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A Nonconforming Finite Element Approximation for the von Karman Equations
ARTICLE in ESAIM MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS · JUNE 2015
Impact Factor: 1.64 · DOI: 10.1051/m2an/2015052 · Source: arXiv
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Gouranga Mallik
Indian Institute of Technology Bombay
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Mathematical Modelling and Numerical Analysis Will be set by the publisher Mod′e lisation Math′e matique et Analyse Num′e rique
A NONCONFORMING FINITE ELEMENT APPROXIMATION FOR THE
VON KARMAN EQUATIONS
Gouranga Mallik1and Neela Nataraj2
Abstract.In this paper,a nonconforming?nite element method has been proposed and
analyzed for the von K′a rm′a n equations that describe bending of thin elastic plates.Optimal
order error estimates in broken energy and H1norms are derived under minimal regularity
assumptions.Numerical results that justify the theoretical results are presented.
1991Mathematics Subject Classi?cation.35J61,65N12,65N30.
.
1.Introduction
Let??R2be a polygonal domain with boundary??.Consider the von K′a rm′a n equations for the de?ection of very thin elastic plates that are modeled by a non-linear system of fourth-order partial di?erential equations with two unknown functions de?ned by:for given f∈L2(?),seek the vertical displacement u and the Airy stress function v such that
?2u=[u,v]+f
?2v=?1
2[u,u]
in?(1.1)
with clamped boundary conditions
u=?u
?ν
=v=
?v
?ν
=0on??,(1.2)
where the biharmonic operator?2and the von K′a rm′a n bracket[·,·]are de?ned by ?2?:=?xxxx+2?xxyy+?yyyy and[η,χ]:=ηxxχyy+ηyyχxx?2ηxyχxy=cof(D2η):D2χ,
cof(D2η)denotes the co-factor matrix of D2ηandνdenotes the unit outward normal to the boundary ??of?.
Depending on the thickness to length ratio,several plate models have been studied in literature; the most important ones being linear models like Kirchho?and Reissner-Mindlin plates for thin and moderately thick plates respectively;and non-linear von K′a rm′a n plate model for very thin plates. Many practical applications deal with the Kirchho?model for thin plates in which the transverse Keywords and phrases:von K′a rm′a n equations,Morley element,plate bending,non-linear,error estimates
1Department of Mathematics,Indian Institute of Technology Bombay Powai,Mumbai,India-400076.Email.
gouranga@math.iitb.ac.in
2Department of Mathematics,Indian Institute of Technology Bombay Powai,Mumbai,India-400076.Email.
neela@math.iitb.ac.in
c EDP Sciences,SMAI1999 This provisional PDF is the accepte
d version. Th
e article should be cited as: ESAIM: M2AN,doi:10.1051/m2an/2015052
2TITLE WILL BE SET BY THE PUBLISHER
shear deformation is negligible.On the other hand,the Reissner-Mindlin plate model for moderately thick plates takes into consideration the shear deformation.The displacements of very thin plates are so large that a non-linear model is essential to consider the membrane action.The assumptions made in the von K′a rm′a n model are similar to those of Kirchho?model except for the linearization of the strain tensor,which in fact,leads to the non-linearity in the model.
For the theoretical study as regards the existence of solutions,regularity and bifurcation phenomena of von K′a rm′a n equations,see[2,4–6,14,19]and the references therein.Due to the importance of the problem in application areas,several numerical approaches have also been attempted in the past. The major challenges are posed by the non-linearity and the higher order nature of the equations. The convergence analysis and error bounds for conforming?nite element methods are analyzed in[12]. The papers[22,25]and[24]investigate and analyze the Hellan-Hermann-Miyoshi mixed?nite element method and a stress-hybrid method,respectively for the von K′a rm′a n equations.In these papers, the authors simultaneously approximate the unknown functions and their derivatives.The papers [12,22,24]deal with the approximation and error bounds for isolated solutions,thereby not discussing the di?culties arising from the non-uniqueness of the solution and the bifurcation phenomena.
