Adaptive Finite Elements and Domain Decomposition Methods

Workshop on
Adaptive Finite Elements and
Domain Decomposition Methods
Milano, June 17–19, 2010
supported by
INdAM Trimestre “Innovative Numerical Methods for PDEs”
Dipartimento di Matematica, Università degli Studi di Milano
organized by L. Pavarino, A. Veeser, C. Verdi
Programme
Thursday, June 17
08:45-09:00
09:00-09:45
10:00-10:45
10:45-11:15
11:15-12:00
12:00-14:00
14:00-14:45
15:00-15:45
15:45-16:15
16:15-17:00
17:15-18:00
20:00-22:00
Opening
Ricardo H. Nochetto, Performance of AFEM with H −1 Data
Kunibert G. Siebert, Analysis of Adaptive Finite Elements for Control
Constrained Optimal Control Problems
Coffee break
Rob P. Stevenson, Convergence and Quasi-Optimality of an Adaptive
Finite Element Method for Controlling L2 Errors
Lunch break
Albert Cohen, High Dimensional Sparse Approximation of StochasticParametric PDEs
Ronald A. DeVore, Greedy Algorithms in the Reduced Basis Method
for Solving Parametric Elliptic PDEs
Coffee break
Alexandre Ern, A Posteriori Error Estimation Based on Potential
and Flux Reconstruction for the Heat Equation
William F. Mitchell, A Summary of hp-Adaptive Finite Element
Strategies
Conference dinner
Friday, June 18
09:00-09:45
10:00-10:45
10:45-11:15
11:15-12:00
12:00-14:00
14:00-14:45
15:00-15:45
15:45-16:15
16:15-17:00
Olof B. Widlund, Domain Decomposition and Irregular Subdomains
– An Update
Marcus Sarkis, FETI-DP Method for DG Discretization of Elliptic
Problems with Discontinuous Coefficients
Coffee break
Axel Klawonn, Coarse Spaces by Projection in Iterative Substructuring Methods
Lunch break
Charbel Farhat, The Discontinuous Enrichment Method for MultiScale Problems and its Domain-Decomposition Iterative Solver
Lourenço Beirao da Veiga, Robust BDDC Preconditioners for
Reissner-Mindlin and Naghdi Thin Structure Problems
Coffee break
Clark R. Dohrmann, Some Domain Decomposition Algorithms for
Mixed Formulations of Elasticity and Incompressible Fluids
Saturday, June 19
09:00-09:45
10:00-10:45
10:45-11:15
11:15-12:00
12:00-12:15
L. Ridgway Scott, Optimal Algorithms Using Optimal Meshes
Alfred Schmidt, Efficient FEM for Simulation of the Thermomechanical Behavior of Steel under Heat Treatment with Phase Changes
Coffee break
Randolph E. Bank, Parallel Algorithms for hp-Adaptive Finite Elements
Closing
Abstracts
Parallel Algorithms for hp-Adaptive Finite Elements
Randolph E. Bank
We will discuss our on-going investigation of generalizing the parallel h-adaptive
method of Bank and Holst to the case of p- and hp-adaptive finite element methods.
Besides a posteriori error estimation and the adaptive procedures themselves, we
will discuss generalization of the domain decomposition solver developed to support
the h-adaptive version of the algorithm. At this early point in the investigation
there remain many outstanding issues, both mathematical and algorithmic, that
will be highlighted.
Robust BDDC Preconditioners for Reissner-Mindlin and Naghdi Thin
Structure Problems
Lourenço Beirao da Veiga
We consider the deformation problem of thin structures following linear dimensionreduced models with both displacement and rotation variables. We construct and
study some Balancing Domain Decomposition Methods by Constraints (BDDC) for
the discrete problems obtained with the MITC (Mixed Interpolation of Tensorial
Components) finite element approximation.
In addition to the standard properties of scalability in the number of subdomains
N , quasi-optimality in the ratio H/h of subdomain/element sizes, and robustness
with respect to discontinuities of the material properties, our goal here is to obtain
robustness also with respect to the additional small parameter t representing the
plate or shell thickness. This is a challenging issue since the condition number
of plates and shell problems typically diverges as O(t−2 ) as t tends to zero. The
proposed BDDC preconditioners are based on a proper selection of primal continuity constraints, the implicit elimination of the interior degrees of freedom in each
subdomain, and the solution of local (and also a global coarse) problems.
