18 research outputs found
Approximation in FEM, DG and IGA: A Theoretical Comparison
In this paper we compare approximation properties of degree spline spaces
with different numbers of continuous derivatives. We prove that, for a given
space dimension, \smooth {p-1} splines provide better a priori error bounds
for the approximation of functions in . Our result holds for all
practically interesting cases when comparing \smooth {p-1} splines with
\smooth {-1} (discontinuous) splines. When comparing \smooth {p-1} splines
with \smooth 0 splines our proof covers almost all cases for , but we
can not conclude anything for . The results are generalized to the
approximation of functions in for , to broken Sobolev
spaces and to tensor product spaces.Comment: 21 pages, 4 figures. Fixed typos and improved the presentatio
Optimal spline spaces for -width problems with boundary conditions
In this paper we show that, with respect to the norm, three classes of
functions in , defined by certain boundary conditions, admit optimal
spline spaces of all degrees , and all these spline spaces have
uniform knots.Comment: 17 pages, 4 figures. Fixed a typo. Article published in Constructive
Approximatio
Sharp error estimates for spline approximation: explicit constants, -widths, and eigenfunction convergence
In this paper we provide a priori error estimates in standard Sobolev
(semi-)norms for approximation in spline spaces of maximal smoothness on
arbitrary grids. The error estimates are expressed in terms of a power of the
maximal grid spacing, an appropriate derivative of the function to be
approximated, and an explicit constant which is, in many cases, sharp. Some of
these error estimates also hold in proper spline subspaces, which additionally
enjoy inverse inequalities. Furthermore, we address spline approximation of
eigenfunctions of a large class of differential operators, with a particular
focus on the special case of periodic splines. The results of this paper can be
used to theoretically explain the benefits of spline approximation under
-refinement by isogeometric discretization methods. They also form a
theoretical foundation for the outperformance of smooth spline discretizations
of eigenvalue problems that has been numerically observed in the literature,
and for optimality of geometric multigrid solvers in the isogeometric analysis
context.Comment: 31 pages, 2 figures. Fixed a typo. Article published in M3A
The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance
In 1994, Nessyahu, Tadmor and Tassa studied convergence rates of monotone
finite volume approximations of conservation laws. For compactly supported,
\Lip^+-bounded initial data they showed a first-order convergence rate in the
Wasserstein distance. Our main result is to prove that this rate is optimal. We
further provide numerical evidence indicating that the rate in the case of
\Lip^+-unbounded initial data is worse than first-order.Comment: 10 pages, 5 figures, 2 tables. Fixed typos. Article published in
Journal of Scientific Computin
A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis
Explicit time integration schemes coupled with Galerkin discretizations of
time-dependent partial differential equations require solving a linear system
with the mass matrix at each time step. For applications in structural
dynamics, the solution of the linear system is frequently approximated through
so-called mass lumping, which consists in replacing the mass matrix by some
diagonal approximation. Mass lumping has been widely used in engineering
practice for decades already and has a sound mathematical theory supporting it
for finite element methods using the classical Lagrange basis. However, the
theory for more general basis functions is still missing. Our paper partly
addresses this shortcoming. Some special and practically relevant properties of
lumped mass matrices are proved and we discuss how these properties naturally
extend to banded and Kronecker product matrices whose structure allows to solve
linear systems very efficiently. Our theoretical results are applied to
isogeometric discretizations but are not restricted to them.Comment: 28 pages, 24 figures. Accepted manuscrip
Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis
In this paper we provide a priori error estimates with explicit constants for
both the -projection and the Ritz projection onto spline spaces of
arbitrary smoothness defined on arbitrary grids. This extends the results
recently obtained for spline spaces of maximal smoothness. The presented error
estimates are in agreement with the numerical evidence found in the literature
that smoother spline spaces exhibit a better approximation behavior per degree
of freedom, even for low smoothness of the functions to be approximated. First
we introduce results for univariate spline spaces, and then we address
multivariate tensor-product spline spaces and isogeometric spline spaces
generated by means of a mapped geometry, both in the single-patch and in the
multi-patch case.Comment: 39 pages, 4 figures. Improved the presentation. Article published in
Numerische Mathemati
Best low-rank approximations and Kolmogorov n-widths
We relate the problem of best low-rank approximation in the spectral norm for
a matrix to Kolmogorov -widths and corresponding optimal spaces. We
characterize all the optimal spaces for the image of the Euclidean unit ball
under and we show that any orthonormal basis in an -dimensional optimal
space generates a best rank- approximation to . We also present a simple
and explicit construction to obtain a sequence of optimal -dimensional
spaces once an initial optimal space is known. This results in a variety of
solutions to the best low-rank approximation problem and provides alternatives
to the truncated singular value decomposition. This variety can be exploited to
obtain best low-rank approximations with problem-oriented properties.Comment: 25 pages, 1 figur
Optimal spline spaces of higher degree for L2 n-widths
In this paper we derive optimal subspaces for Kolmogorov nn-widths in the L2 norm with respect to sets of functions defined by kernels. This enables us to prove the existence of optimal spline subspaces of arbitrarily high degree for certain classes of functions in Sobolev spaces of importance in finite element methods. We construct these spline spaces explicitly in special cases