Over the last few decades,the?nite element methodology has developed in various directions. For higher-order problems,nonconforming methods and discontinuous Galerkin methods are gaining popularity as they have a clear advantage over conforming?nite elements with respect to simplicity in implementation.In this paper,an attempt has been made to study the von K′a rm′a n equations using nonconforming Morley?nite elements.The Morley?nite element method has been proposed and analyzed for the biharmonic equation in[21]and for the Monge-Amp`e re equation in[23].In[26], a two level additive Schwarz method for a non-linear biharmonic equation using Morley elements is discussed under the assumption of smallness of data.The C0interior penalty method,a variant of the discontinuous Galerkin method has been used to analyze the Monge-Amp`e re equation in[9].
The solutions u,v of clamped von K′a rm′a n equations de?ned on a polygonal domain belong to
H20(?)∩H2+α(?)[6],whereα∈(1
2,1]referred to as the index of elliptic regularity is determined by
the interior angles of?.Note that when?is convex,α=1.This paper discusses a nonconforming ?nite element discretization of(1.1)-(1.2)and develops a priori error estimates for the displacement and Airy stress functions in polygonal domains with possible corner singularities.To highlight the contributions of this work,we have
?obtained an approximation of an isolated solution pair(u,v)of(1.1)-(1.2)using nonconforming Morley elements;
?developed optimal order error estimates in broken energy and H1norms under realistic regu-larity assumptions;
?performed numerical experiments that justify the theoretical results.
The advantages of the method are that the nonconforming Morley elements which are based on piece-wise quadratic polynomials are simpler to use and have lesser number of degrees of freedom in com-parison with the conforming Argyris?nite elements with21degrees of freedom in a triangle or the Bogner-Fox-Schmit?nite elements with16degrees of freedom in a rectangle.Moreover,the method is easier to implement than mixed/hybrid?nite element methods.
The di?culties due to non-conformity of the space increases the technicalities in the proofs of error estimates.Moreover,one loses the symmetry property with respect to all the variables in the discrete formulation for nonconforming case.An important aid in the proofs is a companion conforming operator,also known in the literature as the enriching operator which maps the elements in the nonconforming?nite element space to that of the conforming space.Also,as proved in[17]for the biharmonic problem,it is true that when Morley?nite elements are used for the von K′a rm′a n equations, the L2error estimates cannot be further improved.This is evident from the results of the numerical experiments presented in Section5.
The paper is organized as follows.Section1is introductory and Section2introduces the weak for-mulation for the problem.This is followed by description of nonconforming?nite element formulation
TITLE WILL BE SET BY THE PUBLISHER3 in Section3.Section4deals with the existence of the discrete solution and the error estimates in broken energy and H1norms.The results of the numerical experiments are presented in Section5. Conclusions and perspectives are discussed in Section6.The analysis of a more generalized form of (1.1)-(1.2)is dealt with in Appendix A.
Throughout the paper,standard notations on Lebesgue and Sobolev spaces and their norms are employed.We denote the standard L2scalar or vector inner product by(·,·)and the standard norm on H s(?),for s>0by · s.The positive constants C appearing in the inequalities denote generic constants which may depend on the domain?but not on the mesh-size.
2.Weak formulation
The weak formulation corresponding to(1.1)-(1.2)is:given f∈L2(?),?nd u,v∈V:=H20(?) such that
a(u,?1)+b(u,v,?1)+b(v,u,?1)=l(?1)??1∈V(2.1a)
a(v,?2)?b(u,u,?2)=0??2∈V(2.1b) where?η,χ,?∈V,
a(η,χ):=
?
?D2η:D2χdx,b(η,χ,?):=
1
2
?
?
cof(D2η)Dχ·D?dx and l(?):=(f,?).
