Lecture note in computer science

Lecture Notes in Computer Science
Commenced Publication in 1973
Founding and Former Series Editors:
Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board
David Hutchison
Lancaster University, UK
Takeo Kanade
Carnegie Mellon University, Pittsburgh, PA, USA
Josef Kittler
University of Surrey, Guildford, UK
Jon M. Kleinberg
Cornell University, Ithaca, NY, USA
Alfred Kobsa
University of California, Irvine, CA, USA
Friedemann Mattern
ETH Zurich, Switzerland
John C. Mitchell
Stanford University, CA, USA
Moni Naor
Weizmann Institute of Science, Rehovot, Israel
Oscar Nierstrasz
University of Bern, Switzerland
C. Pandu Rangan
Indian Institute of Technology, Madras, India
Bernhard Steffen
University of Dortmund, Germany
Madhu Sudan
Massachusetts Institute of Technology, MA, USA
Demetri Terzopoulos

University of California, Los Angeles, CA, USA
Doug Tygar
University of California, Berkeley, CA, USA
Gerhard Weikum
Max-Planck Institute of Computer Science, Saarbruecken, Germany

4818


Ivan Lirkov Svetozar Margenov
Jerzy Wa´sniewski (Eds.)

Large-Scale
Scientific Computing
6th International Conference, LSSC 2007
Sozopol, Bulgaria, June 5-9, 2007
Revised Papers

13


Volume Editors
Ivan Lirkov
Bulgarian Academy of Sciences
Institute for Parallel Processing
1113 Sofia, Bulgaria
E-mail:
Svetozar Margenov
Bulgarian Academy of Sciences
Institute for Parallel Processing

1113 Sofia, Bulgaria
E-mail:
Jerzy Wa´sniewski
Technical University of Denmark
Department of Informatics and Mathematical Modelling
2800 Kongens Lyngby, Denmark
E-mail:

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Preface

The 6th International Conference on Large-Scale Scientific Computations
(LSSC 2007) was held in Sozopol, Bulgaria, June 5–9, 2007. The conference was
organized by the Institute for Parallel Processing at the Bulgarian Academy of
Sciences in cooperation with SIAM (Society for Industrial and Applied Mathematics). Partial support was also provided from project BIS-21++ funded by
the European Commission in FP6 INCO via grant 016639/2005.
The conference was devoted to the 60th anniversary of Richard E. Ewing.
Professor Ewing was awarded the medal of the Bulgarian Academy of Sciences for
his contributions to the Bulgarian mathematical community and to the Academy
of Sciences. His career spanned 33 years, primarily in academia, but also included
industry. Since 1992 he worked at Texas A&M University being Dean of Science
and Vice President of Research, as well as director of the Institute for Scientific
Computation (ISC), which he founded in 1992. Professor Ewing is internationally well known with his contributions in applied mathematics, mathematical
modeling, and large-scale scientific computations. He inspired a generation of
researchers with creative enthusiasm for doing science on scientific computations. The preparatory work on this volume was almost done when the sad news
came to us: Richard E. Ewing passed away on December 5, 2007 of an apparent
heart attack while driving home from the office.
Plenary Invited Speakers and Lectures:
– O. Axelsson, Mesh-Independent Superlinear PCG Rates for Elliptic
Problems
– R. Ewing, Mathematical Modeling and Scientific Computation in Energy
and Environmental Applications
– L. Gr¨

une, Numerical Optimization-Based Stabilization: From Hamilton-Jacobi-Bellman PDEs to Receding Horizon Control
– M. Gunzburger, Bridging Methods for Coupling Atomistic and Continuum
Models
– B. Philippe, Domain Decomposition and Convergence of GMRES
– P. Vassilevski, Exact de Rham Sequences of Finite Element Spaces on
Agglomerated Elements
– Z. Zlatev, Parallelization of Data Assimilation Modules
The success of the conference and the present volume in particular are the
outcome of the joint efforts of many colleagues from various institutions and
organizations. First, thanks to all the members of the Scientific Committee for
their valuable contribution forming the scientific face of the conference, as well as
for their help in reviewing contributed papers. We especially thank the organizers
of the special sessions. We are also grateful to the staff involved in the local
organization.


VI

Preface

Traditionally, the purpose of the conference is to bring together scientists
working with large-scale computational models of environmental and industrial
problems and specialists in the field of numerical methods and algorithms for
modern high-speed computers. The key lectures reviewed some of the advanced
achievements in the field of numerical methods and their efficient applications.
The conference lectures were presented by the university researchers and practical industry engineers including applied mathematicians, numerical analysts and
computer experts. The general theme for LSSC 2007 was “Large-Scale Scientific
Computing” with a particular focus on the organized special sessions.
Special Sessions and Organizers:
– Robust Multilevel and Hierarchical Preconditioning Methods — J. Kraus,

S. Margenov, M. Neytcheva
– Domain Decomposition Methods — U. Langer
– Monte Carlo: Tools, Applications, Distributed Computing — I. Dimov,
H. Kosina, M. Nedjalkov
– Operator Splittings, Their Application and Realization — I. Farago
– Large-Scale Computations in Coupled Engineering Phenomena with Multiple Scales — R. Ewing, O. Iliev, R. Lazarov
– Advances in Optimization, Control and Reduced Order Modeling —
P. Bochev, M. Gunzburger
– Control Systems — M. Krastanov, V. Veliov
– Environmental Modelling — A. Ebel, K. Georgiev, Z. Zlatev
– Computational Grid and Large-Scale Problems — T. Gurov, A. Karaivanova,
K. Skala
– Application of Metaheuristics to Large-Scale Problems — E. Alba,
S. Fidanova
More than 150 participants from all over the world attended the conference
representing some of the strongest research groups in the field of advanced largescale scientific computing. This volume contains 86 papers submitted by authors
from over 20 countries.
The 7th International Conference LSSC 2009 will be organized in June 2009.
December 2007

Ivan Lirkov
Svetozar Margenov
Jerzy Wa´sniewski


Table of Contents

I

Plenary and Invited Papers

Mesh Independent Convergence Rates Via Differential Operator
Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Owe Axelsson and J´
anos Kar´
atson

