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Preface
Fruitofalongmaturingprocess,freefem,initslastavatar,FreeFem++ ,isahighlevel
integrateddevelopmentenvironment(IDE)fornumericallysolvingpartialdierentialequa-
tions(PDE)indimension2and3.Itistheidealtoolforteachingtheniteelementmethod
but it is also perfectforresearchto quickly test newideas or multi-physics andcomplex
applications.
FreeFem++ hasanadvancedautomaticmeshgenerator,capableofaposteriorimeshadap-
tation;it has ageneralpurposeelliptic solverinterfacedwithfastalgorithms suchas the
multi-frontalmethodUMFPACK,SuperLU.Hyperbolicandparabolicproblemsaresolved
byiterativealgorithmsprescribedbytheuserwiththehighlevellanguageof FreeFem++
. Ithasseveraltriangularniteelements,includingdiscontinuouselements. Finallyevery-
thingisthereinFreeFem++ toprepareresearchqualityreports:colordisplayonlinewith
zoomingandotherfeaturesandpostscriptprintouts.
This manual is s meant t for students at t Master level, , for researchers s at any y level, and for
engineers(includingnancialengineering)withsomeunderstandingofvariationalmethods
forpartialdierentialequations.
11
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12
CONTENTS
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Chapter1
Introduction
Apartialdierentialequationisarelationbetweenafunctionofseveralvariablesandits
(partial)derivatives.Manyproblemsinphysics,engineering,mathematicsandevenbanking
aremodeledbyoneorseveralpartialdierentialequations.
FreeFem++ isasoftwaretosolvetheseequationsnumerically. . Asitsnameimplies,itis
afreesoftware(seethecopyrights forfulldetail)basedontheFiniteElementMethod;it
isnotapackage,itisanintegratedproductwithitsownhighlevelprogramminglanguage.
ThissoftwarerunsonallUNIXOS(withg++3.3orlater,andOpenGL),onWindowXP,
Vistaand7andonMacOS10(powerpc,intel)
Moreover FreeFem++ + ishighly y adaptive. . Manyphenomenainvolve e severalcoupled sys-
tems, for r example: :  uid-structure e interactions,Lorentz forces for aluminium casting and
ocean-atmosphereproblemsarethreesuchsystems. Theserequiredierentniteelement
approximationsandpolynomialdegrees,possiblyondierentmeshes.Somealgorithmslike
Schwarz’domaindecompositionmethodalsorequiredatainterpolationonmultiplemeshes
withinoneprogram. FreeFem++ + canhandlethesediculties,i.e.arbitraryniteelement
spacesonarbitraryunstructuredandadaptedbi-dimensionalmeshes.
ThecharacteristicsofFreeFem++ are:
• Problem m description(realorcomplexvalued)by theirvariationalformulations,with
accesstotheinternalvectorsandmatricesifneeded.
• Multi-variables,multi-equations,bi-dimensionalandthree-dimensionalstaticortime
dependent,linearornonlinearcoupledsystems;howevertheuserisrequiredtodescribe
theiterativeprocedureswhichreducetheproblemtoasetoflinearproblems.
• Easy y geometricinput by analytic description n of boundaries by pieces; however this
partisnotaCADsystem;forinstancewhentwoboundariesintersect,theusermust
specifytheintersectionpoints.
• Automaticmeshgenerator,basedontheDelaunay-Voronoialgorithm;theinnerpoint
densityisproportionaltothedensityofpointsontheboundaries[11].
13
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CHAPTER1. INTRODUCTION
• Metric-basedanisotropicmeshadaptation.Themetriccanbecomputedautomatically
fromtheHessianofanyFreeFem++ function[13].
• Highleveluserfriendly y typed d inputlanguage with an algebraofanalyticand nite
elementfunctions.
• Multipleniteelementmesheswithinoneapplicationwithautomaticinterpolationof
dataondierentmeshesandpossiblestorageoftheinterpolationmatrices.
• Alargevarietyoftriangularniteelements: : linear,quadratic c Lagrangianelements
andmore,discontinuousP1andRaviart-Thomas elements,elementsofanon-scalar
type,themini-element,... (butnoquadrangles).
• Tools s todenediscontinuous Galerkinniteelementformulations s P0,P1dc,P2dc
andkeywords: jump, , mean, intalledges.