Note that b(·,·,·)is derived using the divergence-free rows property[15,23].Since the Hessian matrix D2ηis symmetric,cof(D2η)is symmetric.Consequently,b(·,·,·)is symmetric with respect to the second and third variables,that is,b(η,ξ,?)=b(η,?,ξ).Moreover,since[·,·]is symmetric,b(·,·,·)is symmetric with respect to all the variables in the weak formulation.
An equivalent vector form of the weak formulation which will be also used in the analysis is de?ned as:for F=(f,0)with f∈L2(?),seekΨ=(u,v)∈V:=V×V such that
A(Ψ,Φ)+B(Ψ,Ψ,Φ)=L(Φ)?Φ∈V(2.2) where?Ξ=(ξ1,ξ2),Θ=(θ1,θ2)andΦ=(?1,?2)∈V,
A(Θ,Φ):=a(θ1,?1)+a(θ2,?2),(2.3)
B(Ξ,Θ,Φ):=b(ξ1,θ2,?1)+b(ξ2,θ1,?1)?b(ξ1,θ1,?2)and(2.4)
L(Φ):=(f,?1).(2.5) It is easy to verify that the bilinear forms A(·,·)and B(·,·,·)satisfy the following continuity and coercivity properties.That is,there exist constants C such that
A(Θ,Φ)≤C|||Θ|||
2|||Φ|||
2
?Θ,Φ∈V,(2.6)
A(Θ,Θ)≥C|||Θ|||2
2
?Θ∈V,(2.7)
B(Ξ,Θ,Φ)≤C|||Ξ|||
2|||Θ|||
2
|||Φ|||
2
?Ξ,Θ,Φ∈V,(2.8)
where the product norm|||Φ|||
2:=
A(Φ,Φ)?Φ∈V.In the sequel,the product norm de?ned on
(H s(?))2and(L2(?))2are denoted by|||·|||
s and|||·|||,respectively.
For the results on existence of solution of the weak formulation,we refer to[2,3,14,19].More precisely,the weak solutionΨ=(u,v)of(1.1)-(1.2)can be characterized as the solution of the operator equation IΨ=TΨde?ned on V where T is a compact operator on V and I is an identity operator on V.In[19],it has been proved that there exists at least one solution of the operator equation.Also, the uniqueness of solution under the assumption on smallness of the data function f has been derived.
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a
2
Figure1.Morley element
In this paper,we follow[12]and assume that the solutionΨ=(u,v)is isolated.That is,the linearized problem de?ned by:for given G=(g1,g2)∈(L2(?))2?V ,seekΘ=(θ1,θ2)∈V such that
A(Θ,Φ)=(G,Φ)?Φ∈V(2.9) where A(Θ,Φ):=A(Θ,Φ)+B(Ψ,Θ,Φ)+B(Θ,Ψ,Φ)is well posed and satis?es the a priori bounds
|||Θ|||
2≤C|||G|||,|||Θ|||
2+α
≤C|||G|||(2.10)
whereαis the index of elliptic regularity.
3.Nonconforming Finite Element Method(NCFEM)
In the?rst subsection,the Morley element is de?ned and some preliminaries are introduced.In the second subsection,nonconforming?nite element formulation for von K′a rm′a n equations and the corresponding linearized problem are presented.Some properties and auxiliary results necessary for the analysis are discussed in the third subsection.
3.1.The Morley Element
Let T h be a regular,quasi-uniform triangulation[10,13]ofˉ?into closed triangles.Set h T=
diam(T)?T∈T h and h=max T∈T
h h T.For T∈T h with vertices a i=(x i,y i),i=1,2,3,let
m4,m5and m6denote the midpoints of the edges opposite to the vertices a1,a2and a3respectively (see Figure1).We denote the set of vertices(resp.edges)of T h by V h(resp.E h).For e∈E h,let h e=diam(e).
De?nition3.1.[13]The Morley?nite element is a triplet(T,P T,ΦT)where
?T is a triangle
?P T=P2(T)is the space of all quadratic polynomials on T and
?ΦT={φi}6i=1are the degrees of freedom de?ned by:
φi(v)=v(a i),i=1,2,3andφi(v)=?v
?ν
(m i),i=4,5,6.
