560 research outputs found
Adaptive Spectral Galerkin Methods with Dynamic Marking
The convergence and optimality theory of adaptive Galerkin methods is almost
exclusively based on the D\"orfler marking. This entails a fixed parameter and
leads to a contraction constant bounded below away from zero. For spectral
Galerkin methods this is a severe limitation which affects performance. We
present a dynamic marking strategy that allows for a super-linear relation
between consecutive discretization errors, and show exponential convergence
with linear computational complexity whenever the solution belongs to a Gevrey
approximation class.Comment: 20 page
Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation
This paper presents an a-posteriori goal-oriented error analysis for a
numerical approximation of the steady Boltzmann equation based on a
moment-system approximation in velocity dependence and a discontinuous Galerkin
finite-element (DGFE) approximation in position dependence. We derive
computable error estimates and bounds for general target functionals of
solutions of the steady Boltzmann equation based on the DGFE moment
approximation. The a-posteriori error estimates and bounds are used to guide a
model adaptive algorithm for optimal approximations of the goal functional in
question. We present results for one-dimensional heat transfer and shock
structure problems where the moment model order is refined locally in space for
optimal approximation of the heat flux.Comment: arXiv admin note: text overlap with arXiv:1602.0131
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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
Localized model reduction for parameterized problems
In this contribution we present a survey of concepts in localized model order
reduction methods for parameterized partial differential equations. The key
concept of localized model order reduction is to construct local reduced spaces
that have only support on part of the domain and compute a global approximation
by a suitable coupling of the local spaces. In detail, we show how optimal
local approximation spaces can be constructed and approximated by random
sampling. An overview of possible conforming and non-conforming couplings of
the local spaces is provided and corresponding localized a posteriori error
estimates are derived. We introduce concepts of local basis enrichment, which
includes a discussion of adaptivity. Implementational aspects of localized
model reduction methods are addressed. Finally, we illustrate the presented
concepts for multiscale, linear elasticity and fluid-flow problems, providing
several numerical experiments.
This work has been accepted as a chapter in P. Benner, S. Grivet-Talocia, A.
Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira. Handbook on Model Order
Reduction. Walter De Gruyter GmbH, Berlin, 2019+
Adaptive Discontinuous Galerkin Finite Element Methods
The Discontinuous Galerkin Method is one variant of the Finite Element Methods for solving partial differential equations, which was first introduced by Reed and Hill in 1970’s [27]. Discontinuous GalerkinMethod (DGFEM) differs from the standard Galerkin FEMthat continuity constraints are not imposed on the inter-element boundaries. It results in a solution which is composed of totally piecewise discontinuous functions. The absence of continuity constraints on the inter-element boundaries implies that DG method has a great deal of flexibility at the cost of increasing the number of degrees of freedom. This flexibility is the source of many but not all of the advantages of the DGFEM method over the Continuous Galerkin (CGFEM) method that uses spaces of continuous piecewise polynomial functions and other ”less standard” methods such as nonconforming methods. As DGFEM method leads to bigger system to solve, theoretical and practical approaches to speed it up are our main focus in this dissertation. This research aims at designing and building an adaptive discontinuous Galerkin finite element method to solve partial differential equations with fast time for desired accuracy on modern architecture
Numerical modeling and open-source implementation of variational partition-of-unity localizations of space-time dual-weighted residual estimators for parabolic problems
In this work, we consider space-time goal-oriented a posteriori error
estimation for parabolic problems. Temporal and spatial discretizations are
based on Galerkin finite elements of continuous and discontinuous type. The
main objectives are the development and analysis of space-time estimators, in
which the localization is based on a weak form employing a partition-of-unity.
The resulting error indicators are used for temporal and spatial adaptivity.
Our developments are substantiated with several numerical examples.Comment: Changes in v2: - Updated the title - Reworked space-time function
spaces - Added cG(1) in time partition-of-unity - Added links to the now
published codes used for this work - Added further reference
Cluster-based reduced-order modelling of a mixing layer
We propose a novel cluster-based reduced-order modelling (CROM) strategy of
unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's
group (Burkardt et al. 2006) and and transition matrix models introduced in
fluid dynamics in Eckhardt's group (Schneider et al. 2007). CROM constitutes a
potential alternative to POD models and generalises the Ulam-Galerkin method
classically used in dynamical systems to determine a finite-rank approximation
of the Perron-Frobenius operator. The proposed strategy processes a
time-resolved sequence of flow snapshots in two steps. First, the snapshot data
are clustered into a small number of representative states, called centroids,
in the state space. These centroids partition the state space in complementary
non-overlapping regions (centroidal Voronoi cells). Departing from the standard
algorithm, the probabilities of the clusters are determined, and the states are
sorted by analysis of the transition matrix. Secondly, the transitions between
the states are dynamically modelled using a Markov process. Physical mechanisms
are then distilled by a refined analysis of the Markov process, e.g. using
finite-time Lyapunov exponent and entropic methods. This CROM framework is
applied to the Lorenz attractor (as illustrative example), to velocity fields
of the spatially evolving incompressible mixing layer and the three-dimensional
turbulent wake of a bluff body. For these examples, CROM is shown to identify
non-trivial quasi-attractors and transition processes in an unsupervised
manner. CROM has numerous potential applications for the systematic
identification of physical mechanisms of complex dynamics, for comparison of
flow evolution models, for the identification of precursors to desirable and
undesirable events, and for flow control applications exploiting nonlinear
actuation dynamics.Comment: 48 pages, 30 figures. Revised version with additional material.
Accepted for publication in Journal of Fluid Mechanic
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