• Alargevariety y oflineardirectanditerativesolvers(LU,Cholesky,Crout,CG,GM-
RES, UMFPACK, MUMPS, SuperLU, , ...) ) and d eigenvalue and eigenvector solvers
(ARPARK).
• Near r optimalexecution speed(compared withcompiled C++ implementations pro-
grammeddirectly).
• Onlinegraphics,generationof,.txt,.eps,.gnu, , meshlesforfurthermanipu-
lationsofinputandoutputdata.
• Many examples s and tutorials: : elliptic, , parabolic and hyperbolic problems, Navier-
Stokes ows,elasticity,Fluidstructureinteractions,Schwarz’sdomaindecomposition
method,eigenvalueproblem,residualerrorindicator,...
• Aparallelversionusingmpi
1.1 Installation
1.1.1 Foreveryone:
Firstopenthefollowingwebpage
http://www.freefem.org/ff++/
Andchooseyourplatform:Linux,Windows,MacOSX,orgototheendofthepagetoget
thefulllistofdownloads.
Remark1 :BinariesareavailableforMicrosoftWindows,AppleMacOSXandsomeLinux
systems.
Installbydoubleclickontheappropriatele,underlinuxandMacOStheinstalllearere-
spectivelyindirectory/usr/local/bin,/usr/local/share/freefem++,/usr/local/lib/ff++
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1.1. INSTALLATION
15
Windowsbinariesinstall Firstdownloadthewindowsinstallationexecutable,thendouble
click it. . to o install FreeFem++ + . . Inmost t cases s just answer r yes (or r typr return) ) to o all
questions. Otherwise e in n the AdditionalTask windows, check the box x "Add d application
directorytoyoursystempathyoursystempath."Thisisrequiredotherwisetheprogram
ffglut.exewillnotbefound.
Bynowyoushouldhavetwonewiconsonyourdesktop:
• FreeFem++ + (VERSION).exetheFreeFem++ application.
• FreeFem++ + (VERSION) ExamplesalinktotheFreeFem++ folderofexamples.
where(VERSION)istheversionoftheles(forexample3.3-0-P4).
Bydefault,theinstalledlesarein
C:\Programs Files\FreeFem++
Inthisdirectory,youhaveallthe.dlllesandotherapplications:FreeFem++-nw.exe,ffglut.exe,
...theFreeFem++ applicationwithoutgraphicwindows.
Thesyntaxforthecommand-linetoolsarethesameasthoseofFreeFem.exe.
MacOSXbinariesinstall DownloadtheMacOSXbinaryversionle,extractalltheles
withadoubleclickontheiconofthele,gothethedirectoryandputtheFreeFem+.app
applicationinthe/Applicationsdirectory.IfyouwantaterminalaccesstoFreeFem++
justcopytheleFreeFem++ inadirectoryofyour$PATHshellenvironmentvariable.
IfyouwanttoautomaticallylaunchtheFreeFem++.app,doubleclickona.edpleicon.
Under thender pick a .edpin directory examples++-tutorialfor example,select
menuFile -> > Get t Infoanchange Open n with:(chooseFreeFem++.app)andclick
onbuttonchange All....
Wheretogofromhere AnintegratedenvironmentcalledFreeFem++-cs,writtenbyAn-
toineLeHyaric,isprovidedwithFreeFem++ .Unlessyouwishtoprolenowyourownde-
velopmentenvironment,youmayproceedtothenextparagraph"HowtouseFreeFem++".
1.1.2 Forthepros: : Installationfromsources
Thissectionisforthosewhoforsomereasondonotwishtousethebinariesandhenceneed
torecompileFreeFem++ orinstallitfromthesourcecode:
Thedocumentationarchive :Thedocumentationisalsoopensource;toregenerateityou
needaL
A
T
E
XenvironmentcapableofcompilingaCVSarchive;underMS-Windowsyouwill
havetousemingw/msys
http://www.mingw.org
andunderMacOSXwehaveusedApple’sDeveloperTools"Xcode"andL
A
T
E
X fromhttp:
//www.ctan.org/system/mac/texmac.