The nonconforming Morley element space associated with the triangulation T h is de?ned by
V h:=
?∈L2(?):?|T∈P2(T)?T∈T h,?is continuous at the vertices{a i}3i=1of the triangle and the normal derivatives of?at the midpoint of the edges{m i}6i=4are continuous,
?=0at the vertices on??,
??
?ν
=0at the midpoint of the edges on??
.
TITLE WILL BE SET BY THE PUBLISHER 5
For ?∈V h and Φ=(?1,?2)∈V h :=V h ×V h ,the mesh dependent semi-norms which are equivalent to the norms denoted as |?|2,h and |||Φ|||2,h ,respectively,are de?ned by:
|?|22,h :=
T ∈T h |?|22,T ,|||Φ|||2
2,h :=|?1|22,h +|?2|22,h .Also,for a non-negative integer m,1≤p <∞and ?∈W m,p (?;T h )
|?|2m,p,h :=
T ∈T h |?|2m,p,T , ? 2m,p,h := T ∈T h ? 2m,p,T ,and for p =∞
|?|m,∞,h :=max T ∈T h |?|m,∞,T , ? m,∞,h :=max T ∈T h
? m,∞,T ,where |·|m,p,T and · m,p,T denote the usual semi-norm and norm in the Banach space W m,p (T )and W m,p (?;T h )denotes the broken Sobolev space with respect to the mesh T h .For Φ=(?1,?2)
with ?1,?2∈W m,p (?;T h ),de?ne |||Φ|||2m,p,h :=|?1|2m,p,h +|?2|2m,p,h .When p =2,the notation is
abbreviated as |·|m,h and · m,h .
3.2.Nonconforming Finite Element Formulation
The NCFEM formulation corresponding to (2.1a)-(2.1b)can be stated as:for f ∈L 2(?),seek (u h ,v h )∈V h such that
a h (u h ,?1)+
b h (u h ,v h ,?1)+b h (v h ,u h ,?1)=l h (?1)??1∈V h
(3.1a)a h (v h ,?2)?b h (u h ,u h ,?2)=0??2∈V h
(3.1b)where ?η,χ,?∈V h ,
a h (η,χ):=
T ∈T h ?T D 2η:D 2χdx ,b h (η,χ,?):=12 T ∈T h ?T cof(D 2η)Dχ·D?dx and l h (?):= T ∈T h ?
T
f?dx .As in the continuous formulation,the discrete form b h (·,·,·)is symmetric with respect to the second and third variables.However,unlike in the conforming case [12],b h (·,·,·)is not symmetric with respect to the ?rst and second variables or the ?rst and third variables.The equivalent vector form corresponding to (3.1a)-(3.1b)is given by:seek Ψh =(u h ,v h )∈V h such that
A h (Ψh ,Φ)+
B h (Ψh ,Ψh ,Φ)=L h (Φ)
?Φ∈V h (3.2)where ?Ξ=(ξ1,ξ2),Θ=(θ1,θ2)and Φ=(?1,?2)∈V h ,
A h (Θ,Φ):=a h (θ1,?1)+a h (θ2,?2),
(3.3)B h (Ξ,Θ,Φ):=b h (ξ1,θ2,?1)+b h (ξ2,θ1,?1)?b h (ξ1,θ1,?2)and
(3.4)L h (Φ):=
T ∈T h ?
T f?1dx .(3.5)
The nonconforming ?nite element formulation corresponding to (2.9)reads as:for given G ∈(L 2
(?))2,?nd Θh ∈V h such that A h (Θh ,Φ)=(G,Φ)?Φ∈V h (3.6)
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where A h(Θh,Φ):=A h(Θh,Φ)+B h(Ψ,Θh,Φ)+B h(Θh,Ψ,Φ)and A h(·,·),B h(·,·,·)are de?ned in (3.3)and(3.4),respectively.
