48 research outputs found
Convergence rates for adaptive finite elements
In this article we prove that it is possible to construct, using
newest-vertex bisection, meshes that equidistribute the error in -norm,
whenever the function to approximate can be decomposed as a sum of a regular
part plus a singular part with singularities around a finite number of points.
This decomposition is usual in regularity results of Partial Differential
Equations (PDE). As a consequence, the meshes turn out to be quasi-optimal, and
convergence rates for adaptive finite element methods (AFEM) using Lagrange
finite elements of any polynomial degree are obtained
Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients
Elliptic partial differential equations (PDEs) with discontinuous diffusion
coefficients occur in application domains such as diffusions through porous
media, electro-magnetic field propagation on heterogeneous media, and diffusion
processes on rough surfaces. The standard approach to numerically treating such
problems using finite element methods is to assume that the discontinuities lie
on the boundaries of the cells in the initial triangulation. However, this does
not match applications where discontinuities occur on curves, surfaces, or
manifolds, and could even be unknown beforehand. One of the obstacles to
treating such discontinuity problems is that the usual perturbation theory for
elliptic PDEs assumes bounds for the distortion of the coefficients in the
norm and this in turn requires that the discontinuities are matched
exactly when the coefficients are approximated. We present a new approach based
on distortion of the coefficients in an norm with which
therefore does not require the exact matching of the discontinuities. We then
use this new distortion theory to formulate new adaptive finite element methods
(AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in
the sense of distortion versus number of computations, and report insightful
numerical results supporting our analysis.Comment: 24 page
Adaptive Algorithms
Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations
A modified simplicial algorithm for convex maximization based on an extension of ω -subdivision
The simplicial algorithm is a popular branch-and-bound approach to the convex maximization problem with multiple local maxima. In this paper, we discuss some difficulties revealed when implementing this algorithm under the ω-subdivision rule. To overcome those, we modify the bounding process and extend the ω-subdivision rule. We also report numerical results for the simplicial algorithm according to the new subdivision rule
Adaptive Numerical Methods for PDEs
This collection contains the extended abstracts of the talks given at the Oberwolfach Conference on “Adaptive Numerical Methods for PDEs”, June 10th - June 16th, 2007. These talks covered various aspects of a posteriori error estimation and mesh as well as model adaptation in solving partial differential equations. The topics ranged from the theoretical convergence analysis of self-adaptive methods, over the derivation of a posteriori error estimates for the finite element Galerkin discretization of various types of problems to the practical implementation and application of adaptive methods
High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates
We present a new AFEM for the Laplace-Beltrami operator with arbitrary
polynomial degree on parametric surfaces, which are globally and
piecewise in a suitable Besov class embedded in with . The idea is to have the surface sufficiently well resolved in
relative to the current resolution of the PDE in . This gives
rise to a conditional contraction property of the PDE module. We present a
suitable approximation class and discuss its relation to Besov regularity of
the surface, solution, and forcing. We prove optimal convergence rates for AFEM
which are dictated by the worst decay rate of the surface error in
and PDE error in .Comment: 51 pages, the published version contains an additional glossar
The Markov-Dubins Problem with Free Terminal Direction in a Nonpositively Curved Cube Complex
State complexes are nonpositively curved cube complexes that model the state spaces of reconfigurable systems. The problem of determining a strategy for reconfiguring the system from a given initial state to a given goal state is equivalent to that of finding a path between two points in the state complex. The additional requirement that allowable paths must have a prescribed initial direction and minimal turning radius determines a Markov-Dubins problem with free terminal direction (MDPFTD).
Given a nonpositively curved, locally finite cube complex X, we consider the set of unit-speed paths which satisfy a certain smoothness condition in addition to the boundary conditions and curvature constraint that define a MDPFTD. We show that this set either contains a path of minimal length, or is empty.
We then focus on the case that X is a surface with a nonpositively curved cubical structure. We show that any solution to a MDPFTD in X must consist of finitely many geodesic segments and arcs of constant curvature, and we give an algorithm for determining those solutions to the MDPFTD in X which are CL paths, that is, made up of an arc of constant curvature followed by a geodesic segment. Finally, under the assumption that the 1-skeleton of X is d-regular, we give sufficient conditions for a topological ray in X of constant curvature to be a rose curve or a proper ray
Convergence of adaptive discontinuous galerkin methods
We develop a general convergence theory for adaptive discontinu-
ous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and
LDG schemes as well as all practically relevant marking strategies. Another
key feature of the presented result is, that it holds for penalty parameters only
necessary for the standard analysis of the respective scheme. The analysis
is based on a quasi interpolation into a newly developed limit space of the
adaptively created non-conforming discrete spaces, which enables to generalise
the basic convergence result for conforming adaptive finite element methods by
Morin, Siebert, and Veeser [A basic convergence result for conforming adaptive
finite elements, Math. Models Methods Appl. Sci., 2008, 18(5), 707–737]