145 research outputs found
Analysis-suitable adaptive T-mesh refinement with linear complexity
We present an efficient adaptive refinement procedure that preserves
analysis-suitability of the T-mesh, this is, the linear independence of the
T-spline blending functions. We prove analysis-suitability of the overlays and
boundedness of their cardinalities, nestedness of the generated T-spline
spaces, and linear computational complexity of the refinement procedure in
terms of the number of marked and generated mesh elements.Comment: We now account for T-splines of arbitrary polynomial degree. We
replaced the proof of Dual-Compatibility by a proof of Analysis-suitability,
added a section where we address nestedness of the corresponding T-spline
spaces, and removed the section on finite overlap the spline supports. 24
pages, 9 Figure
Stable Multiscale Petrov-Galerkin Finite Element Method for High Frequency Acoustic Scattering
We present and analyze a pollution-free Petrov-Galerkin multiscale finite
element method for the Helmholtz problem with large wave number as a
variant of [Peterseim, ArXiv:1411.1944, 2014]. We use standard continuous
finite elements at a coarse discretization scale as trial functions,
whereas the test functions are computed as the solutions of local problems at a
finer scale . The diameter of the support of the test functions behaves like
for some oversampling parameter . Provided is of the order of
and is sufficiently small, the resulting method is stable
and quasi-optimal in the regime where is proportional to . In
homogeneous (or more general periodic) media, the fine scale test functions
depend only on local mesh-configurations. Therefore, the seemingly high cost
for the computation of the test functions can be drastically reduced on
structured meshes. We present numerical experiments in two and three space
dimensions.Comment: The version coincides with v3. We only resized some figures which
were difficult to process for certain printer
Quantitative Anderson localization of Schr\"odinger eigenstates under disorder potentials
This paper concerns spectral properties of linear Schr\"odinger operators
under oscillatory high-amplitude potentials on bounded domains. Depending on
the degree of disorder, we prove the existence of spectral gaps amongst the
lowermost eigenvalues and the emergence of exponentially localized states. We
quantify the rate of decay in terms of geometric parameters that characterize
the potential. The proofs are based on the convergence theory of iterative
solvers for eigenvalue problems and their optimal local preconditioning by
domain decomposition.Comment: accepted for publication in M3A
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,
201
A localized orthogonal decomposition method for semi-linear elliptic problems
In this paper we propose and analyze a new Multiscale Method for solving
semi-linear elliptic problems with heterogeneous and highly variable
coefficient functions. For this purpose we construct a generalized finite
element basis that spans a low dimensional multiscale space. The basis is
assembled by performing localized linear fine-scale computations in small
patches that have a diameter of order H |log H| where H is the coarse mesh
size. Without any assumptions on the type of the oscillations in the
coefficients, we give a rigorous proof for a linear convergence of the H1-error
with respect to the coarse mesh size. To solve the arising equations, we
propose an algorithm that is based on a damped Newton scheme in the multiscale
space
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