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16
CHAPTER1. INTRODUCTION
TheC++archive :FreeFem++ + mustbecompiledfromthesourcearchive,asindicated
in
http://www.freefem.org/ff++/index.htm
Toextractlesfromthecompressedarchivefreefem++-(VERSION).tar.gztoadirec-
torycalled
freefem++-(VERSION)
enterthefollowingcommandsinashellwindow:
tar zxvf freefem++-(VERSION).tar.gz
cd freefem++-(VERSION)
To compile and install FreeFem++ + , , just t followthe INSTALLand READMEles. . The
followingprogramsareproduced,dependingonthesystemyouarerunning:
1. FreeFem++ + ,standardversion,withagraphicalinterfacebasedonGLUT/OpenGL
(useglutvisualizationtool)ornotjustadd -nw w parameter.
2. ffglutthevisualizationtoolsthroughapipeoffreefem++(remark: ifglutisnot
inthesystempath,youwillhavenoplot)
3. FreeFem++-nw,postscriptplotoutputonlyandmedit(batchversion,nographics
windowsviaffglut)
4. FreeFem++-mpi,parallelversion,postscriptoutputonly
5. /Applications/FreeFem++.app,theDragandDropCoCoaMacOSXApplica-
tion
6. bamg,thebamgmeshgenerator
7. cvmsh2,ameshleconvertor
8. drawbdmesh,ameshleviewer
9. ffmeditthefreefem++versionofmeditsoftware(thankstoP.Frey)
ThesyntaxoftoolsFreeFem++ ,FreeFem++-nw,FreeFem++-mpi,onthecommand-line
are
• FreeFem++ + [-?] ] [-v nn] [-fglut file1] [-glut file2] [-f]
edpfilepath where
the
• orFreeFem++-nw w -? [-v nn] [-fglut t file1] ] [-glut file2] [-f]
edpfilepath where
the
-?showtheusage.
-fglut filenametostoreallthedataforgraphicinle e filename,andtoreplaydo
ffglut filename.
-glut
ffglutprogamtochangethevisualisatorprogram’s.
1.1. INSTALLATION
17
-nwnocalltoglutandmedit
-v nnsetthelevelofverbositytonnbeforeexecutionofthescript.
-nenoedpscriptoutput
-waitwaitanreturnintextwindowbeforeclosingFreeFem++
-nowaitwaitanreturnintextwindowbeforeclosingFreeFem++
-nenoedpscriptoutput
-cdChangedirectorytoscriptdir(thescriptpathmustbyglobal)
ifnolepaththenyougetadialogboxtochoosetheedpleonwindowssysteme.
Thenotation[]means"optional".
Remark2 Inmost t casesyoucansetthe level ofoutput(verbosity)tovaluennbyadding
theparameters-v nnonthecommandline.
Asaninstallationtest,underunix:gointothedirectoryexamples++-tutorialandrun
FreeFem++ ontheexamplescriptLaplaceP1.edpwiththecommand:
FreeFem++ LaplaceP1.edp
Ifyouareusingneditasyourtexteditor,doonetimenedit -import edp.neditto
havecoloringsyntaxforyour .edples.
Linkwithothertexteditors
notepad++ athttp://notepad-plus.sourceforge.net/uk/site.htm
• OpenNotepad++andEnterF5
• Inthenewwindowenterthecommandlaunchff++ + "$(FULL_CURRENT_PATH)"
• ClickonSave,andenterFreeFem++ + inthebox"Name",nowchoosetheshort
cutkeytolaunchdirectlyFreeFem++(forexamplealt+shift+R)
• ToaddColorSyntaxCompatiblewithFreeFem++InNotepad++,
{ InMenu"Parameters"->"Configuration n of the
Color Syntax"
proceedasfollows:
{ Inthelist"Language"selectC++
{ Add"edp"intheeld"add d ext"
{ Select"INSTRUCTION N WORD"inthelist"Description"andintheeld
"supplementary key word",cutandpastthefollowinglist:
P0P1P2P3P4P5P1dcP2dcP3dcP4dcP5dcRT0RT1RT2RT3RT4
RT5macroplotint1dint2dsolvemovemeshadaptmeshtrunccheckmovemesh
onfuncbuildmeshsquareEigenvalueminmaximagexecLinearCGNLCG
NewtonBFGSLinearGMREScatchtryintalledgesjumpaveragemeanload
savemeshconvectabssincostanatanasinacoscotansinhcoshtanhcotanh
atanhasinhacoshpowexploglog10sqrtdxdyendlcout
18
CHAPTER1. INTRODUCTION
{ Select t "TYPEWORD"inthe list"Description"and... . ""supplementary
keyword",cutandpastthefollowinglist
mesh real fespace e varf f matrix x problem string g border r complex x ifstream m of-
stream
{ ClickonSave e & Close.Nownodepad++iscongured.