We first consider the case of a flat structure, therefore addressing the ReissnerMindlin plate bending model. In such case we can prove that the proposed BDDC
algorithm is scalable and, most important, robust with respect to the plate thickness. While this result is due to an underlying mixed formulation of the problem,
both the interface plate problem and the preconditioner are positive definite. Numerical results also show that the proposed algorithm seems to be quasi-optimal
and robust with respect to discontinuities of the plate material properties.
In the last part of the talk, we address shells of general geometry, namely the
Naghdi model. In such case we do not have a theoretical bound and consider
different choices of the primal constraints, driven by heuristic considerations and
previous experience. Several numerical tests seem to indicate that the proposed
BDDC preconditioners are scalable, quasi-optimal, robust with respect to discontinuities of the shell material properties, and almost robust with respect to the shell
thickness.
This is joint work with Claudia Chinosi, Carlo Lovadina and Luca F. Pavarino.
High Dimensional Sparse Approximation of Stochastic-Parametric
PDEs
Albert Cohen
Various mathematical problems are challenged by the fact they involve functions
of a very large number of variables. Such problems arise naturally in learning
theory, partial differential equations or numerical models depending on parametric
or stochastic variables. They typically result in numerical difficulties due to the
so-called ”curse of dimensionality”. We shall explain how these difficulties may
be handled in the context of stochastic-parametric PDE’s based on two important
concepts: (i) variable reduction and (ii) sparse approximation.
Greedy Algorithms in the Reduced Basis Method for Solving
Parametric Elliptic PDEs
Ronald A. DeVore
The goal in numerically solving parametric pdes is to exploit the smooth dependence
of the solution on the parameters in order to simultaneously solve the parametric
family efficiently. One particularly prominent technique is the reduced basis method
which attempts to find a good choice of n parameters and their solutions u1 , . . . , un
such that for any other parameter the solution is close to a linear combination of
u1 , . . . , un . In practice n is very small. Once these n functions are found one can
use the Galerkin projection onto the span of u1 , . . . , un as an efficient solver for any
other choice of a parameter. One particular algorithm for finding u1 , . . . , un can be
described as a greedy algorithm in a Hilbert space for simultaneously approximating
a compact set of functions. We shall discuss what is known about the rate of
convergence of this greedy procedure.
Some Domain Decomposition Algorithms for Mixed Formulations of
Elasticity and Incompressible Fluids
Clark R. Dohrmann
In this talk, we present a collection of domain decomposition algorithms for mixed
finite element formulations of elasticity and incompressible fluids. The key component of each of these algorithms is the coarse space. Here, the coarse spaces are
obtained in an algebraic manner by harmonically extending coarse boundary data.
Various aspects of the coarse spaces are discussed for both continuous and discontinuous interpolation of pressure. Further, both classical overlapping Schwarz and
hybrid iterative substructuring preconditioners are described. Numerical results
are presented for almost incompressible elasticity and the Navier Stokes equations
which demonstrate the utility of the methods for both structured and irregular
mesh decompositions. We also discuss a simple residual scaling approach which
often leads to significant reductions in iterations for these algorithms.
A Posteriori Error Estimation Based on Potential and Flux
Reconstruction for the Heat Equation
Alexandre Ern
We derive a posteriori error estimates for the discretization of the heat equation in
a unified and fully discrete setting comprising the discontinuous Galerkin, various
finite volume, and mixed finite element methods in space and the backward Euler
scheme in time. The estimates are based on a H 1 -conforming reconstruction of
the potential, continuous and piecewise affine in time, and a locally conservative
H(div)-conforming reconstruction of the flux, piecewise constant in time. They
yield a guaranteed and fully computable upper bound on the error measured in the
energy norm augmented by a dual norm of the time derivative. Local-in-time lower
bounds are also derived.
This is joint work with Martin Vohralik.
The Discontinuous Enrichment Method for Multi-Scale Problems and
its Domain Decomposition Iterative Solver
Charbel Farhat
Wave propagation problems pertain to many technologies including sonar, radar,
geophysical exploration, medical imaging, nondestructive testing, and structural
design. As technology expands in the nano-world, evanescent waves are playing a
major role in the design various optical, chemical, and biological sensors. Advectiondiffusion arises in many important flow and transport problems such as contaminant transport through aquifers in hydromechanics, heat and mass transport in
chemical engineering, and ltration processes and transport of drugs in biomedical
engineering. Two important attributes that are shared by all of the aforementioned
problems and applications are: (a) their multi-scale nature, and (b) the computational complexity required to solve them numerically by standard approximation
methods. Indeed, the analysis by the standard finite element method of wave
propagation problems in the medium frequency regime is either computationally
unfeasible or simply unreliable, particularly in the presence of evanescent waves.
