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Numerical Differential Equations Theory and Technique
ODE Methods Finite Differences Finite Elements and Collocation John Loustau
Finite element method (FEM) is a numerical technique for approximation of the solution of partial differential equations. This book on FEM and related simulation tools starts with a brief introduction to Sobolev spaces and elliptic scalar problems and continues with explanation of finite element spaces and estimates for the interpolation error. Construction of finite elements on simplices, quadrilaterals, hexahedral is discussed by the authors in self-contained manner. The last chapter focuses on object-oriented finite element algorithms and efficient implementation techniques. Readers can also find the concepts of scalar parabolic problems and high dimensional parabolic problems in different chapters. Besides, the book includes some recent advances in FEM, including nonconforming finite elements for boundary value problems of higher order and approaches for solving differential equations in high-dimensional domains. There are plenty of solved examples and mathematical theorems interspersed throughout the text for better understanding.
Sashikumaar Ganesan is Assistant Professor at the Indian Institute of Science (IISc). He heads the research group on Computational Mathematics at the Department of Computational and Data Sciences at IISc. His areas of interest include numerical analysis, finite elements in fluid dynamics, moving boundary value problems, numerical methods for turbulent flows, variational multiscale methods, population balance modelling, scalable algorithms and high performance computing.
Lutz Tobiska is Professor at the Institute for Analysis and Computational Mathematics, Ottovon-Guericke University Magdeburg, Germany. His research interests include finite elements in fluid dynamics, parallel algorithms, multigrid methods and adaptive methods for convection diffusion equations.
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3
3
6
5.3Conforming finiteelementmethods62
5.4Nonconforming finiteelementmethods70
6
6.2Agene
6.3Weakf
6.4 Semidiscretization byfiniteelements
6 5Time discretizati
6.6Finiteelements
7
7.
8
8 2Weakformulationof the Stokes problem134
8.3Conforming discretizationsof the Stokes problem139
8.4Nonconforming discretizationsof the Stokes problem146
8.5ThenonconformingCrouzeix–Raviartelement150
8.6Further inf–sup stable finiteelement pairs154
8.7 Equal order stabilizedfiniteelements
8.8Navier–Stokes problem with mixedboundary conditions166
9 7Object-oriented C++ programming 195 Bibliography
Preface
The purposeof this book isto present—inacoherentandlucidway—themathematical
theory and algorithmsof thefiniteelementmethod, which isthemost widely-used methodfor
thesolutionofpartialdifferential equations inthefield of ComputationalScience. We believe
thatthe fullpotential of thefiniteelementmethod can be realised onlywhenthetheoretical
background and the implemented algorithmsareconsidered asaunit.
Theselectionof the basicmathematical theory of finiteelements inthis book is based on lecturesgiven inthe“Finite ElementIand II”coursesofferedfor severalyearsat
IndianInstituteofScience, Bangalore. Furthermore,thefiniteelementalgorithms presented are based onthe knowledgeand experiencegained through the developmentof our in-house
finiteelement package for morethan10 years. Thetheory and algorithmsof finiteelements
that we describe hereareself-contained;our aim isthat beginners will find ourbook to be
bothreadableand useful.
Westart inChapter 1 with a brief introductionto Sobolevspacesand thenecessarybasics
offunctional analysis. This willhelp those readers whoareunfamiliarwithfunctional analysis.
Thegoal of Chapter 2 istoexplainthefiniteelementmethod to beginners inthesimplest
possible way. Theconceptsofweak solutions,variationalformulationof second-order elliptic boundary value problems, incorporationofdifferent boundary conditions inavariational
ThenexttwoChaptersgivethe basictheory of thefiniteelementmethod. InChapter 3, theconstructionof finiteelementsonsimplices, quadrilaterals,andhexahedrals is discussed in detail. Furthermore, linear, bilinear and isoparametrictransformationsareexplained,and mapped finiteelementsareconsidered. Chapter 4 deals with the interpolationtheory of affine equivalentfiniteelements in Sobolevspaces Wealso discussthe interpolationoffunctionsthat are lesssmooth; in particular, Scott-Zhang interpolation is described inthisChapter.
Finiteelements for moreadvanced scalarproblemsare presented inChapters5and 6. As anexampleof afiniteelementmethodfor elliptic higher-orderproblems,conformingand nonconformingfiniteelementmethods for the biharmonicequationarestudied inChapter 5. Next,thefiniteelementmethodfor scalarparabolic problems is presented inChapter 6. This includesa discussionof thestandard θ -methodsanddiscontinuousGalerkinmethods for temporaldiscretisation Inaddition,splittingschemes inthecontextof finiteelementmethods forhigh-dimensional scalarparabolic problemsareexamined.
viii preface
Finiteelementmethods for systemsof equations insolid mechanicsand fluid mechanics
are presented inChapters7and 8. In particular,finiteelementmethods for systemsof
equations in linear elasticity andfor Mindlin-Reissnerplate problemsare discussed in
Chapter 7. Chapter 8 deals in detailwith finiteelementmethods for the Stokesand
equations, we discussthe implementationofdifferent boundary conditions,amixed
variationalformulation,thenecessary condition for thestability of thefiniteelementscheme, and finiteelementsthatsatisfy thatstability condition. Furthermore,the derivationof a priori
error estimates forStokes problems,conformingand nonconformingfiniteelement discretizations,and anarray of inf-sup finiteelement pairs intwoand three dimensionsare presented. Asanalternativetomixed finiteelementmethods,stabilized finiteelements with equal-order interpolationsthatcircumventthe inf-sup stability are discussed. Hintsonthe choiceofpressurespacesand on linearizationstrategies for theNavier-Stokes problemare given.
Finally, inChapter 9,finiteelementalgorithmsand implementations based on object-oriented conceptsare provided. All finiteelementalgorithmsand computingtoolsthat areneeded inanefficientandrobustobject-oriented finiteelement packageare presented in detail.