3

Bridging Methods for Coupling Atomistic and Continuum Models . . . . . .
Santiago Badia, Pavel Bochev, Max Gunzburger,
Richard Lehoucq, and Michael Parks

16

Parallelization of Advection-Diffusion-Chemistry Modules . . . . . . . . . . . . .
Istv´
an Farag´
o, Krassimir Georgiev, and Zahari Zlatev

28

Comments on the GMRES Convergence for Preconditioned Systems . . . .
Nabil Gmati and Bernard Philippe

40

Optimization Based Stabilization of Nonlinear Control Systems . . . . . . . .
Lars Gr¨
une

52

II

Robust Multilevel and Hierarchical Preconditioning
Methods

On Smoothing Surfaces in Voxel Based Finite Element Analysis of
Trabecular Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Peter Arbenz and Cyril Flaig

69

Application of Hierarchical Decomposition: Preconditioners and Error
Estimates for Conforming and Nonconforming FEM . . . . . . . . . . . . . . . . . .
Radim Blaheta

78

Multilevel Preconditioning of Rotated Trilinear Non-conforming Finite
Element Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ivan Georgiev, Johannes Kraus, and Svetozar Margenov

86

A Fixed-Grid Finite Element Algebraic Multigrid Approach
for Interface Shape Optimization Governed by 2-Dimensional
Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dalibor Luk´

aˇs and Johannes Kraus

96


VIII

Table of Contents

The Effect of a Minimum Angle Condition on the Preconditioning of
the Pivot Block Arising from 2-Level-Splittings of Crouzeix-Raviart
FE-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Josef Synka

III

105

Monte Carlo: Tools, Applications, Distributed
Computing

Development of a 3D Parallel Finite Element Monte Carlo Simulator
for Nano-MOSFETs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Manuel Aldegunde, Antonio J. Garc´ıa-Loureiro, and Karol Kalna
Numerical Study of Algebraic Problems Using Stochastic Arithmetic . . .
Ren´e Alt, Jean-Luc Lamotte, and Svetoslav Markov

115
123

Monte Carlo Simulation of GaN Diode Including Intercarrier
Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A. Ashok, D. Vasileska, O. Hartin, and S.M. Goodnick

131

Wigner Ensemble Monte Carlo: Challenges of 2D Nano-Device
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
M. Nedjalkov, H. Kosina, and D. Vasileska

139

Monte Carlo Simulation for Reliability Centered Maintenance
Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cornel Resteanu, Ion Vaduva, and Marin Andreica

148

Monte Carlo Algorithm for Mobility Calculations in Thin Body Field
Effect Transistors: Role of Degeneracy and Intersubband Scattering . . . .
V. Sverdlov, E. Ungersboeck, and H. Kosina

157

IV

Operator Splittings, Their Application and Realization

A Parallel Combustion Solver within an Operator Splitting Context for
Engine Simulations on Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Laura Antonelli, Pasqua D’Ambra, Francesco Gregoretti,
Gennaro Oliva, and Paola Belardini

167

Identifying the Stationary Viscous Flows Around a Circular Cylinder
at High Reynolds Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Christo I. Christov, Rossitza S. Marinova, and Tchavdar T. Marinov

175

On the Richardson Extrapolation as Applied to the Sequential Splitting
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
´
Istv´
an Farag´
o and Agnes
Havasi

184


Table of Contents

A Penalty-Projection Method Using Staggered Grids for Incompressible
Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C. F´evri`ere, Ph. Angot, and P. Poullet

IX

192

Qualitatively Correct Discretizations in an Air Pollution Model . . . . . . . .
K. Georgiev and M. Mincsovics

201

Limit Cycles and Bifurcations in a Biological Clock Model . . . . . . . . . . . .
B´
alint Nagy

209

Large Matrices Arising in Traveling Wave Bifurcations . . . . . . . . . . . . . . . .
Peter L. Simon

217

V

Recent Advances in Methods and Applications for
Large Scale Computations and Optimization of
Coupled Engineering Problems

Parallel Implementation of LQG Balanced Truncation for Large-Scale
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jose M. Bad´ıa, Peter Benner, Rafael Mayo,
Enrique S. Quintana-Ort´ı, Gregorio Quintana-Ort´ı, and
Alfredo Rem´
on

227

Finite Element Solution of Optimal Control Problems Arising in
Semiconductor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pavel Bochev and Denis Ridzal

235

Orthogonality Measures and Applications in Systems Theory in One
and More Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Adhemar Bultheel, Annie Cuyt, and Brigitte Verdonk

243

DNS and LES of Scalar Transport in a Turbulent Plane Channel Flow
at Low Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jordan A. Denev, Jochen Fr¨
ohlich, Henning Bockhorn,
Florian Schwertfirm, and Michael Manhart

251

Adaptive Path Following Primal Dual Interior Point Methods for Shape
Optimization of Linear and Nonlinear Stokes Flow Problems . . . . . . . . . .
Ronald H.W. Hoppe, Christopher Linsenmann, and Harbir Antil

259

Analytical Effective Coefficient and First-Order Approximation to

Linear Darcy’s Law through Block Inclusions . . . . . . . . . . . . . . . . . . . . . . . .
Rosangela F. Sviercoski and Bryan J. Travis

267


X

VI

Table of Contents

Control Systems

Optimal Control for Lotka-Volterra Systems with a Hunter
Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Narcisa Apreutesei and Gabriel Dimitriu
Modeling Supply Shocks in Optimal Control Models of Illicit Drug
Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Roswitha Bultmann, Jonathan P. Caulkins, Gustav Feichtinger, and
Gernot Tragler

277

285

Multicriteria Optimal Control and Vectorial Hamilton-Jacobi
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nathalie Caroff

293

Descent-Penalty Methods for Relaxed Nonlinear Elliptic Optimal
Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ion Chryssoverghi and Juergen Geiser

300

Approximation of the Solution Set of Impulsive Systems . . . . . . . . . . . . . .
Tzanko Donchev

309

Lipschitz Stability of Broken Extremals in Bang-Bang Control
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ursula Felgenhauer

317

On State Estimation Approaches for Uncertain Dynamical Systems
with Quadratic Nonlinearity: Theory and Computer Simulations . . . . . . .
Tatiana F. Filippova and Elena V. Berezina

