of M{\aa}lqvist and Peterseim
strongly simplified analysis is based on a reformulation of their method in
terms of variational multiscale methods and on the theory of iterative methods,
more precisely, of additive Schwarz or subspace decomposition methods.Comment: published electronically in Mathematics of Computation on January 19,
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Numerical Homogenization of Fractal Interface Problems
We consider the numerical homogenization of a class of fractal elliptic
interface problems inspired by related mechanical contact problems from the
geosciences. A particular feature is that the solution space depends on the
actual fractal geometry. Our main results concern the construction of
projection operators with suitable stability and approximation properties. The
existence of such projections then allows for the application of existing
concepts from localized orthogonal decomposition (LOD) and successive subspace
correction to construct first multiscale discretizations and iterative
algebraic solvers with scale-independent convergence behavior for this class of
problems
NUMERICAL HOMOGENIZATION OF FRACTAL INTERFACE PROBLEMS
We consider the numerical homogenization of a class of fractal elliptic interface
problems inspired by related mechanical contact problems from the geosciences. A particular
feature is that the solution space depends on the actual fractal geometry. Our main
results concern the construction of projection operators with suitable stability and approximation
properties. The existence of such projections then allows for the application of
existing concepts from localized orthogonal decomposition (LOD) and successive subspace
correction to construct first multiscale discretizations and iterative algebraic solvers with
scale-independent convergence behavior for this class of problems
Regularity and approximability of electronic wave functions
The electronic Schrödinger equation describes the motion of N-electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, with three spatial dimensions for each electron. Approximating these solutions is thus inordinately challenging, and it is generally believed that a reduction to simplified models, such as those of the Hartree-Fock method or density functional theory, is the only tenable approach. This book seeks to show readers that this conventional wisdom need not be ironclad: the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of one or two electrons. The text is accessible to a mathematical audience at the beginning graduate level as well as to physicists and theoretical chemists with a comparable mathematical background and requires no deeper knowledge of the theory of partial differential equations, functional analysis, or quantum theory
Numerical homogenization of fractal interface problems
We consider the numerical homogenization of a class of fractal elliptic interface problems inspired by related mechanical contact problems from the geosciences. A particular feature is that the solution space depends on the actual fractal geometry. Our main results concern the construction of projection operators with suitable stability and approximation properties. The existence of such projections then allows for the application of existing concepts from localized orthogonal decomposition (LOD) and successive subspace correction to construct first multiscale discretizations and iterative algebraic solvers with scale-independent convergence behavior for this class of problems
The mixed regularity of electronic wave functions multiplied by explicit correlation factors
The electronic Schrödinger equation describes the motion of N
electrons under Coulomb interaction forces in a field of clamped
nuclei. The solutions of this equation, the electronic wave functions,
depend on 3N variables, three spatial dimensions for each electron.
Approximating them is thus inordinately challenging. As is shown in
the author's monograph [Yserentant, Lecture Notes in Mathematics 2000,
Springer (2010)], the regularity of the solutions, which
increases with the number of electrons, the decay behavior of their
mixed derivatives, and the antisymmetry enforced by the Pauli
principle contribute properties that allow these functions to be
approximated with an order of complexity which comes arbitrarily
close to that for a system of two electrons. The present paper
complements this work. It is shown that one can reach almost the
same complexity as in the one-electron case adding a simple
regularizing factor that depends explicitly on the interelectronic
distances
On the expansion of solutions of Laplace-like equations into traces of separable higher dimensional functions
This paper deals with the equation - Δ u + μ u = f on high-dimensional spaces R^m where μ is a positive constant. If the right-hand side f is a rapidly converging series of separable functions, the solution u can be represented in the same way. These constructions are based on approximations of the function 1/ r by sums of exponential functions. The aim of this paper is to prove results of similar kind for more general right-hand sides f ( x ) = F ( T x ) that are composed of a separable function on a space of a dimension n greater than m and a linear mapping given by a matrix T of full rank. These results are based on the observation that in the high-dimensional case, for ω in most of the R^n, the euclidian norm of the vector T^tnω in the lower dimensional space R^m behaves like the euclidian norm of ω .TU Berlin, Open-Access-Mittel – 202