Similarly, high Reynolds number flows are well beyond the current reach of standard
approximation methods and numerical considerations continue to limit the size of
the Peclet number one can simulate using the standard finite element method. The
higher-order (or p-type) finite element method alleviates some of these problems,
but only to some extent. Alternative approximation methods based on the idea of
using special, problem dependent approximation functions have recently emerged
to address these issues. The Discontinuous Enrichment Method (DEM) is such an
alternative. It distinguishes itself from other enrichment methods by its domain decomposition approach and legacy, and its ability to evaluate the important system
matrices analytically thereby bypassing the typical accuracy and cost issues associated with high-order quadrature rules. DEM also provides a unique multi-scale
approach to computation by employing fine scales that contain solutions of the underlying homogeneous partial differential equation in a discontinuous framework.
The theoretical and computational underpinnings of this method whose development started a decade ago will be overviewed and specialized to wave propagation,
flow, and transport problems. An associated scalable domain decomposition based
iterative solver will also be presented. Then, recent applications to underwater
acoustic scattering problems in the medium frequency regime, geo-acoustic scattering at interfaces between fluid and solid media, and various high Peclet number
advection diffusion problems will be discussed. One to two orders of magnitude
accuracy and/or CPU time improvement over the standard higher-order finite element method will be demonstrated in three dimensions.
Coarse Spaces by Projection in Iterative Substructuring Methods
Axel Klawonn
The choice of the coarse space is of vital importance to the numerical scalability
of domain decomposition methods for elliptic partial differential equations. An
efficient implementation of the coarse problem is necessary to obtain parallel scalability on massively parallel systems. In this talk we will discuss certain choices
for coarse spaces in the dual-primal Finite Element Tearing and Interconnecting
(FETI-DP) method with a special emphasis on their implementation. For three
dimensional problems, edge and/or face constraints, either implemented by using
optional Lagrange multipliers or combined with a transformation of basis, have been
used to build the coarse problem. Recently, as an alternative or in addition, projector preconditioning has been used to enhance the FETI-DP coarse problem for
contact problems. In this talk we will discuss several new results for this approach.
This work is motivated by an earlier joint project with Marta Jarosova and Oliver
Rheinbach. The new results are obtained in joint work with Oliver Rheinbach.
A Summary of hp-Adaptive Finite Element Strategies
William F. Mitchell
Adaptive finite element methods have been studied for nearly 30 years now. Most
of the work has focused on h-adaptive methods where the mesh size, h, is adapted
locally by means of a local error estimator with the goal of placing the smallest
elements in the areas where they will do the most good. h-adaptive methods for
elliptic partial differential equations are quite well understood now, and widely used
in practice.
Recently, the research community has begun to focus more attention on hpadaptive methods where in addition to h-adaptivity one locally adapts the degree
of the polynomials, p. One attraction of these methods is that they can achieve
exponential rates of convergence. But the design of an optimal strategy to determine when to use p-refinement, when to use h-refinement, and what p’s to use in
h-refined elements is an open area of research. Many such hp-adaptive strategies
have been proposed over the past two decades. In this talk, we will briefly describe
many of these strategies and present numerical results to demonstrate their effectiveness.
Performance of AFEM with H −1 Data
Ricardo H. Nochetto
In contrast to most of the existing theory of adaptive finite element methods
(AFEM), we design an AFEM for −∆u = f with right hand side f in H −1 instead of L2 . This has two important consequences. First we formulate our AFEM
in the natural space for f , which is nonlocal. Second, we show that decay rates
for data oscillation are dominated by those for the solution u in the energy norm.
This allows us to conclude that the performance of AFEM is solely dictated by the
approximation class of u.
This is joint work with A. Cohen and R. DeVore.
FETI-DP Method for DG Discretization of Elliptic Problems with
Discontinuous Coefficients
Marcus Sarkis
In the talk a discontinuous Galerkin (DG) approximation of elliptic problems with
discontinuous coefficients will be discussed. The problem is considered in polygonal
region Ω which is a union of disjoint polygonal subregions Ωi . The discontinuities
of the coefficients occur across ∂Ωi . The problem is approximated by a conforming
finite element method (FEM) on matching triangulation in each Ωi and nonmatching one across ∂Ωi . This kind of triangulation and composite discretization are
motivated first of all by the regularity of solution of the problem being discussed.