326

Using the Escalator Boxcar Train to Determine the Optimal
Management of a Size-Distributed Forest When Carbon Sequestration
Is Taken into Account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Renan Goetz, Natali Hritonenko, Angels Xabadia, and Yuri Yatsenko

334

On Optimal Redistributive Capital Income Taxation . . . . . . . . . . . . . . . . .
Mikhail I. Krastanov and Rossen Rozenov

342

Numerical Methods for Robust Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P.Hr. Petkov, A.S. Yonchev, N.D. Christov, and M.M. Konstantinov

350

Runge-Kutta Schemes in Control Constrained Optimal Control . . . . . . . .
Nedka V. Pulova

358

Optimal Control of a Class of Size-Structured Systems . . . . . . . . . . . . . . . .
Oana Carmen Tarniceriu and Vladimir M. Veliov

366


Table of Contents

VII

XI

Environmental Modelling

Modelling Evaluation of Emission Scenario Impact in Northern Italy . . .
Claudio Carnevale, Giovanna Finzi, Enrico Pisoni, and
Marialuisa Volta
Modelling of Airborne Primary and Secondary Particulate Matter with
the EUROS-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Felix Deutsch, Clemens Mensink, Jean Vankerkom, and
Liliane Janssen
Comparative Study with Data Assimilation Experiments Using Proper
Orthogonal Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gabriel Dimitriu and Narcisa Apreutesei
Effective Indices for Emissions from Road Transport . . . . . . . . . . . . . . . . . .
Kostadin G. Ganev, Dimiter E. Syrakov, and Zahari Zlatev

377

385

393
401

On the Numerical Solution of the Heat Transfer Equation in the
Process of Freeze Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
K. Georgiev, N. Kosturski, and S. Margenov

410

Results Obtained with a Semi-lagrangian Mass-Integrating Transport
Algorithm by Using the GME Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Joppich and Sabine Pott

417

The Evaluation of the Thermal Behaviour of an Underground
Repository of the Spent Nuclear Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Roman Kohut, Jiˇr´ı Star´y, and Alexej Kolcun

425

Study of the Pollution Exchange between Romania, Bulgaria, and
Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Maria Prodanova, Dimiter Syrakov, Kostadin Ganev, and
Nikolai Miloshev
A Collaborative Working Environment for a Large Scale Environmental
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cihan Sahin, Christian Weihrauch, Ivan T. Dimov, and
Vassil N. Alexandrov
Advances on Real-Time Air Quality Forecasting Systems for Industrial
Plants and Urban Areas by Using the MM5-CMAQ-EMIMO . . . . . . . . . .
Roberto San Jos´e, Juan L. P´erez, Jos´e L. Morant, and
Rosa M. Gonz´
alez

VIII

433

442

450

Computational Grid and Large-Scale Problems

Ultra-fast Semiconductor Carrier Transport Simulation on the Grid . . . .
Emanouil Atanassov, Todor Gurov, and Aneta Karaivanova

461


XII

Table of Contents

Simple Grid Access for Parameter Study Applications . . . . . . . . . . . . . . . .
P´eter D´
ob´e, Rich´
ard K´
apolnai, and Imre Szeber´enyi
A Report on the Effect of Heterogeneity of the Grid Environment on a
Grid Job . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ioannis Kouvakis and Fotis Georgatos

470

476

Agents as Resource Brokers in Grids — Forming Agent Teams . . . . . . . . .
Wojciech Kuranowski, Marcin Paprzycki, Maria Ganzha,
Maciej Gawinecki, Ivan Lirkov, and Svetozar Margenov

484

Parallel Dictionary Compression Using Grid Technologies . . . . . . . . . . . . .
D´enes N´emeth

492

A Gradient Hybrid Parallel Algorithm to One-Parameter Nonlinear
Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D´
aniel Pasztuhov and J´
anos T¨
or¨
ok
Quantum Random Bit Generator Service for Monte Carlo and Other
Stochastic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Radomir Stevanovi´c, Goran Topi´c, Karolj Skala,
Mario Stipˇcevi´c, and Branka Medved Rogina
A Hierarchical Approach in Distributed Evolutionary Algorithms for
Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Daniela Zaharie, Dana Petcu, and Silviu Panica

IX

500

508

516

Application of Metaheuristics to Large-Scale Problems

Optimal Wireless Sensor Network Layout with Metaheuristics: Solving
a Large Scale Instance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Enrique Alba and Guillermo Molina

527

Semi-dynamic Demand in a Non-permutation Flowshop with
Constrained Resequencing Buffers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gerrit F¨
arber, Said Salhi, and Anna M. Coves Moreno

536

Probabilistic Model of Ant Colony Optimization for Multiple Knapsack
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stefka Fidanova

545

An Ant-Based Model for Multiple Sequence Alignment . . . . . . . . . . . . . . .
Fr´ed´eric Guinand and Yoann Pign´e
An Algorithm for the Frequency Assignment Problem in the Case of
DVB-T Allotments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.A. Kateros, P.G. Georgallis, C.I. Katsigiannis,
G.N. Prezerakos, and I.S. Venieris

553

561


Table of Contents

XIII

Optimizing the Broadcast in MANETs Using a Team of Evolutionary
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Coromoto Le´
on, Gara Miranda, and Carlos Segura

569

Ant Colony Models for a Virtual Educational Environment Based on a
Multi-Agent System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ioana Moisil, Iulian Pah, Dana Simian, and Corina Simian

577

Simulated Annealing Optimization of Multi-element Synthetic Aperture
Imaging Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Milen Nikolov and Vera Behar

585

Adaptive Heuristic Applied to Large Constraint Optimisation
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Kalin Penev

593

Parameter Estimation of a Monod-Type Model Based on Genetic
Algorithms and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Olympia Roeva

601

Analysis of Distributed Genetic Algorithms for Solving a Strip Packing
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Carolina Salto, Enrique Alba, and Juan M. Molina

609

Computer Mediated Communication and Collaboration in a Virtual
Learning Environment Based on a Multi-agent System with Wasp-Like
Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dana Simian, Corina Simian, Ioana Moisil, and Iulian Pah
Design of 2-D Approximately Zero-Phase Separable IIR Filters Using
Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F. Wysocka-Schillak