3.3.Auxiliary Results
In this subsection,some auxiliary results which are essential for the analysis are stated.
Lemma3.2.(Integral average)[7]The projection P e:L2(T)?→P0(e)de?ned by P e?=1
h e
?
e
?ds,
satis?es
??P e? 0,e≤Ch1/2
T
|?|1,T??∈H1(T).(3.7) Lemma3.3.(Interpolant)[11,13,20]LetΠh:V?→V h be the Morley interpolation operator de?ned by:
(Πh?)(p)=?(p)?p∈V h,
?e ?Πh?
?ν
ds=
?
e
??
?ν
ds?e∈E h.
Then for?∈H2+α(?),α∈(0,1],it holds:
??Πh? m,p,h≤Ch1+α?m+2p ? 2+α,0≤m≤2,1≤p<∞.
For simplicity of notation,the interpolant ofΦ∈V is denoted byΠhΦand belongs to V h. Lemma 3.4.(Enrichment function)[11]Let V c be chosen as Hsieh-Clough-Tocher macro element space[11,13]which is a conforming relative of the Morley?nite element space V h.For any?∈V h, there exists E h?∈V c?V such that
T∈T h
h?4
T
|??E h?|20,T+h?2
T
|??E h?|21,T
+|E h?|22,h≤C|?|22,h.(3.8)
Again,forΦ∈V h,the enrichment function corresponding toΦdenoted by E hΦ,belongs to V.
In the next lemma,we establish an imbedding result.A similar result has been proved in[26, Lemma3.1]for the case of convex polygonal domains.However,for the sake of completeness,we provide a detailed proof for the case of polygonal domains.Note that only the edge estimation in (3.12)is di?erent from the proof in[26].
Lemma3.5.(An imbedding result)For?∈V h,it holds:
|?|1,4,h≤C|?|2,h.
Proof.The tangential and normal derivative of?∈V h are continuous at the midpoint of each edges of T∈T h.That is?x,?y∈S h where S h is the nonconforming Crouzeix-Raviart?nite element space de?ned by
S h:=
w∈L2(?):w|T∈P1(T)?T∈T h,w is continuous at the midpoints of the triangle edges and w=0at the midpoint of the edges on??
.
It is enough to prove|w|0,4,h≤|w|1,h?w∈S h.
Consider the auxiliary problem:givenθ∈H?1(?),seekξsuch that
??ξ=θin?,ξ=0on??.(3.9)
TITLE WILL BE SET BY THE PUBLISHER7 The solution satis?es the following a priori bounds
ξ 1≤C θ ?1, ξ 1+γ≤C θ ,(3.10)
whereγ∈(1
2,1]denotes the elliptic regularity of the problem(3.9).Let I hξ∈S h be an interpolant
which satis?es the estimate[8,10]
|ξ?I hξ|0,h+h|ξ?I hξ|1,h≤Ch1+γ ξ 1+γ.(3.11) A multiplication of(3.9)with w and a use of Green’s formula leads to
(θ,w)=(??ξ,w)=
T∈T h (?ξ,?w)?
T∈T h
?
?T
?ξ
?ν
w ds(3.12)
The boundary term can be estimated as follows:
T∈T h ?
?T
?ξ
?ν
w ds=
T∈T h
e??T
?
e
?ξ
?ν
(w?P e w)ds.
Since
?
e (w?P e w)ds=0?e∈E h and
?
?ν
I hξis a constant over each edge,we obtain
T∈T h
e??T
?
e
?ξ
?ν
(w?P e w)ds=
T∈T h
e??T
?
e
?
?ν
(ξ?I hξ)(w?P e w)ds
≤
T∈T h
e??T
ξ?I hξ 1,e w?P e w 0,e.
A use of trace theorem,Lemma3.2and(3.11)leads to the estimate
?
T∈T h
?