The discrete problem is formulated using DG method with interior penalty terms
on ∂Ωi .
In the talk a FETI-DP method for the resulting discrete problem will be designed and analyzed. It is proved that the method is almost optimal and its rate
of convergence is independent of the parameters of triangulation, the number of
substructures and the jumps of coefficients.
The results presented were obtained in joint work with Prof. Maksymilian Dryja
(Warsaw).
Efficient FEM for Simulation of the Thermomechanical Behavior of
Steel under Heat Treatment with Phase Changes
Alfred Schmidt
The talk presents simulations of the thermomechanical behaviour of steel pieces
during quenching. The model leads to a coupled system of parabolic and elliptic
PDEs, together with ODEs for the solid-solid phase transition which occur. Elastic
and permanent deformations will be considered. Especially in 3D, efficient simulation methods are needed. We show the application of a FEM discretization with
aspects of adaptivity and domain decomposition methods.
This is joint work with Bettina Suhr and Jost Vehmeyer.
Optimal Algorithms Using Optimal Meshes
L. Ridgway Scott
We discuss two problems involving adaptive meshes. The first relates to non-nested
multigrid in two and three dimensions. We review what is known theoretically and
describe some recent work related to optimal implementation. The second involves
meshes in arbitrary dimensions. We show that there are meshes in which the
number of nodes grows linearly in the dimension, and give some evidence via a
quantum mechanics example that an h-P strategy can be effective to obtain good
convergence behavior on these meshes.
Analysis of Adaptive Finite Elements for
Control Constrained Optimal Control Problems
Kunibert G. Siebert
Adaptive finite elements are successfully used since the 1970s. The typical adaptive
iteration is a loop of the form
SOLVE → ESTIMATE → MARK → REFINE,
this is: solve for the finite element solution on the current grid, compute an a posteriori error estimator, mark with its help elements to be refined, and refine the
current grid into a new one.
Traditional a posteriori error analysis was mainly concerned with the step ESTIMATE by deriving computable error bounds for the true error. During the last
years there is an increasing interest in proving convergence of the above iteration.
This means, we show that the sequence of discrete solutions converges to the exact
one.
In this talk we present a unified framework for the a posteriori error analysis
of control constrained optimal control problems solely based on estimators for the
linear state equation and the adjoint equation. We would like to stress that there
exists a well-established theory of different kinds of estimators for a huge class
of linear problems. This a posteriori error bound then in turn allows us to prove
convergence of the adaptive iteration with the techniques of Morin, Siebert & Veeser
and Siebert.
Convergence and Quasi-Optimality of an Adaptive Finite Element
Method for Controlling L2 Errors
Rob P. Stevenson
In this talk, a contraction property is proved for an adaptive finite element method
for controlling the global L2 error. Furthermore, it is shown that the method
converges in L2 with the best possible rate. The method that is analyzed is the
standard adaptive method for solving Poisson’s equation except that, if necessary,
additional refinements are made to keep the meshes sufficiently mildly graded. This
modification does not compromise the quasi-optimality of the resulting algorithm.
This is joint work with Alan Demlow.
Domain Decomposition and Irregular Subdomains – An Update
Olof B. Widlund
A number of results were obtained, several years ago, for domain decomposition
algorithms with quite irregular subdomains for H 1 problems in the plane. This work
was in collaboration with Clark Dohrmann, Axel Klawonn, and Oliver Rheinbach.
The subdomains do not have to be even Lipschitz but were only required to belong
to class of John or Jones (uniform) domains.
In this talk, we will, after reviewing some of the older results, consider an overlapping Schwarz method with a coarse space more similar to that of the classical
case than those studied previously. We will also discuss new results for H(curl)
in two dimensions and what the prospects are for extensions to three dimensions.
These new results are all obtained in collaboration with Clark Dohrmann.