X

618

626

Contributed Talks

Optimal Order Finite Element Method for a Coupled Eigenvalue
Problem on Overlapping Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.B. Andreev and M.R. Racheva

637

Superconvergent Finite Element Postprocessing for Eigenvalue
Problems with Nonlocal Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . .
A.B. Andreev and M.R. Racheva

645

Uniform Convergence of Finite-Difference Schemes for
Reaction-Diffusion Interface Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ivanka T. Angelova and Lubin G. Vulkov

654


XIV

Table of Contents

Immersed Interface Difference Schemes for a Parabolic-Elliptic Interface
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ilia A. Brayanov, Juri D. Kandilarov, and Miglena N. Koleva
Surface Reconstruction and Lagrange Basis Polynomials . . . . . . . . . . . . . .
Irina Georgieva and Rumen Uluchev

661

670

A Second-Order Cartesian Grid Finite Volume Technique for Elliptic
Interface Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Juri D. Kandilarov, Miglena N. Koleva, and Lubin G. Vulkov

679

MIC(0) DD Preconditioning of FEM Elasticity Systems on
Unstructured Tetrahedral Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nikola Kosturski

688

Parallelizations of the Error Correcting Code Problem . . . . . . . . . . . . . . . .
C. Le´
on, S. Mart´ın, G. Miranda, C. Rodr´ıguez, and J. Rodr´ıguez

696

Benchmarking Performance Analysis of Parallel Solver for 3D Elasticity
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ivan Lirkov, Yavor Vutov, Marcin Paprzycki, and Maria Ganzha

705

Re-engineering Technology and Software Tools for Distributed
Computations Using Local Area Network . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.P. Sapozhnikov, A.A. Sapozhnikov, and T.F. Sapozhnikova

713

On Single Precision Preconditioners for Krylov Subspace Iterative
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hiroto Tadano and Tetsuya Sakurai

721

A Parallel Algorithm for Multiple-Precision Division by a
Single-Precision Integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Daisuke Takahashi

729

Improving Triangular Preconditioner Updates for Nonsymmetric Linear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jurjen Duintjer Tebbens and Miroslav T˚
uma

737

Parallel DD-MIC(0) Preconditioning of Nonconforming Rotated
Trilinear FEM Elasticity Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Yavor Vutov

745

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

753

Mesh Independent Convergence Rates Via
Differential Operator Pairs
Owe Axelsson1 and J´
anos Kar´
atson2
1

2

Department of Information Technology, Uppsala University,
Sweden & Institute of Geonics AS CR, Ostrava, Czech Republic
Department of Applied Analysis, ELTE University, Budapest, Hungary

Abstract. In solving large linear systems arising from the discretization
of elliptic problems by iteration, it is essential to use efficient preconditioners. The preconditioners should result in a mesh independent linear
or, possibly even superlinear, convergence rate. It is shown that a general
way to construct such preconditioners is via equivalent pairs or compactequivalent pairs of elliptic operators.

1

Introduction

Preconditioning is an essential part of iterative solution methods, such as conjugate gradient methods. For (symmetric or unsymmetric) elliptic problems, a
primary goal is then to achieve a mesh independent convergence rate, which
can enable the solution of extremely large scale problems. An efficient way to
construct such a preconditioner is to base it on an, in some way, simplified differential operator. The given and the preconditioning operators should then form
an equivalent pair, based on some inner product. Then the finite element discretization of these operators form the given matrix and its preconditioner, that
is, if the given elliptic boundary value problem

Lu = f
is suitably discretized to an algebraic system Lh uh = fh , then another, equivalent
operator S considerably simpler than L, is discretized in the same FEM subspace
to form a preconditioner Sh , and the system which is actually solved is
Sh−1 Lh uh = Sh−1 fh .
By use of equivalent pairs of operators, one can achieve a mesh independent linear convergence rate. If, in addition, the operator pairs are compactequivalent, then one can achieve a mesh independent superlinear convergence
rate. The purpose of this presentation is to give a comprehensive background
to the above, and to illustrate its applications for some important classes of
elliptic problems. Mesh independence and equivalent operator pairs have been
rigorously dealt with previously in [12,15], while superlinear rate of convergence
and compact-equivalent pairs have been treated in [6,8] (see also the references
I. Lirkov, S. Margenov, and J. Wa´
sniewski (Eds.): LSSC 2007, LNCS 4818, pp. 3–15, 2008.
c Springer-Verlag Berlin Heidelberg 2008


4

O. Axelsson and J. Kar´
atson

therein). Since in general the problems dealt with will be nonsymmetric, we first
recall some basic results on generalized conjugate gradient methods, which will
be used here. Equivalent and compact-equivalent pairs of operators are then discussed. Then some applications are shown, including a superlinear convergence
result for problems with variable diffusion coefficients.

2

Conjugate Gradient Algorithms and Their Rate of
Convergence

Let us consider a linear system
Au = b

(1)

with a given nonsingular matrix A ∈ R
, f ∈ R and solution u. Letting ., .
be a given inner product on Rn and denoting by A∗ the adjoint of A w.r.t. this
inner product, in what follows we assume that
n×n

n

A + A∗ > 0,
i.e., A is positive definite w.r.t. ., . . We define the following quantities, to be
used frequently in the study of convergence:
λ0 := λ0 (A) := inf{ Ax, x :

x = 1} > 0,

Λ := Λ(A) := A ,

(2)

where . denotes the norm induced by the inner product ., . .
2.1

Self-adjoint Problems: The Standard CG Method

If A is self-adjoint, then the standard CG method reads as follows [3]: let u0 ∈ Rn
be arbitrary, d0 := −r0 ; for given uk and dk , with residuals rk := Auk − b, we
let
uk+1 = uk + αk dk , dk+1 = −rk+1 + βk dk ,
(3)
rk , dk
rk+1 2
, βk =
.
Adk , dk
rk 2
To save computational time, normally the residual vectors are also formed by
recursion:
rk+1 = rk + αk Adk ,
(4)
where αk = −

further we use rk , dk = − rk 2 for αk , i.e, αk = rk 2 / Adk , dk . In the study
of convergence, one considers the error vector ek = u − uk and is generally
interested in its energy norm
ek

A

= Aek , ek

1/2

.