We believethatthetheoretical material of Chapters1to4and the implementationmatters of Chapter 9shouldbe presented inany courseonfiniteelements. Theselectionof theother Chapters isamatter of tasteanddependson which application is inthe foreground. We have madeaneffortto writeeach chapterbetweenChapters5and 8asaself-contained unit,so—for example—a readerwho ismainly interested in partialdifferential equationsofhigher order can
concentrateonourdiscussionof the biharmonicequation.
We wish tothank all our colleaguesandfriends, including D.Braess,V. John, P. Knobloch, Q. Lin,G. Matthies,H.-G.Roos,F.Schieweck,M.Stynes, R. Verf ¨ urth,Z. Zhangand A. Zhou, for theirhelp and constructive discussionsonseveral of thesetopics. Also, we wouldliketo thank ourwives Sangeethaand Inge for theirpatientand continuousencouragement. More importantly, wearegrateful toall ourfundingagencies(in particular theAlexander von Humboldt(AvH) foundationand GermanAcademic Exchange Service(DAAD)) for their generoussupport. Finally, we wouldliketoexpressour appreciationtoCambridgeUniversity Pressand IndianInstituteofScience(IISc) Press for their cooperation inthe productionand publishingof this book.
INTRODUCTIONTOSOBOLEV SPACES
Inthischapterwe recall some basicson functional analysisandprov idea brief introductionto
Sobolevspaces. For amore detailed and comprehensivestudy, we refer toAdams(1975).
1.1.Banach andhilbertspaces
Definition 1.1. Let X bea reallinear space. Amapping · : X → R iscalled anormon X if
• x = 0 ⇔ x = 0 for all x ∈ X ,
• λx =|λ| x for all x ∈ X , λ ∈ R,
• x + y ≤ x + y for all x , y ∈ X
The pair (X , · ) iscalled anormed space.
Setting y =−x inthe inequalit y and usingtheother two proper ties weget x ≥ 0 for all x ∈ X .
Definition 1.2. Asequence(xn )n∈N iscalled Cauchy sequence, iffor all ε> 0thereexistsan index n0 (ε )such that for all m, n > n0 (ε ) it holds xm xn <ε .
Definition 1.3. Asequence(xn )n∈N convergesto x ∈ X iffor all ε> 0there isan index n0 (ε ) such that for all n > n0 (ε ) it holds xn x <ε .
Definition 1.4. Anormed space(X , · ) iscalled complete if every Cauchy sequence in X converges in X .
Definition 1.5. Acompletenormed space iscalledBanach space.
Example1.1. Let ⊂ Rd , d isthe dimension, beanopenandboundeddomain We denote by L p ( ),1 ≤ p < ∞,thesetof all measurable functions f : → R forwhich
|f (x )|p dx < ∞.
Similarly,thesetof all measurable functions f : → R satisfy ing
esssup{|f (x )| : x ∈ } < ∞
w illbe denotedby L ∞ ( ). Then, L p ( ), p ∈ [1, ∞] isa reallinear space.Setting for 1 ≤ p < ∞
f L p ( ) := |f (x )|p dx 1/p and f L ∞ ( ) := esssup{|f (x )| : x ∈ }, respectively, L p ( ) becomesanormed space for any p ∈ [1, ∞] ifwe identifyfunctions which areequal up toasetof measurezero. This identification isnecessarybecause weonlyhave for a set M ⊂ of measurezero
|f (x )|p dx = 0 ⇒ f (x ) = 0 for all x ∈ \M .
Strictly speaking,theelements in L p ( ), p ∈ [1, ∞]areclassesof equivalent functions(equal up toasetof measurezero). Then,thefirstcondition in Definition1.1 becomestrue. Thesecond condition follows directlyfromthe definitionof · L p ( ) and thethird condition is justthe inequalit y stated inLemma1.1.
Lemma 1.1 (Minkowski’s inequalit y). Forf , g ∈ L p ( ),p ∈ [1, ∞],wehave f + g L p ( ) ≤ f L p ( ) + g L p ( ) .
Theorem 1.1. L p ( ),p ∈ [1, ∞],isaBanachspace.
Lemma 1.2 (Hölders ’ s inequalit y). Forf ∈ L p ( ) andg ∈ L q ( ) withp , q ∈ [1, ∞], 1/p + 1/q = 1,wehavefg ∈ L 1 ( ) and fg L 1 ( ) ≤ f L p ( ) g L q ( )
Definition 1.6. Twonorms · 1 and · 2 onanormed space X arecalled equivalent if there exist positiveconstants C1 , C2 such that C1 x 1 ≤ x 2 ≤ C2 x 1 for all x ∈ X .
Notethattopolog icalproper ties donotchange whensw itchingtoanequivalentnorm, for example,asequence isconvergent in(X , · 1 ) iff it isconvergent in(X , · 2 ).
Definition 1.7. Let(X , · X ) beanormed space Amapping g : X → R iscalledlinear if g (α x + β y ) = α g (x ) + β g (y ) for all α , β ∈ R, x , y ∈ X .
A linear mapping g : X → R iscontinuous if there isaconstant C such that |g (x )|≤ C x X for all x ∈ X
Definition 1.8. We definethesumof twocontinuous linearfunctionals, g1 and g2 ,and the multiplicationof acontinuous linearfunctional g w ith a real number α by
(g1 + g2 )(x ):= g1 (x ) + g2 (x )and (α g )(x ):= α g (x ), α ∈ R, x ∈ X .
Then, itcan beshow nthatthesetof all continuous linearfunctionalscreatea reallinear space, the dual space X ∗ . Inthe follow ing, for theelements g ∈ X ∗ weusethenotation g , x := g (x ).
Lemma 1.3. ThesetX ∗ ofcontinuouslinearfunctionalsx → g , x onXisaBanachspacewith respecttothenorm g X ∗ := sup 0=x ∈X | g , x | x X .
Proof. Usingthe definitionof the dual norm · X ∗ , wesee g X ∗ = 0
Fur ther, we have
=x
X
.