Participants
Ana Alonso Rodriguez, Università di Trento, Italy
Paola Antonietti, Politecnico di Milano, Italy
Blanca Ayuso, Universidad Autonoma de Madrid, Spain
Randolph E. Bank, University of California, USA
Stefan Baumgartner, University of Vienna, Austria
Lourenco Beirao da Veiga, Università di Milano, Italy
Silvia Bertoluzza, IMATI del CNR, Pavia, Italy
Luca Bonaventura, Politecnico di Milano, Italy
Francesca Bonizzoni, Politecnico di Milano, Italy
Franco Brezzi, IMATI del CNR, Pavia, Italy
Jed Brown, ETH Zurich, Switzerland
Andrea Cangiani, Università di Milano Bicocca, Italy
Claudio Canuto, Politecnico di Torino
Paola Causin, Università di Milano, Italy
Claudia Chinosi, Università del Piemonte Orientale, Italy
Durkbin Cho, Università di Pavia, Italy
Albert Cohen, Universit Pierre et Marie Curie, Paris, France
Piero Colli Franzone, Università di Pavia, Italy
Carlo D’Angelo, Politecnico di Milano, Italy
Franco Dassi, Politecnico di Milano, Italy
Alan Demlow, University of Kentucky, USA
Ronald DeVore, Texas A&M University, USA
Clark R. Dohrmann, Sandia National Laboratories, USA
Alexandre Ern, Université Paris-Est, France
Nur Fadel, Politecnico di Milano, Italy
Charbel Farhat, Stanford University, USA
Paolo Ferrandi, Politecnico di Milano, Italy
Peter Fick, Delft University of Technology, The Netherlands
Francesca Fierro, Università di Milano, Italy
Luca Formaggia, Politecnico di Milano, Italy
Alessio Fumagalli, Politecnico di Milano, Italy
Loredana Gaudio, Politecnico di Milano, Italy
Francesca Gardini, Università di Pavia, Italy
Lucia Gastaldi, Università di Brescia, Italy
Tamas Horvath, Szechenyi University, Gyor, Hungary
Carlo Janna, Università di Padova, Italy
Antoine Celestin Kengni Jotsa, Universita‘ dell’Insubria, Italy
Axel Klawonn, Universität Duisburg - Essen, Germany
Dmitriy Leykekhman, University of Connecticut, USA
Carlo Lovadina, Università di Pavia, Italy
Donatella Marini, Università di Pavia, Italy
Dave May, ETH Zurich, Switzerland
Ilario Mazzieri, Politecnico di Milano, Italy
Alessandro Melani, Politecnico di Milano, Italy
Stefano Micheletti, Politecnico di Milano, Italy
Giovanni Migliorati, Politecnico di Milano, Italy
William Mitchell, National Institute of Standards and Technology, USA
Elfatini Mohamed, Université de Pau et des pays de l’Adou, France
Francesco Mora, Università di Milano, Italy
Giovanni Naldi, Università di Milano, Italy
Fabio Nobile, Politecnico di Milano, Italy
Ricardo H. Nochetto, University of Maryland, USA
Marcus Page, University of Vienna, Austria
Damiano Pasetto, Università di Padova, Italy
Luca Pavarino, Università di Milano, Italy
Kevin Payne, Università di Milano, Italy
Simona Perotto, Politecnico di Milano, Italy
Ilaria Perugia, Università di Pavia, Italy
Simone Pezzuto, Politecnico di Milano, Italy
Matteo Pischiutta, Politecnico di Milano, Italy
Matteo Pozzoli, Politecnico di Milano, Italy
Elisabetta Repossi, Politecnico di Milano, Italy
Fabrizio Rossi, Politecnico di Milano, Italy
Simone Rossi, EPFL, Switzerland
Giancarlo Sangalli, Università di Pavia, Italy
Simona Sanfelici, Università di Parma, Italy
Maura Salvatori, Università di Milano, Italy
Marcus Sarkis, Worcester Polytechnic Institute, USA
Simone Scacchi, Università di Milano, Italy
Alfred Schmidt, Universität Bremen, Germany
Ridgway Scott, University of Chicago, USA
Kunibert G. Siebert, Universität Duisburg - Essen, Germany
Kathrin Smetana, Universität Mnster, Germany
Rob Stevenson, University of Amsterdam, The Netherlands
Lorenzo Tamellini, Politecnico di Milano, Italy
Francesca Tantardini, Università di Milano, Italy
Andreas Veeser, Università di Milano, Italy
Marco Verani, Politecnico di Milano, Italy
Claudio Verdi, Università di Milano, Italy
Christian Vergara, Politecnico di Milano, Italy
Umberto Villa, Emory University, USA
Christian Waluga, RWTH Aachen University, Germany
Olof B. Widlund, Courant Institute, NYU, USA
Elena Zampieri, Università di Milano, Italy
Stefano Zampini, Università di Milano, Italy
Paolo Zunino, Politecnico di Milano, Italy