(5)

Now we briefly summarize the minimax property of the CG method and two
convergence estimates, based on [3]. We first note that the construction of the
algorithm implies ek = Pk (A)e0 with some Pk ∈ πk1 , where πk1 denotes the set


Mesh Independent Convergence Rates Via Differential Operator Pairs

5

of polynomials of degree k, normalized at the origin. Moreover, we have the
optimality property
ek A = min1 Pk (A)e0 A .
(6)
Pk ∈πk

If 0 < λ1 ≤ . . . ≤ λn are the eigenvalues of A, then (6) implies
ek
e0

A
A

≤ min1 max |Pk (λ)| ,

(7)

Pk ∈πk λ∈σ(A)}

which is a basis for the convergence estimates of the CG method.
Using elementary estimates via Chebyshev polynomials, we obtain from (7)
the linear convergence estimate
ek
e0

1/k
A
A

≤2

1/k

√
√
λ − λ
√ n √ 1 = 21/k
λn + λ1

κ(A) − 1
κ(A) + 1

(k = 1, 2, ..., n), (8)

where κ(A) = λn /λ1 is the standard condition number.
To show a superlinear convergence rate, another useful estimate is derived if
we consider the decomposition
A=I +E

(9)

k

and choose Pk (λ) := j=1 1 − λλj in (7), where λj := λj (A) are ordered
according to |λ1 − 1| ≥ |λ2 − 1| ≥ ... Then a calculation [3] yields
ek
e0

1/k
A
A

≤

2
k λ0

k

λj (E)

(k = 1, 2, ..., n).

(10)

j=1

Here by assumption |λ1 (E)| ≥ |λ2 (E)| ≥ .... If these eigenvalues accumulate
in zero then the convergence factor is less than 1 for k sufficiently large and

moreover, the upper bound decreases, i.e. we obtain a superlinear convergence
rate.
2.2

Nonsymmetric Systems

For nonsymmetric matrices A, several CG algorithms exist (see e.g. [1,3,11]).
First we discuss the approach that generalizes the minimization property (6) for
nonsymmetric A and avoids the use of the normal equation, see (18) below.
A general form of the algorithm, which uses least-square residual minimization,
is the generalized conjugate gradient–least square method (GCG-LS method)
[2,3]. Its full version uses all previous search directions when updating the new
approximation, whose construction also involves an integer t ∈ N, further, we
let tk = min{k, t} (k ≥ 0). Then the algorithm is as follows: let u0 ∈ Rn be
arbitrary, d0 := Au0 − b; for given uk and dk , with rk := Auk − b, we let


6

O. Axelsson and J. Kar´
atson

⎧
tk
k
(k)
(k)
⎪
⎪
u

=
u
+
α
d
and
d
=
r
+
βk−j dk−j ,
⎪
k+1
k
k−j
k+1
k+1
k−j
⎪
⎪
j=0
j=0
⎪
⎪
⎪
(k)
⎪
⎨ where βk−j
= − Ark+1 , Adk−j / Adk−j 2 (j = 0, . . . , sk )
⎪

⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩

(k)

and the numbers αk−j
k
j=0

(j = 0, . . . , k) are the solution of

(k)

αk−j Adk−j , Adk−l = − rk , Adk−l

(11)

(0 ≤ l ≤ k).

There exist various truncated versions of the GCG-LS method that use only a
bounded number of search directions, such as GCG-LS(k), Orthomin(k), and
GCR(k) (see e.g. [3,11]). Of special interest is the GCG-LS(0) method, which
requires only a single, namely the current search direction such that (11) is

replaced by
uk+1 = uk + αk dk , where αk = − rk , Adk / Adk

2

;

dk+1 = rk+1 + βk dk , where βk = − Ark+1 , Adk / Adk

2

.

(12)

Proposition 1. (see, e.g., [2]). If there exist constants c1 , c2 ∈ R such that
A∗ = c1 A + c2 I , then the truncated GCG-LS(0) method (12) coincides with the
full version (11) .
The convergence estimates in the nonsymmetric case often involve the residual
rk = Aek = Auk − b.

(13)

as this is readily available. It follows from [2] that
rk+1 ≤

1−

λ0
Λ

2

k/2

r0

(k = 1, 2, ..., n).

(14)

The same estimate holds for the GCR and Orthomin methods together with
their truncated versions, see [11].
An important occurrence of the truncated GCG-LS(0) algorithm (12) arises
when the decomposition
A=I +E
(15)
holds for some antisymmetric matrix E, which often comes from symmetric part
preconditioning. In this case A∗ = 2I − A and hence Proposition 1 is valid
[2]. The convergence of this iteration is then determined by E, and using that
E has imaginary eigenvalues, one can easily verify as in [7] that 1 − (λ0 /Λ)2 =
E 2 /(1+ E 2). Hence (14) yields that the GCG-LS(0) algorithm (12) converges
as
rk
r0

1/k

≤

E
1+ E

2

(k = 1, 2, ..., n).

(16)


Mesh Independent Convergence Rates Via Differential Operator Pairs

7

On the other hand, if A is normal and we have the decomposition (9), then
the residual errors satisfy a similar estimate to (10) obtained in the symmetric
case, see [3]:
1/k

rk
r0

≤

2
kλ0

k

λj (C)

(k = 1, 2, ..., n).

(17)

j=1

Again, this shows superlinear convergence if the eigenvalues λj (C) accumulate in
zero. If A is non-normal then, as shown in [3], the superlinear estimate remains
uniform in a family of problems if the order of the largest Jordan block is bounded
as n → ∞.
Another common way to solve (1) with nonsymmetric A is the CGN method,
where we consider the normal equation
A∗ Au = A∗ b

(18)

and apply the symmetric CG algorithm (3) for the latter [13]. In order to preserve
the notation rk for the residual Auk − b, we replace rk in (3) by sk and let
rk = A−∗ sk , i.e., we have sk = A∗ rk . Further, A and b are replaced by A∗ A
and A∗ b, respectively. From this we obtain the following algorithmic form: let
u0 ∈ Rn be arbitrary, r0 := Au0 − b, s0 := d0 := A∗ r0 ; for given dk , uk , rk and
sk , we let
⎧
zk = Adk ,
⎪
⎪
⎪
⎪
⎪

⎪
⎪ αk = − rk , zk ,
⎪
⎪
⎨
zk 2
⎪
⎪
sk+1 = A∗ rk+1 ,
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
⎩ βk = sk+1 ,
sk 2

uk+1 = uk + αk dk ,

rk+1 = rk + αk zk ;
(19)

dk+1 = sk+1 + βk dk .