?T
?ξ
?ν
w ds
≤Chγ ξ 1+γ|w|1,h.(3.13)
Therefore,the a priori bounds in(3.10)yields
(θ,w)≤(|ξ|1+Chγ ξ 1+γ)|w|1,h≤C( θ ?1+hγ θ )|w|1,h.(3.14) A choice ofθ=w3in(3.14)leads to
|w|40,4,h≤C( w3 ?1+hγ w3 )|w|1,h.(3.15) A use of inverse inequality yields
w3 = w 3
L6(?)≤Ch?14 w 3
L4(?)
.(3.16)
Also,H¨o lder’s inequality and the imbedding result L4(?) →H10(?)lead to
(w3,ξ)= w 3
L4(?) ξ L4(?)≤C w 3
L4(?)
|ξ|1=? w3 ?1≤C w 3
L4(?)
.(3.17)
Hence,a use of(3.16)and(3.17)in(3.15)leads to the required result
|w|0,4,h≤C(1+hγ?14)|w|1,h≤C|w|1,h.
8TITLE WILL BE SET BY THE PUBLISHER The next lemma follows from[11,Lemmas4.2&4.3].
Lemma3.6.(Bounds for A h(·,·))(i)Letχ∈(H2+α(?))2andΦ∈V h.Then,it holds
A h(χ,E hΦ?Φ)≤Chα|||χ|||
2+α|||Φ|||
2,h
.
(ii)Further,forχ∈(H2+α(?))2andΦ∈(H20(?))2∩(H2+α(?))2,it holds
A h(χ,ΠhΦ?Φ)≤Ch2α|||χ|||
2+α|||Φ|||
2+α
.
A use of the de?nition of
B h(·,·,·),generalized H¨o lder’s inequality and Lemma3.5leads to a bound given by
B h(Ξ,Θ,Φ)≤
C b|||Ξ|||
2,h |||Θ|||
2,h
|||Φ|||
2,h
,(3.18)
where C b is a positive constant independent of h.
Lemma3.7.(A bound for B h(·,·,·))ForΞ∈(H2+α(?))2andΘ,Φ∈V+V h,there holds
B h(Ξ,Θ,Φ)≤C|||Ξ|||
2+α|||Θ|||
1,4,h
|||Φ|||
1,h
≤C|||Ξ|||
2+α
|||Θ|||
2,h
|||Φ|||
1,h
.
Proof.Consider
b h(η,χ,?)=1
2
T∈T h
?
T
((ηyyχx?ηxyχy)?x+(ηxxχy?ηxyχx)?y)dx.(3.19)
Forη∈H2+α(?),a use of generalized H¨o lder’s inequality and the imbedding result H2+α(?) →W2,4(?)leads to an estimate of the?rst term on the right hand side of(3.19)as
1 2
T∈T h
?
T
ηyyχx?x dx
≤
T∈T h
|η|42,4,T
14
T∈T h
|χ|41,4,T
14
T∈T h
|?|21,2,T
12
≤ η W2,4(?)|χ|1,4,h|?|1,h≤C η 2+α|χ|1,4,h|?|1,h.
Similar bounds hold true for the remaining three terms in(3.19).Hence the required result follows using the de?nition of B h(·,·,·)and Lemma3.5. https://www.doczj.com/doc/5b17448825.html,ing a proof similar to that of Lemma3.7,it can be deduced that forΞ∈(H2+α(?))2 andΘ,Φ∈V+V h,there holds
B h(Ξ,Θ,Φ)≤C|||Ξ|||
2+α|||Θ|||
1,h
|||Φ|||
1,4,h
≤C|||Ξ|||
2+α
|||Θ|||
1,h
|||Φ|||
2,h
.
Using the de?nition of b h(·,·,·),an integration by parts and a use of(3.7),the following lemma holds true.
Lemma3.9.(An intermediate result)Forη∈(V∩H2+α(?))+V h andχ,?∈V h,it holds
b h(η,χ,?)=b h(χ,η,?)+1
2
T∈T h
?