, x | x X =|λ| sup 0=x ∈X | g , x | x X =|
| g X ∗ andfor thesumof two functionals it holds g1 + g2 X ∗ = sup 0=x ∈X | g1 + g2 , x | x X = sup 0=x ∈X | g1 , x + g2 , x | x X ≤ sup 0=x ∈X | g1 , x | x X + sup 0=x ∈X | g2 , x | x X = g1 X ∗ + g2 X ∗
It remainstoshow that(X ∗ , · X ∗ ) iscomplete. Let(gn )n∈N beaCauchy sequence in X ∗ ,that
is, for all ε> 0there isan index n0 (ε )such that for all m, n > n0 (ε ) it holds gm gn X ∗ <ε
Then, for afixed x ∈ X ,thesequenceofreal numbers( gn , x )n∈N isaCauchy sequence in R dueto
| gm , x − gn , x |=| gm gn , x |≤ gm gn X ∗ x X <ε x X
for all m, n > n0 (ε ). Any Cauchy sequenceofreal numbers isconvergent in R,thatmeans limn→∞ gn , x = g (x ) for all x ∈ X The linearit y of g follows from g (α x + β y ) = lim n→∞ gn , α x + β y = lim n→∞ α gn , x + β gn , y = α lim n→∞ gn , x + β lim n→∞ gn , y = α g (x ) + β g (y )
Concerningthecontinuit y of g we recall thatCauchy sequencesare bounded,thus
| gn , x |≤ gn X ∗ x X ≤ C x X for all x ∈ X .
The limit n →∞ showsthatthe linear mapping g : X → R iscontinuous
Definition 1.9. Let V bea linear space. Amapping( , ): V × V → R iscalled an innerproduct on V if
• (u , v ) = (v , u ) for all u , v ∈ V ,
• (α u + β v , w ) = α (u , w ) + β (v , w ) for all u , v , w ∈ V , α , β ∈ R,
• (u , u ) > 0 for u = 0.
Anormon V is inducedby setting u := √(u , u ). If (V , · V ) iscomplete wecall V aHilber t space.
Example1.2. V = L 2 ( )equippedw ith the innerproduct
(f , g ):= f (x )g (x ) dx
isaHilber tspace.
Theorem 1.2 (Riesz representationtheorem). LetVbeaHilbertspace.Thereisauniquelinear mappingR : V ∗ → VfromthedualspaceV ∗ intotheHilbertspaceVsuchthatforallg ∈ V ∗ and v ∈ V
(Rg , v ) = g , v and Rg V = g V ∗ .
Fixedpointtheoremsareoftenused toestablish theuniquesolvabilit y of anabstractoperator equation. Letusconsider the follow ing problem ina Banach space V w ith anoperator P : V → V :
Find u ∈ V such that u = Pu. (1.1)
Solutions u ∈ V ofProblem(1.1)arecalled fixedpointsof themapping P : V → V .
Definition 1.10. Themapping P : V → V iscalled contractiveor to beacontraction if there is a positiveconstant ρ< 1such that Pv1 Pv2 V ≤ ρ v1 v2 V for all v1 , v2 ∈ V
Theorem 1.3 (Banach’sfixedpointtheorem). LetVbeaBanachspaceandP : V → Vbea contraction.Then,wehavethefollowingstatements:
(1) Thereisauniquefixedpointu∗ ∈ VsolvingProblem (1.1). (2) Thesequenceun = Pun 1 ,n ≥ 1,convergestothefixedpointu ∗ ∈ Vforanyinitialguess u0 ∈ V.
(3) Itholdstheerrorestimate
un u ∗ V ≤ ρ n 1 ρ Pu0 u0 V foralln ∈ N.
23:26:21
Proof. Westar t by estimatingthe distanceof ui ∈ V to ui 1 ∈ V ui ui 1 V = Pui 1 Pui 2
Fromthat weget for m > n ≥ 1
<ε ,
consequently,(un ) isaCauchy sequenceand convergestosome u ∗ ∈ V . Weshow that u ∗ isa
fixedpointof P . Indeed,
Next we provethatthere isatmostonefixedpoint. Assumethattherearetwofixedpoints
u ∗ 1 = u ∗ 2 Then, weconclude u ∗
Since ρ< 1 weget u ∗ 1 = u ∗ 2 . Finally, we recall theestimate for m > n ≥ 1 un u ∗ V ≤ un um V + um u ∗ V ≤ ρ n 1 ρ Pu0 u0 V + um u ∗ V
from which the laststatement follows by m →∞.
1.2.Weakderivatives
First we introduceageneralizationof the integ ration bypar ts formula know n for smooth functions u , v :[a , b ] → R: b a u v dx =
Westar t w ith acharacterizationof geometric proper tiesof a domain ⊂ Rd
Definition 1.11. The domain ⊂ Rd issaid to haveaLipschitzcontinuous boundary, iffor
everypointof the boundary = ∂ thereexistsa local coordinatesystem in which the boundary correspondstosome hypersurface w ith the domain ly ingononesideof thatsurfaceand
∂ can locallybe representedby theg raph of aLipschitzcontinuousmapping .
Theorem 1.4 (Gausstheorem). Let ⊂ Rd beaboundeddomainwithLipschitzcontinuous boundary andouternormal n.Then,for w ∈ C 1 ( )d wehave div w dx = d i =1
wi
xi dx = w n dγ
Let j ∈{1,2, ... , d } beafixed index. The formulaof integ ration bypar ts follows by setting wi = uv for i = j and wi = 0 for i = j ,and usingthe product rule.
Lemma 1.5 (Integ ration bypar ts). Foru , v ∈ C 1 ( ) wehave ∂
xj dx , j = 1, , d .
Nowwe havethetoolstogeneralizetheconceptofdifferentiabilit y.
Definition 1.12. Thesuppor tof a function f : → R is definedby supp f = {x ∈ : f (x ) = 0}.