Crimson Editoravailbleathttp://www.crimsoneditor.com/andadaptedasfollows:
• GototheTools/Preferences/Fileassociationmenuandaddthe.edpex-
tensionset
• In the same panelinTools/User Tools, add a FreeFem++ + item m (1st line)
withthepathtofreefem++.exeonthesecondlineand$(FilePath)and
$(FileDir)onthirdandfourthlines.Tickthe8.3box.
• forcolorsyntax,extractlefromcrimson-freefem.zipandputlesinthe
correspondingsub-folderofCrimsonfolder(C:\Program Files\Crimson Editor
).
winedt forWindows:thisisthebestbutitcouldbetrickytosetup.Downloaditfrom
http://www.winedt.com
this isa multipurposetexteditorwithadvancedfeatures suchassyntaxcoloring;a
macroisavailableonwww.freefem.orgtolocalizewinedttoFreeFem++ without
disturbingthewinedtfunctionalmodeforLateX,TeX,C,etc. Howeverwinedtisnot
freeafterthetrialperiod.
TeXnicCenter forWindows:thisistheeasiestandwillbethebestoncewendavolunteer
toprogramthecolorsyntax.Downloaditfrom
http://www.texniccenter.org/
ItisalsoaneditorforTeX/LaTeX.Ithasa"‘tool"’menuwhichcanbeconguredto
launchFreeFem++ programsasin:
• SelecttheTools/Customizeitemwhichwillbringupadialogbox.
• SelecttheToolstabandcreateanewitem:callitfreefem.
• inthe3linesbelow,
1. searchforFreeFem++.exe
2. selectMainlewithfurtheroptionthenFullpathandclickalsoonthe8.3
box
3. selectmainlefulldirectorypathwith8.3
nedit ontheMacOS,Cygwin/Xfreeandlinux,toimportthecolorsyntaxdo
nedit -import edp.nedit
1.2. HOWTOUSEFREEFEM++
19
Smultron on n theMac, available at http://smultron.sourceforge.net. . It t comes
ready withcolorsyntaxfor.edple. . Toteachit t tolaunchFreeFem++ + les,doa
"commandB"(i.e. themenuTools/Handle e Command/newcommand) andcreatea
commandwhichdoes
/usr/local/bin/FreeFem++-CoCoa %%p
Figure1.1:IntegratedenvironmentforFreeFem++developmentwithWindows
1.2 HowtouseFreeFem++
UnderMacOSXwithGraphicInterfaces Totestan.edple,justdraganddropthele
iconontheMacOSapplicationiconFreeFem++.app.Youcanalsolaunchthisapplication
andusethemenu:File!Open.
20
CHAPTER1. INTRODUCTION
OneofthebestwayshoweverontheMacistouseatexteditorlikeSmultron.app(see
above).
Figure 1.2: : The e 3panelsof the integratedenvironmentbuilt withthefraise e Editor
withFreeFem++ in n action. . The e Tools menu has anitem to launch FreeFem++ + by y a
Ctrl+1command.
InTerminalmode ChoosethetypeofapplicationfromFreeFem++ + ,FreeFem++-nw,
FreeFem++-mpi,...accordingtoyourneeds.Addatleastthepathname;forexample
FreeFem++ your-edp-file-path
1.3 Environmentvariables,andtheinitle
FreeFem++ readsauser’sinitlenamed d freefem++.preftoinitializeglobalvariables:
verbosity,includepath,loadpath.
Remark3 Thevariableverbositychangesthelevelofinternalprinting(0,nothing(un-
lesstherearesyntaxerrors),1few,10lots,etc. ...),thedefaultvalueis2.
The include lesaresearched fromthe includepathlist andthe load lesare searched
fromloadpathlist.
Thesyntaxoftheleis:
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