?T
(ηxχy?ηyχx)??·τds
whereτis the unit tangent to the boundary?T of the triangle T.Moreover,
?η,χ,?∈V h,1
2
T∈T h
?
?T
(ηxχy?ηyχx)??·τds≤Ch|η|2,h|χ|1,∞,h|?|2,h.(3.20)
TITLE WILL BE SET BY THE PUBLISHER9 Remark3.10.A use of Lemma3.9,Remark3.8and imbedding result H2+α(?) →W1,∞(?)leads to: forΞh,Φh∈V h andξ∈(H2+α(?))2,
|B h(Ξh,ξ,Φh)|≤|B h(ξ,Ξh,Φh)|+Ch|||Ξh|||
2,h |||ξ|||
2+α
|||Φh|||
2,h
.(3.21)
The next lemma which will be used to establish the well posedness of the linearized problem(3.6), follows easily under the assumption thatΨis an isolated solution of(2.2).
Lemma3.11.(Well posedness of dual problem)IfΨis an isolated solution of(2.2),then the dual problem de?ned by:given Q∈(H?1(?))2,?ndζ∈V such that
A(Φ,ζ)=(Q,Φ)?Φ∈V(3.22) is well posed and satis?es the a priori bounds:
|||ζ|||
2≤C|||Q|||
?1
,|||ζ|||
2+α
≤C|||Q|||
?1
,(3.23)
whereαdenotes the elliptic regularity index and|||Q|||
?1:=sup
?∈(H1
(?))2
(Q,?)
|||?|||
1
.
Since the Morley?nite element space V h is not a subspace of V and the discrete form b h(·,·,·) is non-symmetric with respect to?rst and second or?rst and third variables,we encounter addi-tional di?culties in establishing the well posedness of the discrete problem(3.6)in comparison to the conforming case.
Theorem3.12.(Well posedness of discrete linearized problem)IfΨis an isolated solution of(2.2), then for su?ciently small h,the discrete linearized problem(3.6)is well-posed.
Proof.The space V h being?nite dimensional,uniqueness of solution of(3.6)implies existence of solution.Uniqueness follows if an a priori bound for the solution of(3.6)can be established.That is, we aim to prove that
|||Θh|||
2,h
≤C|||G|||(3.24) for su?ciently small h.ForΦ∈V h,using Lemma3.7and Remark3.10,the following G?arding’s type inequality holds true:
A h(Φ,Φ)=A h(Φ,Φ)+
B h(Ψ,Φ,Φ)+B h(Φ,Ψ,Φ)
≥|||Φ|||2
2,h ?C|||Ψ|||
2+α
|||Φ|||
2,h
|||Φ|||
1,h
?Ch|||Φ|||
2,h
|||Ψ|||
2+α
|||Φ|||
2,h
.(3.25)
SubstituteΦ=Θh in(3.6)and use(3.25)to obtain
|||Θh|||
2,h ≤C(h|||Ψ|||
2+α
|||Θh|||
2,h
+|||Ψ|||
2+α
|||Θh|||
1,h
+|||G|||).(3.26)
Note that
|||Θh|||
1,h ≤|||Θh?E hΘh|||
1,h
+|||E hΘh|||
1,h
≤Ch|||Θh|||
2,h
+|||E hΘh|||
1
.(3.27)
Now we estimate|||E hΘh|||
1
.Choose Q=??E hΘh andΦ=E hΘh in(3.22)and use(3.6)to obtain
|||E hΘh|||2
1
=A(E hΘh,ζ)=A h(E hΘh,ζ?Πhζ)+A h(E hΘh,Πhζ)
=A h(E hΘh,ζ?Πhζ)+A h(E hΘh?Θh,Πhζ)+(G,Πhζ)
=A h(E hΘh?Θh,ζ)+A h(Θh,ζ?Πhζ)+B h(Ψ,E hΘh,ζ?Πhζ)+B h(E hΘh,Ψ,ζ?Πhζ)
+B h(Ψ,E hΘh?Θh,Πhζ)+B h(E hΘh?Θh,Ψ,Πhζ)+(G,Πhζ)(3.28)
10TITLE WILL BE SET BY THE PUBLISHER
A use of Lemmas3.3,3.4,3.6,(3.18),(3.23),Remarks3.8and3.10leads to
|||E hΘh|||2
1≤C
hα|||Θh|||
2,h
|||ζ|||
2+α
+hα|||Ψ|||
2
|||Θh|||
2,h
|||ζ|||
2+α
+h|||Ψ|||
2+α
|||Θh|||
2,h
|||ζ|||
2
+|||G||||||ζ|||
2
≤C
hα|||Θh|||
2,h
+|||G|||
|||??E hΘh|||
?1
≤C
hα|||Θh|||
2,h
+|||G|||
|||E hΘh|||
1
.