Thesetof allfunctionsthatare differentiableof any orderw ith compactsuppor t in w illbe denotedby C ∞ 0 ( ).
For = (0,1),the function x → x (1 x ) doesnot belongto C ∞ 0 ( ) because itssuppor t is[0,1] ⊂ (0,1). Indeed, functions in C ∞ 0 ( )arezero intheneig hbourhood of the boundary
∂ . Asaconsequence,a function f ∈ C ∞ 0 ( )can beextended toasmoothfunction defined on Rd by setting f = 0outsideof .
Definition 1.13. We define
L 1 loc ( ):={f : → R : f ∈ L 1 (A) for all compact A ⊂ }.
Forboundeddomains , we have L 2 ( ) ⊂ L 1 ( ) ⊂ L 1 loc ( )
Example1.3. Consider = (0,1). Themapping x → 1/x belongsto L 1 loc ( ) butnotto L 1 ( ). Themapping x → 1/√x belongsto L 1 ( ) butnotto L 2 ( ).
Definition 1.14. Let α = (α1 , , αd ) beamulti-index,that is, αi arenon-negative integersand
|α |= α1 +···+ αd . A function, f ∈ L 1 loc ( ) hasa weakderivative v = D α f ∈ L 1 loc ( ) iffor any
ϕ ∈ C ∞ 0 ( )
v ϕ dx = ( 1)|α | fD α ϕ dx .
Thenotionof a weakderivativegeneralizestheclassicalpar tialderivatives. Indeed,a par tial derivative intheclassical sensesatisfiestheabovecondition bypar tial integ ration.
Example1.4. Let f :( 1, +1) → R w ith f (x ) = 2 √|x | Then,thegeneralizedderivative
Df ∈ L 1 loc ( 1, +1) isg iven by
Proof. Apply ing integ ration byp
( 1, +1)
where weused t
1.3.Sobolevspaces
Definition 1.15. We denote by W m,p ( ),1 ≤ p ≤∞, m ≥ 0,thesetof allfunctions, f ∈ L p ( ) w ithweakderivatives D α f ∈ L p ( )u
isaBanachspace.Inparticular,theSobolevspaceH m ( ):= W m,2 ( ) isaHilbertspacewith respecttotheinnerproduct (u , v )H m ( ) :=
Proof. Here, weg ivethearguments for 1 ≤ p < ∞. For thecase p =∞, we refer toAdams (1975). Thesets W m,p ( )are reallinear spaces(belowweshow thatthesum(f + g )of two
elements f , g ∈ W m,p ( ) isanelementof W m,p ( )). Next we havetoshow thatthemapping f → f W m,p ( ) isanorm,that is,thethreeconditions in Definition1.1 hold. If f = 0,then
f W m,p ( ) = 0 bydefinitionof · W m,p ( ) . Fur ther,since0 = f W m,p ( ) ≥ f L p ( ) wecon-
clude f = 0 inthesenseof L p ( ),that is, functions which areequal up toasetof measurezero
are identified. Thus,thefirstcondition in Definition1.1 holds. Thesecond condition follows
directlyfromthe definition
λf W m,p ( ) = ⎛
|α |≤m |D α λf (x )|p dx ⎞
1/p =
|λ|p |α |≤m |D α f (x )|p dx ⎞ ⎠ 1/p =|λ|
|α |≤m |D α f (x )|p dx ⎞ ⎠ 1/p =|λ| f W m,p ( ) .
Let f , g ∈ W m,p ( )and |α |≤ m. TheMinkowski’s inequalit y,Lemma1.1, impliesthat
D α (f + g ) = D α f + D α g ∈ L p ( )and
D α (f + g ) L p ( ) ≤ D α f L p ( ) + D α g L p ( ) .
UsingtheMinkowski’s inequalit yfor sums, weget
f + g p W m,p ( ) = |α |≤m
D α (f + g ) p L p ( ) ≤ |α |≤m
|α |≤m
D α f p L p ( )
D α f L p ( ) + D α g L p ( ) p
1/p + ⎛ ⎝|α |≤m
D α g p L p ( ) ⎞ ⎠ 1/p ⎤ ⎥ ⎦ p ≤ f W m,p ( ) + g W m,p ( ) p
from which thethird condition follows. It remainsto provethecompletenessof thenormed
space W m,p ( ). Let(fn )n∈N beaCauchy sequence in W m,p ( ). Then,(D α fn )n∈N isaCauchy sequence in L p ( ) for any multi-index α w ith |α |≤ m.Since L p ( ) isa Banach space,there is f α ∈ L p ( ) w ith D α fn → f α in L p ( ) It remainstoshow that f α = D α f where f = f 0 Weshow this for α = (1,0, ,0). For any ϕ ∈ C ∞ 0 ( ) we have (f α ϕ + fD α ϕ )dx ≤ |f α D α fn ||ϕ |dx + (D α fn ϕ + fn D α ϕ )dx + |f fn ||D α ϕ |dx . Thesecond termonthe rig ht hand sideequalszerosince D α fn isthe weakderivativeof fn The convergenceof fn to f and D α fn to f α in L p ( ), respectively, imply theconvergence in L 1 ( ), thusthefirstand third termonthe rig ht hand sidetend tozero for n →∞. The left hand side
isnon-negativeand independentof n,consequentlyweconcludethat it iszero, which means
f α isthe weakderivativeof f .
The proper ties for the innerproduct in H m ( ),see Definition1.9, followfrom its definition
and theobser vationthat(v , v )H m ( ) = v 2 W m,2 ( ) for all v ∈ W m,2 ( ).
Definition 1.16. We denote by W m,p 0 ( )theclosureof C ∞ 0 ( ) inthenorm · W m,p ( ) .
Analogously, H m 0 ( ) istheclosureof C ∞ 0 ( ) in H m ( ).