The convergence estimates for this algorithm follow directly from the symmetric case. Using ek A∗ A = Aek = rk and that (2) implies κ(A∗ A)1/2 =
κ(A) ≤ Λ/λ0 , we obtain
rk

r0

1/k

≤ 21/k

Λ − λ0
Λ + λ0

(k = 1, 2, ..., n).

(20)

On the other hand, having the decomposition (9), using the relation (A∗ A)−1
∗
∗
∗
= A−1 2 ≤ λ−2
0 and A A = I +(C +C +C C), the analogue of the superlinear
∗
∗
estimate (10) for equation A Au = A b becomes
rk
r0

1/k

≤

2

kλ20

k

λi (C ∗ + C) + λi (C ∗ C)
i=1

(k = 1, 2, ..., n). (21)


8

3

O. Axelsson and J. Kar´
atson

Equivalent Operators and Linear Convergence

We now give a comprehensive presentation of the equivalence property between
pairs of operators, followed by a basic example for elliptic operators. First a brief
outline of some theory from [12] is given.
Let B : W → V and A : W → V be linear operators between the Hilbert
spaces W and V . Let B and A be invertible and let D := D(A) ∩ D(B) be dense,
where D(A) denotes the domain of an operator A. The operator A is said to be
equivalent in V -norm to B on D if there exist constants K ≥ k > 0 such that
k≤

Au
Bu

V
V

≤K

(u ∈ D \ {0}).

(22)

The condition number of AB −1 in V is then bounded by K/k. Similarly, the W norm equivalence of B −1 and A−1 implies this bound for B −1 A. If Ah and Bh are
finite element approximations (orthogonal projections) of A and B, respectively,
then the families (Ah ) and (Bh ) are V -norm uniformly equivalent with the same
bounds as A and B.
In practice for elliptic operators, it is convenient to use H 1 -norm equivalence,
since this avoids unrealistic regularity requirements (such as u ∈ H 2 (Ω)). We
then use the weak form satisfying
Aw u, v

1
HD

= Au, v

(u, v ∈ D(A)),

L2

(23)

1
where HD
(Ω) is defined in (26). The fundamental result on H 1 -norm equivalence
in [15] reads as follows: if A and B are invertible uniformly elliptic operators, then
−1
1
A−1
w and Bw are H -norm equivalent if and only if A and B have homogeneous
Dirichlet boundary conditions on the same portion of the boundary.
In what follows, we use a simpler Hilbert space setting of equivalent operators
from [8] that suffices to treat most practical problems. We recall that for a
symmetric coercive operator, the energy space HS is the completion of D(S)
under the inner product u, v S = Su, v , and the coercivity of S implies HS ⊂
H. The corresponding S-norm is denoted by u S , and the space of bounded
linear operators on HS by B(HS ).

Definition 1. Let S be a linear symmetric coercive operator in H. A linear
operator L in H is said to be S-bounded and S-coercive, and we write L ∈
BCS (H), if the following properties hold:
(i) D(L) ⊂ HS and D(L) is dense in HS in the S-norm;
(ii) there exists M > 0 such that
| Lu, v | ≤ M u

S

v

Lu, u ≥ m u

2

S

S

(u, v ∈ D(L));

(iii) there exists m > 0 such that
(u ∈ D(L)).


Mesh Independent Convergence Rates Via Differential Operator Pairs

9

The weak form of such operators L is defined analogously to (23), and produces
a variationally defined symmetrically preconditioned operator:
Definition 2. For any L ∈ BCS (H), let LS ∈ B(HS ) be defined by
LS u, v

S

(u, v ∈ D(L)).

= Lu, v

Remark 1. (i) Owing to Riesz representation theorem the above definition makes
sense. (ii) LS is coercive on HS . (iii) If R(L) ⊂ R(S) (where R(. ) denotes the
range), then LS D(L) = S −1 L.
The above setting leads to a special case of equivalent operators:
Proposition 2. [9] Let N and L be S-bounded and S-coercive operators for the

same S. Then
(a) NS and LS are HS -norm equivalent,
(b) NS−1 and L−1
S are HS -norm equivalent.
Definition 3. For given L ∈ BCS (H), we call u ∈ HS the weak solution of
equation Lu = g if LS u, v S = g, v (v ∈ HS ). (Note that if u ∈ D(L)
then u is a strong solution.)
Example. A basic example of equivalent elliptic operators in the S-bounded
and S-coercive setting is as follows. Let us define the operator
Lu ≡ −div (A ∇u) + b · ∇u + cu

for u|ΓD = 0,

∂u
+ αu|ΓN = 0,
∂νA

(24)

∂u
= A ν · ∇u and ν denotes the outer normal derivative, with the
∂νA
following properties:
where

Assumptions 3.1
(i) Ω ⊂ Rd is a bounded piecewise C 1 domain; ΓD , ΓN are disjoint open
measurable subsets of ∂Ω such that ∂Ω = Γ D ∪ Γ N ;
(ii) A ∈ C 1 (Ω, Rd×d ) and for all x ∈ Ω the matrix A(x) is symmetric; b ∈
C 1 (Ω)d , c ∈ L∞ (Ω), α ∈ L∞ (ΓN );

(iii) there exists p > 0 such that A(x)ξ · ξ ≥ p |ξ|2 for all x ∈ Ω and ξ ∈ Rd ;
cˆ := c − 12 div b ≥ 0 in Ω and α
ˆ := α + 12 (b · ν) ≥ 0 on ΓN ;
(iv) either ΓD = ∅, or cˆ or α
ˆ has a positive lower bound.
Let S be a symmetric elliptic operator on the same domain Ω:
Su ≡ −div (G ∇u) + σu

for u|ΓD = 0,

∂u
∂νG

+ βu|ΓN = 0,

(25)

with analogous assumptions on G, σ, β. Let
1
HD
(Ω) = {u ∈ H 1 (Ω), u|ΓD = 0},

u, v

S

(G ∇u·∇v+σuv)+

=
Ω

βuv dσ
ΓN

(26)
which is the energy space HS of S. Then the following result can be proved:


10

O. Axelsson and J. Kar´
atson

Proposition 3. [9]. The operator L is S-bounded and S-coercive in L2 (Ω).
The major results in this section are mesh independent convergence bounds
corresponding to some preconditioning concepts. Let us return to a general
Hilbert space H. To solve Lu = g, we use a Galerkin discretization in Vh =
span{ϕ1 , . . . , ϕn } ⊂ HS , where ϕi are linearly independent. Let
n

Lh :=

LS ϕi , ϕj

S

i,j=1

and, for the discrete solution, solve
Lh c = bh

(27)

with bh = { g, ϕj }nj=1 . Since L ∈ BCS (H), the symmetric part of Lh is positive
definite.
First, let L be symmetric itself. Then its S-coercivity and S-boundedness
turns into the spectral equivalence relation
m u

2
S

≤ LS u, u

S

≤M u

(u ∈ HS ).

2
S

(28)

Then Lh is symmetric too. Let
n

Sh =

ϕi , ϕj

S

(29)
i,j=1

be the stiffness matrix of S, to be used as preconditioner for Lh . This yields the
preconditioned system
−1
S−1
(30)
h Lh c = Sh bh .
Now S−1
h Lh is self-adjoint w.r.t. the inner product c, d

Sh

:= Sh c · d.

Proposition 4. (see, e.g., [10]). For any subspace Vh ⊂ HS ,
κ(S−1
h Lh ) ≤

M
m

(31)

independently of Vh .

Consider now nonsymmetric problems with symmetric equivalent preconditioners. With Sh from (29) as preconditioner, we use the bounds (2) for the
GCG-LS and CGN methods:
λ0 = λ0 (S−1
h Lh ) := inf{Lh c · c : Sh c · c = 1},

−1
Λ = Λ(S−1
h Lh ) := Sh Lh

Sh

.

These bounds can be estimated using the S-coercivity and S-boundedness
m u

2
S

≤ LS u, u S ,

| LS u, v S | ≤ M u

S

v

S

(u, v ∈ HS ).

(32)


Mesh Independent Convergence Rates Via Differential Operator Pairs

11

Proposition 5. [9]. For any subspace Vh ⊂ HS ,
Λ(S−1
M
h Lh )
≤
−1
m
λ0 (Sh Lh )

(33)

independently of Vh .
Consequently, by (14), the GCG-LS algorithm (11) for system (30) satisfies
rk
r0

1/k
Sh
Sh

≤ 1−

m
M

2 1/2

(k = 1, 2, ..., n),

(34)

which holds as well for the GCR and Orthomin methods together with their
truncated versions; further, by (20), the CGN algorithm (19) for system (30)
satisfies
1/k
rk Sh
M −m
≤ 21/k
(k = 1, 2, ..., n).
(35)
r0 Sh
M +m
Finally, let now Sh := (Lh + LTh )/2 be the symmetric part of Lh . Here Lh =
Sh +Qh with Qh := (Lh −LTh )/2, and LS = I +QS where QS is antisymmetric in
−1
−1
HS , further, S−1
h Lh = Ih +Sh Qh where Sh Qh is antisymmetric w.r.t. the inner
product ., . Sh . Then the full GCG algorithm reduces to the simple truncated
version (12), further, we obtain the mesh independent estimate S−1
h Qh S h ≤
QS , whence by (16),

rk
r0

4

1/k
Sh
Sh

≤

QS
1 + QS

2

(k = 1, 2, ..., n).

(36)

Compact-Equivalent Operators and Superlinear
Convergence

We now present the property of compact-equivalence between operator pairs,
based on [8], which is a refinement of the equivalence property and provides
mesh independent superlinear convergence. We use the Hilbert space setting of
Definition 1 and include a main example (which, moreover, is a characterization)
for elliptic operators.
Definition 4. Let L and N be S-bounded and S-coercive operators in H. We
call L and N compact-equivalent in HS if

LS = μNS + QS

(37)

for some constant μ > 0 and compact operator QS ∈ B(HS ).
Remark 2. If R(L) ⊂ R(N ), then compact-equivalence of L and N means that
N −1 L is a compact perturbation E of constant times the identity in the space
HS , i.e., N −1 L = μI + E.


12

O. Axelsson and J. Kar´
atson

One can characterize compact-equivalence for elliptic operators. Let us take two
operators as in (24):
∂u
+ α1 u|ΓN = 0,
∂νA1
∂u
= 0,
+ α2 u|ΓN = 0
∂νA2

L1 u ≡ −div (A1 ∇u) + b1 · ∇u + c1 u

for u|ΓD = 0,

L2 u ≡ −div (A2 ∇u) + b2 · ∇u + c2 u

for u|ΓD

where we assume that L1 and L2 satisfy Assumptions 3.1. Then the following
fundamental result holds:
Proposition 6. [8]. The elliptic operators L1 and L2 are compact-equivalent
1
in HD
(Ω) if and only if their principal parts coincide up to some constant μ > 0,
i.e. A1 = μA2 .
Now we discuss preconditioned CG methods and corresponding mesh independent superlinear convergence rates. Let us consider an operator equation Lu = g
in a Hilbert space H for some S-bounded and S-coercive operator L, and its
Galerkin discretization as in (27). Let us first introduce the stiffness matrix Sh
as in (29) as preconditioner.
Proposition 7. [8]. If L and S are compact-equivalent with μ = 1, then the
CGN algorithm (19) for system (30) yields
rk
r0

1/k
Sh
Sh

≤ εk

(k = 1, 2, ..., n),

(38)

where εk → 0 is a sequence independent of Vh .