Inthe follow ing, for f ∈ W m,p (ω ), m ≥ 0,1 ≤ p ≤∞,thenorm w illbeshor tlydenotedby f m,p,ω := f W m,p (ω ) and aseminorm is introducedby |f |m,p,ω := ⎛
ω |α |=m |D α f (x )|p dx ⎞ ⎠ 1/p if 1 ≤ p < ∞
and |f |m,p,ω := max |α |=m esssupx ∈ |D α f (x )| if p =∞.
Incase, ω = and/or p = 2 weomit and p , respectively. Thus, f m = f m,2, , f m,p = f m,p, , |f |m =|f |m,2, , |f |m,p =|f |m,p,
Proof. Supposethe Poincaré inequalit y istrue forfunctions from C ∞ 0 ( ). Then, for any v ∈ W 1,p 0 ( )there isasequence vn ∈ C ∞ 0 ( ) w ith vn → v in W 1,p 0 ( ) We have
v 0,p ≤ v vn 0,p + vn 0,p ≤ v vn 0,p + cP |vn |1,p ≤ (1 + cP ) v vn 1,p + cP |v |1,p .
Since v vn 1,p tendstozero for n →∞,the Poincaré inequalit y istrue forfunctions from
W 1,p 0 ( )
Inorder to provethe Poincaré inequalit yforfunctions v ∈ C ∞ 0 ( ) weextend v : → R by setting v (x ) = 0 for all x ∈ Rd \ . Let ⊂ [ a , +a ] × Rd 1 ,then
v (x ) = x1 a ∂v ∂x1 (s , x2 , ... , xd ) ds
and Hölder’s inequalit yw ith 1/p + 1/q = 1 implies
|v (x )| ≤ x1 a ∂v ∂x1 (s , x2 , ... , xd ) p ds 1/p x1 a 1q ds 1/q , |v (x )|p ≤ (2a )p/q +a a ∂v ∂x1 (s , x2 , , xd ) p ds , +a a |v (x )|p dx1 ≤ (2a )1+p/q +a a ∂v ∂x1 (s , x2 , ... , xd ) p ds , Rd |v (x )|p dx ≤ (2a )p Rd ∂v ∂x1 (x ) p dx .
Using v (x ) = 0 for x ∈ Rd \ the inequalit yfollows.
The Poincaré inequalit y allowstoshow thattheseminorm |·|m,p isequivalenttothenorm
· m,p on W m,p 0 ( ). Let v ∈ W m,p 0 ( ),then D α v ∈ W 1,p 0 ( ) for |α |≤ m 1. Weapply
successively the inequalit y
|D α v |0,p ≤ cP |D α v |1,p for |α |= 0,1, ... , m 1
toestimatethe lowerderivatives by the hig hest derivativeand obtain |v |m,p ≤ v m,p =
|D α v |p 0,p ⎞
1/p ≤ C |v |m,p ,
|α |≤m
that is,theequivalenceof theseminormtothenorm.
An impor tanttoolw illbethe Sobolevembeddingtheorems which statethat functions from W m,p ( ) belongto W m 1,p∗ for some p ∗ > p Moreover, itcan beshow nthat for suitablenumbers m, p functions from W m,p ( ) belongtoclassical spacesof continuousand continuously differentiable functions, respectively.
Definition 1.17. We introducethespaceof Hölder continuous functions w ith Hölder exponent β ∈ (0,1]
C 0,β ( ):= u ∈ C 0 ( ):sup x ,y ∈ ,x =y
|u (x ) u (y )| |x y |β < ∞ equippedw ith thenorm
v C 0,β ( ) = sup x ∈ |u (x )|+ sup x ,y ∈ ,x =y |u (x ) u (y )| |x y |β
Exercise1.1. Consider f :[ 1, +1] → R g iven by f (x ) = √|x |.Provethat f ∈ C 0,1/2 ([ 1, +1]) and thatthere isno β ∈ (1/2, +1] w ith f ∈ C 0,β ([ 1, +1])
Exercise1.2. Let f : R → R beg iven by f (x ) = ⎧
⎩ x 2 sin 1 x 2 for x = 0, 0 for x = 0
Show that f is differentiableatanypoint x ∈ R anddeterminethefirst derivative f (x ). Is f Lipschitzcontinuouson R?
Theorem 1.5 (Sobolevembeddingtheorem). Let ⊂ Rd beadomainwithLipschitz continuousboundary.Then,wehavethefollowingcontinuousembeddingsform ≥ 0, 1 ≤ p ≤∞, W m,p ( ) → L p∗ ( ) with 1 p ∗ = 1 p m d , ifm < d p ,
W m,p ( ) → L q ( ) forallq ∈ [1, ∞), ifm = d p ,
W m,p ( ) → C 0,m d /p ( ), if d p < m < d p + 1,
W m,p ( ) → C 0,α ( ) forall α ∈ (0,1), ifm = d p + 1,
W m,p ( ) → C 0,1 ( ), ifm > d p + 1.
Proof. See,Adams(1975).
The lastthreeembeddings haveto beunderstood inthe follow ingsense A function v ∈ W m,p ( )can bemodified onasetof measurezerotoacontinuous function ˜ v and the ‘embeddingoperator’ iscontinuous,that is, incaseof W m,p ( ) → C 0,α ( ) we have
˜ v C 0,α ( ) ≤ C v W m,p ( ) for all v ∈ W m,p ( ).
Often we donot distinguishbetween v and ˜ v ,andwrite v instead of ˜ v .
Example1.5. Theorem1 5showsthat H 1 functions in1d arecontinuous This isnotthecase inthe2d caseasshow n inthe follow ingexample. Let ={(x1 , x2 ) ∈ R2 : x 2 1 + x 2 2 < 1} and f (r ) = log( log(1/r )) for r = 0. Using polar coordinates(x1 , x2 ) = (r cos ϕ , r sin ϕ ) wecompute
|f |2 1 = |∇ f |2 dx = 2π 0 1 0 ∂f ∂r 2 r dr dϕ = 2π 1 0 dr r log 2 (1/r ) < ∞.
Thus, v ∈ H 1 ( ) but hasasingularit y at r = 0.
Example1.6. Consider theunit ball, ={x ∈ Rd : x 2 1 +···+ x 2 d < 1} and the function f : \{0}→ R w ith f (x ) = ( d i =1 x 2 i )α/2 = r α . We wantto knowforwhich α the function
f belongsto H 1 ( ). Inthecase, d = 2 weuse polar coordinatestoget
|f |2 1 = |∇ f |2 dx = 2π 0 1 0 ∂f ∂r 2 r dr dϕ = 2πα 2 1 0 dr r 1 2α < ∞
prov ided that α> 0. Inthe3d case, d = 3and spherical coordinates(x1 , x2 , x3 ) = (r cos ϕ sin θ , r sin ϕ sin θ , r cos θ ) imply
|f |2 1 = |∇ f |2 dx = 2π 0 π 0 1 0
prov ided that α> 1/2. Thisshowsthat functions f : ⊂ Rd → R belong ingto H 1 ( )can haveasingularit y of t ype r α for α> 1/2 inthe3d case, however,not inthe2d case where α> 0 isneeded. Notethatthesingularit y considered in Example1.5 is weaker thanthesingularit y r α w ith α< 0
Ingeneral, functions in Sobolevspacesare defined up toasetof measurezero.Sincethe d -dimensional measureof = ∂ iszero, itmakesnosense,onthefirstg lance,tospeak on boundary valuesoffunctions v ∈ W m,p ( ). However,onecanshow thatthere isa positive
constant C ( )such that
v 0, := v 2 dγ 1/2 ≤ C ( ) v 1, for all v ∈ C ∞ ( ).
Since C ∞ ( ) is dense in H 1 ( ),this inequalit y statesthatthere isacontinuous linear mapping
tr : H 1 ( ) → L 2 ( ),called thetraceoperator. Inthissense, wesay that functions from H 1 ( ) havetraces(‘boundary values ’) in L 2 ( ). Thisalsog ivesanother characterizationof H 1 0 ( )as H 1 0 ( ) ={v ∈ H 1 ( ): v = 0on }.
Notethatthe imagetr(H 1 ( )) belongsto L 2 ( ) but isa proper subspace in L 2 ( ). For a precise characterizationof the imagetr(W m,p ( )), fractionalSobolevspaces W s ,p ( ),0 < s ,1 ≤ p < ∞ haveto be introduced (seeAdams(1975)).
Theorem 1.6. Let 1 ≤ p < ∞ and ⊂ Rd beaboundeddomainwithLipschitzcontinuous boundary .Thenthetraceoperator, tr : W 1,p ( ) → W 1 1/p,p ( ) issurjectiveandW 1,p 0 ( ) = {v ∈ W 1,p ( ):tr(v ) = 0}.
Proof. SeeTheorem7.53 inAdams(1975).
Asaspecial case weset H 1/2 ( ):= W 1/2,2 ( ). Then,thefirststatementof Theorem1.6
meansthateveryfunction in H 1/2 ( ) isthetraceof a function in H 1 ( ) Thist ypeofresults is
useful incase weseek solutionsofpar tialdifferential equations ina domain thatsatisfy g iven valuesonthe boundary = ∂ . Ingeneral, we have
Corollar y1.1. Let 1 ≤ p < ∞ and ⊂ Rd beaboundeddomainwithLipschitzcontinuous boundary .Then,thereisapositiveconstantCsuchthatforallg ∈ W 1 1/p,p ( ) thereexists
Inthischapter, we introducetheconceptofweak (variational) formulationsandweak solutions of elliptic boundary value problemsof second order. The fundamentalresultsonexistence, uniquenessandwell-posednessofweak solutionsare presented. Asa basictoolfor afinite elementapproximation,thestandard Galerkinmethodfor elliptic problemsand their abstract error estimateare introduced.
Let ⊂ Rd , d isthe dimension, beanopenandboundeddomain w ith Lipschitzcontinuous boundary = ∂ . Weconsider thesecond order ellipticequation d i ,j =1 ∂ ∂xj aij (x ) ∂u ∂xi + d i =1 bi (x ) ∂u ∂xi + c (x )u = f (x ) x ∈ ,(2.1)
where u isanunknow nscalarfunction. Here,the diffusion,advection/convectionandreaction coefficients, aij (x ), bi (x )and c (x ), respectively,and thesourceterm f (x )areassumed to be sufficiently smooth. For continuouslydifferentiable functions u : → R,themixedderivatives satisfy
∂2 u
∂xj ∂xi = ∂2 u ∂xi ∂xj , i , j = 1, , d
Thus, wecanassume w ithout losinggeneralit y thatthematrix aij issy mmetric,that is, aij = aji , i , j = 1, ... , d . Then,the differential equation(2.1) iscalled ellipticat x ∈ if thematrix aij (x ) is positive definite. Weassumethatthe differential equation isuniformly elliptic in ,that is, there isa positiveconstant α0 such that
i ,j =1 aij (x )ξi ξj ≥ α0
i =1 ξ 2 i for all x ∈ , ξ ∈ Rd .
This requirement isslig htly stronger thanassumingthatthe differential equation iselliptic for all x ∈
• Neumann boundary condition d i ,j =1 aij ∂u ∂xi nj = gN on ,
where n = (n1 , ... , nd )T istheoutward normal vector on and gN : → R isag iven
flux,
• Robin boundary condition d i ,j =1 aij ∂u ∂xi nj = σ (u∞ u )on ,
where σ> 0and u∞ areg iven functionson . Inthesimplestcase,thesolutionof theelliptic Equation(2.1)subjecttooneof theabove boundary conditions issoug ht. Inapplications, mixedboundary conditionsmay alsoappear,and in whichdifferentt ypesof conditionsare posed on different par tsof the boundary. Theconceptofweak solutionsof a boundary value problem w illbe discussed next.