A similar result holds for the GCG-LS method, provided however that QS is a
normal compact operator in HS and the matrix S−1
h Qh is Sh -normal [6]. These
properties hold, in particular, for symmetric part preconditioning. The sequence
εk contains similar expressions of eigenvalues as (17) or (21) related to QS , which
we omit for brevity.
For elliptic operators, we can derive a corresponding result. Let L be the
elliptic operator in (24) and S be the symmetric operator in (25). If the principal
parts of L and S coincide, i.e., A = G, then L and S are compact-equivalent by
Proposition 6, and we have μ = 1. Hence Proposition 7 yields a mesh independent
superlinear convergence rate. Further, by [8], an explicit order of magnitude in
which εk → 0 can be determined in some cases. Namely, when the asymptotics for
∂u
symmetric eigenvalue problems Su = μu, u|ΓD = 0, r ∂ν
+ βu |Γ = μu
A
N

satisfies μi = O(i2/d ), as is the case for Dirichlet problems, then
εk ≤ O

log k
k

if d = 2

and εk ≤ O

1
k 2/d

if d ≥ 3.

(39)


Mesh Independent Convergence Rates Via Differential Operator Pairs

5

13

Applications of Symmetric Equivalent Preconditioners

We consider now symmetric preconditioning for elliptic systems defined on a
domain Ω ⊂ RN . Let
Li ≡ −div(Ai ∇ui ) + bi · ∇ui +

l

Vij uj = gi
j=1

ui = 0 on ∂ΩD ,

∂ui
∂νA

(40)

+ αi ui = 0 on ∂ΩN , i = 1, 2, · · · l.

Here it is assumed that bi ∈ C 1 (Ω)N , gi ∈ L2 (Ω) and Vij ∈ L∞ (Ω), and
the matrix V = {Vij }li,j=1 satisfies the coercivity property pointwise in Ω,
λmin (V + V T )− max divbi ≥ 0, pointwise in Ω, where λmin denotes the smallest
eigenvalue.
1
Then system (40) has a unique solution u ∈ HD
(Ω)l .
As preconditioning operator we use the l-tuple S = (S1 , · · · , Sl ) of independent
∂ui
operators, Si ui ≡ −div(Ai ∇ui ) + hi u, where ui = 0 on ∂ΩD ,
+ βi ui = 0
∂νA
on ∂ΩN and βi ≥ 0, i = 1, 2, · · · l.
1
Now we choose a FEM subspace Vh ⊂ HD
(Ω)l and look for the solution uh of
the corresponding system Lh c = b using a preconditioner Sh being the stiffness
matrix of S.
One can readily verify that there occurs a superlinear convergence of the
preconditioned CGM which, furthermore, is mesh independent.
An application where such systems arise is in meteorology, where the chemical reaction terms have been linearized in a Newton nonlinear iteration method
([16]). Another important application of equivalent pairs of elliptic operators
arises for the separable displacement preconditioning method for elasticity systems, formulated in displacement variables. There, the equivalence of the given
and the separable displacement operators can be proven using Korn’s inequality,
see [4,5] for details and further references.
We have shown that a superlinear convergence takes place for operator pairs
(i.e., the given and its preconditioner) which are compact-equivalent. The main
theorem states that the principal, i.e., the dominating (second order) parts of the

operators must be identical, apart from a constant factor. This seems to exclude
an application for variable coefficient problems, where for reasons of efficiency we
choose a preconditioner which has constant, or piecewise constant coefficients, assuming we want to use a simple operator such as the Laplacian as preconditioner.
However, we show now how to apply some method of scaling or transformation
to reduce the problem to one with constant coefficients in the dominating part.
We use then first a direct transformation of the equation. Let
Lu ≡ −div(a∇u) + b · ∇u + cu = g,
where a ∈ C (Ω), a(x) ≥ p > 0.
Here a straightforward computation shows that
1
1
c
g
Lu = −div(∇u) + (b − ∇a)∇u + u = ,
a
a
a
a
i.e.,the principal part consists simply of the Laplacian operator, −Δ.
1

(41)


14

O. Axelsson and J. Kar´
atson

A case of special importance occurs when a is written in the form a = e−φ , φ ∈

C (Ω) and b = 0. Then −∇a = e−φ ∇φ = a∇φ and (41) takes the form
1

1
Lu = −Δu + ∇φ∇u + eφ cu = eφ g.
a
This is a convection-diffusion equation with a so called potential vector field,
v = ∇φ. Such problems occur frequently in practice, e.g. in modeling of semiconductors.
When the coefficient a varies much over the domain Ω one can apply transformations of both the equation and the variable,
to √
reduce variations of gradients
√
( ∇u ) of O(max(a)/ min(a)) to O(max a/ min( a)). Let then u = a1/2 v and
assume that a ∈ C 2 (Ω). Then a computation shows that
−a−1/2

∂
∂u
∂
∂v
1
∂a
(a
) = −a−1/2
(a−1/2
− a−1/2
v
∂xi ∂xi
∂xi
∂xi

2
∂xi
∂2v
∂ 2 (a−1/2 )
= − 2 + a−1/2
,
∂xi
∂x2i

and a−1/2 b∇u = a−1 b∇v − 12 (b · ∇u/a2 )v. Hence
a−1/2 (Lu − g) = Δv + b · ∇v + cv − g,

(42)

where b = a−1 b, c = a−1 c − 12 b · ∇u/a2 + a−1/2 Δ(a−1/2 ) and g = a−1/2 g.
Remark 3. It is seen that when b = 0 both the untransformed (41) and transformed (42) operators are selfadjoint.
The relation N v ≡ a−1/2 Lu shows that
N v, v

L2 (Ω

= a−1/2 Lu, a1/2 u

L2 (Ω

= Lu, u

L2 (Ω

holds for all u ∈ D(L). The positivity of the coefficient a shows hence that

u H 1 and v H 1 are equivalent, and N inherits the H 1 -coercivity of L, i.e.,
the relation Lu, u L2 (Ω ≥ m u 2H 1 (Ω) is replaced by N v, v L2 (Ω ≥ m v 2H 1 (Ω)
for a certain constant m > 0. This shows the we may apply e.g. the Laplacian
operator (−Δ) as preconditioner to N which, being a compact equivalent pair,
implies a superlinear and meshindependent rate of convergence of CGM.

Acknowledgement
The second author was supported by the Hungarian Research Office NKTH
¨
under Oveges
Program and by the Hungarian Research Grant OTKA No.K
67819.