2.2.Weaksolution
We beg in w ith the Poisson problem by imposing aij (x ) = δij , i , j = 1, , d , bi (x ) = 0, i = 1, , d , c (x ) = 0 intheelliptic Equation(2.1). Using homogeneous Dirichlet boundary condition, it becomes
u = f in , u = 0on ,(2.2)
wheretheLaplaceoperator isg iven by
u :=
d i =1
∂2 u
∂x 2 i .
We knowfromthetheory ofpar tialdifferential equations, Equation(2.2) hasauniqueclassical
solution u ∈ C 2 ( ) ∩ C ( ) prov ided that f ∈ C ( )and the boundary issmooth enoug h.
However, both thesmoothnessassumptiononthesource f and onthe boundary areoftennot
fulfilled in practical applications. For example,the follow ing problem u = f in = ( 2,2)d , u = 0on
w ith f = 1 + sgn (1 −|x |) doesnot haveaclassical solution u ∈ C 2 ( ) ∩ C ( )sincethesource
f isonlypiecew isecontinuous.
Par tialdifferential equations w ith ‘non-smooth’dataappear inmanypractical applications.
Sinceaclassical solutionof theseequations doesnotexist ingeneral, wearemotivated to relax
the differentiabilit yrequirementsof thesolutionand the data.
Let u ∈ C 2 ( ) ∩ C ( ) beaclassical solutionofEquation(2.2),and assumethat f ∈ C ( ).
Uponmultiply ing Equation(2.2) by a function v ∈ C ∞ 0 ( )and integ ratingover , weobtain
uv dx = fv dx .
Integ rating bypar ts,theLaplaceterm becomes
uv
ni v ds = ∇ u ·∇ v dx .
Here,the boundary integ ral vanishessincethetest function, v vanishesonthe boundary and weend upw ith
∇ u ·∇ v dx = fv dx for all v ∈ C ∞ 0 ( ).
Moreover,theclassical solution u alsosatisfies
∇ u ·∇ v dx = fv dx for all v ∈ H 1 0 ( ), (2.3)
since C ∞ 0 ( ) is dense in H 1 0 ( ). Now,notethatthesolution u need not be in C 2 ( ) ∩ C ( ) for the integ rals in Equation(2.3)to bemeaningful,and it isenoug h toseek thesolution u in thespace H 1 0 ( ).
Definition 2.1. A function u ∈ H 1 0 ( )satisfy ing Equation(2.3) iscalled a weaksolution or a generalizedsolution ofEquation(2.2). Notethatthe par tialderivatives inthe integ ralsof Equation(2.3) haveto beunderstood as weakderivatives.
a (u , v ):= ∇ u ·∇ v dx = fv dx =: F (v ) for all v ∈ V (2 4)
It isnot difficulttocheck thatthemappings a : V × V → R and F : V → R are bilinear and
linear, respectively. Fur ther, it isclearfrom Equation(2.3)that if u istheclassical solutionof
Problem(2.2)then it isalsoa weak solutionofProblem(2.2). However,theconverse isnot
true whenthe weak solution isnotsmooth enoug h.Roug hly speaking, we increasethesetof possiblesolutions(totheclassof generalized solutions) w ithout losingtheuniqueness(as we w ill see below)
Next, weshallprovetheexistenceand uniquenessof a weak solutionof theelliptic
Problem(2.2)usingtheLax–Milg ramtheorem.
Definition 2.2. Let V beaHilber tspace A bilinearform a : V × V → R issaid to be continuous (bounded) if thereexistaconstant β> 0such that |a (u , v )|≤ β u V v V for all u , v ∈ V
and coercive or V -elliptic, if thereexistaconstant α> 0such that
a (u , u ) ≥ α u 2 V for all u ∈ V .
Under theadditional assumptionthatthe bilinearform issy mmetric,that is,
a (u , v ) = a (v , u ) for all u , v ∈ V , themapping(u , v ) → a (u , v ) definesan innerproducton V and theassociated norm v →
√a (v , v ) iscalled energ y norm.
Theorem 2.1 (Lax–Milg ram). LetVbeaHilbertspace, a ( , ) beacontinuous, coercivebilinear form, andF ( · ) beacontinuouslinearfunctional.Then, thereexistsauniqueu ∈ Vsuchthat a (u , v ) = F (v ) forallv ∈ V (2 5)
Proof. For each u ∈ V themapping v → a (u , v ) is linear and continuouson V ,thusthereexists aunique functional Au ∈ V ∗ (V ∗ isthe dual spaceof V )such that a (u , v ) = Au , v for all v ∈ V (2 6)
Fur ther, we have Au V ∗ := sup 0=v ∈V | Au , v |
Consequently,the linear mapping A : V → V ∗ iscontinuous w ith theoperator norm A L(V ,V ∗ ) ≤ β .
Now, fromTheorem1.2(Riesz representationtheorem),there isaunique linear mapping R : V ∗ → V such that for each f ∈ V ∗ (Rf , v ) = f , v , Rf V = f V ∗ for all v ∈ V . (2.7)
solution u ∈ V . We provethis by show ingthat, for anappropriatevalueof ρ> 0,themapping
T : V → V w ith
T (v ):= v ρ (RAv Rf ) for all v ∈ V
iscontractive. If T isacontraction,thenthereexistsaunique u ∈ V such that T (u ) = u ρ (RAu Rf ) = u ,that is, RAu = Rf Toshow that T isacontraction, let v = v1 v2 , for any v1 , v2 ∈ V . Then, Tv1 Tv2 2 V = v1 v2 ρ (RAv1 RAv2 ) 2 V = v ρ (RAv ) 2 V = v 2 V 2ρ (RAv , v ) + ρ 2 RAv 2 V = v 2 V 2ρ Av , v + ρ 2 Av , RAv = v 2 V 2ρ a (v , v ) + ρ 2 a (v , RAv ) ≤ v 2 V 2ρα v 2 V + ρ 2 β v V RAv V ≤ (1 2ρα + ρ 2 β 2 ) v 2 V = (1 2ρα + ρ 2 β 2 ) v1 v2 2 V .
Therefore,themapping T iscontractive prov ided (1 2ρα + ρ 2 β 2 ) < 1,that is, ρ ∈ (0,2α/β 2 ), and thiscompletesthe proof.
Remark 2.1. It follows fromthecoerciv it y of the bilinearformand (2.5)that u V ≤ 1 α f V ∗ ,
where α isthecoerciv it y constant. Therefore,the weakform(2.5) is well-posed inthesensethat it hasauniquesolution, whichdependscontinuously onthe data f .
Proof. To provetheexistenceof the weak solution, weapply Theorem2 1(Lax–Milg ram)
Therefore, it isenoug h toshow that a ( , )and F ( ) defined in Problem(2.4)satisfy the assumptionsof theLax–Milg ramtheorem.
For afixed u ∈ V ,themapping v → a (u , v ) is linear,and similarly, for afixed v ∈ V the mapping u → a (u , v ) is linear. Therefore, a (· , ·) is bilinear. Apply ing Schwarz inequalit y,first for integ ralsand then for sums, we have |a (u , v )|= ∇ u ·∇ v dx ≤ d i =1
u ∂xi
v ∂xi dx
u |1 |v |1 ≤ u V v V .
Thus,the bilinearform a (· , ·) iscontinuous w ith β = 1. Next,thecoerciv it y of a (· , ·) follows from
where inthe laststep,thenormequivalenceof theseminormtothenorm in H 1 0 ( ) has been
used. Now,coerciv it yfollows by setting α = 1/(1 + c 2 P ). It remainsto provethat F ( · ) is linear andbounded. Linearit y of F is immediate,the boundednessof F follows for f ∈ L 2 ( ) by means ofSchwarz inequalit y |F (v )|= fv dx ≤ f 0 v 0 ≤ f 0 v V
Hence,the bilinearform a (· , ·) iscontinuousand coercive,the linearform F ( · ) iscontinuous.
Theexistenceof auniquesolution forProblem(2 2) which satisfies Problem(2 4) follows from
Theorem2.1(Lax–Milg ram).
Example2.1. Wenow consider amoregeneral ellipticequation. Let ⊂ Rd bea bounded
domain w ith a boundary split intomutuallydisjoint par tsofDirichlet,NeumannandRobin t ype. Consider theellipticequationof second orderw ith mixedboundary conditions
u + b ·∇ u + cu = f in , u = 0on D , ∂u ∂n = g on N , ∂u ∂n = σ (u∞ u )on R (2 8)
The data, b = (b1 , , bd ), c , σ areassumed to besufficiently smooth, f ∈ L 2 ( ), g ∈ L 2 ( N ), and u∞ ∈ L 2 ( R ). Fur thermore, wesupposethatthe(d 1)-dimensional measureof D is positive.
Wenowdefine
V := H 1 D = v ∈ H 1 ( ): v = 0on D .
Notethatonthesubspace V ⊂ H 1 ( ),theseminorm |·|1 isequivalenttothenorm · 1
As inthecaseof the Poissonequation, wemultiply the differential equation by atest function v ∈ V which vanisheson D , integ rateover , integ rate bypar tstheLaplaceterm,and finally incorporatetheNeumannandRobint ype boundary conditions
uv dx = ∇ u ·∇ v dx N ∪ R ∂u ∂n v ds = ∇ u ·∇ v dx + R σ uv ds R σ u∞ v ds N gv ds .
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nerve, peripheral paralysis of (Bell's palsy), 1202
Faradic reactions, significance, in diagnosis of nervous lesions, 66
Fasting girls, 351
saints,
351
Fatty heart, in chronic alcoholism, 611
Febrile insomnia,
379
Feeding, forced, in neurasthenia,
359
Feet, condition of, in advanced tabes dorsalis, 838 in diffuse sclerosis,
Festination in paralysis agitans, 436
Fever, thermic, 388 and cerebral anæmia, 779
Fibrillary contractions in nervous diseases, 47
Fifth nerve, neuralgias of, 1233
Flexibility, wax-like, in catalepsy, 321 , 337
Fontanels, state of, in chronic hydrocephalus, 741 , 742 in tubercular meningitis, 727
Frauds of hysterical patients, 233 ,
Frequency of epileptic attacks, 483
Friedreich's disease, 870
Front-tap, in diffuse spinal sclerosis, 888
Functional nervous diseases, diagnosis of, 63
Gait in chronic lead-poisoning,
686 in diffuse sclerosis, 888 in general paralysis of the insane, 194 in paralysis agitans, 435 in spastic spinal paralysis, 863 , 864 in tabes dorsalis,
Galvanic reactions, significance of, in diagnosis of nervous lesions, 66
Galvanism, use in insanity, 137
Ganglionic elements, condition of, in abscess of the brain, 793
Gangrene in vaso-motor neuroses, 1252
Definition, history, and synonyms, 1257
Duration, course, and nature, 1261
Etiology and diagnosis, 1262 , 1263 of Reynaud's disease, 1259 , 1260
Prognosis, 1262
Symmetrical, 1257
Symptoms, 1258-1260
Treatment, 1262
Electricity, 1262
Massage, rest, and tonics, use, 1262
Opium, use, 1262
Gastric crises in tabes dorsalis, 835
Disorders, as a cause of epilepsy,
Vertigo, 420
Gastralgia (see Neuralgia ).
Gelsemium, use, in neuralgia, 1225 in painless facial spasm, 462
General paralysis of the insane, 176
Genital organs, disorders of, in chronic alcoholism, 614
Genito-urinary disorders of the chloral habit, 662 , 663 of the opium habit, 654 , 658
Genius, relations of, to insanity, 115
Glandular system, atrophy of, 1268
Gliomatous tumors of the brain, 1046
Gold, use of, in hemiplegia, 970 and sodium chloride, use of, in hysteria, 278
Gout, relation of, to chronic lead-poisoning, 685 to hysteria,