Therefore,
|||E hΘh|||
1,h ≤C(hα|||Θh|||
2,h
+|||G|||).(3.29)
Now,(3.26)-(3.29)yield
|||Θh|||
2,h ≤C?hα|||Θh|||
2,h
+C|||G|||.
That is,|||Θh|||
2,h ≤C|||G|||for a choice of h≤h1=(1
2C?
)1αwithα∈(1
2
,1].
Remark3.13.IfΨis an isolated solution of(2.2),then for su?ciently small h,the discrete linearized dual problem:given G∈(L2(?))2,?ndζh∈V h such that
A h(Φ,ζh)=(G,Φ)?Φ∈V h(3.30) is well posed.The proof is similar to that of Theorem3.12and hence is skipped.
4.Existence,Uniqueness and Error Estimates
In view of Theorem3.12and Remark3.13,the bilinear form A h(·,·):V h×V h→R de?ned by
A h(Θ,Φ)=A h(Θ,Φ)+
B h(Ψ,Θ,Φ)+B h(Θ,Ψ,Φ)(4.1) is nonsingular on V h×V h.
The next lemma establishes that the perturbed bilinear form?
A h(·,·),constructed usingΠhΨis also nonsingular.Though a similar result is proved in[12]for the conforming case,we provide a proof here for the sake of completeness.
Lemma4.1.(Nonsingularity of perturbed bilinear form)LetΠhΨbe the interpolation ofΨas de?ned in Lemma3.3.Then,for su?ciently small h,the perturbed bilinear form de?ned by
?
A h(Θ,Φ)=A h(Θ,Φ)+
B h(ΠhΨ,Θ,Φ)+B h(Θ,ΠhΨ,Φ)(4.2) is nonsingular on V h×V h,if(4.1)is nonsingular on V h×V h.
Proof.The bilinear form A h:V h×V h?→R is bounded and satis?es
sup
|||Θ|||
2,h =1
A h(Θ,Φ)≥β|||Φ|||2,h,sup
|||Φ|||
2,h
=1
A h(Θ,Φ)≥β|||Θ|||2,h,
whereβ>0is a constant.For?Ψ∈V+V h,a use of the above properties of A h(·,·)and continuity of B h(·,·,·)(see(3.18))yields
sup
|||Φ|||
2,h =1
A h(Θ,Φ)+
B h(Ψ??Ψ,Θ,Φ)+B h(Θ,Ψ??Ψ,Φ)
≥sup
|||Φ|||
2,h =1
A h(Θ,Φ)?sup
|||Φ|||
2,h
=1
B h(?Ψ,Θ,Φ)+B h(Θ,?Ψ,Φ)
≥β|||Θh|||
2,h ?2C b
?
Ψ
2,h
|||Θ|||
2,h
≥
β
2
|||Θ|||
2,h
,
TITLE WILL BE SET BY THE PUBLISHER 11
provided ?Ψ 2,h
≤β4C b .Such a choice is justi?ed for su?ciently small h ≤h 2(say),by setting ?Ψ=Ψ?Πh Ψand using Lemma 3.3.Similarly,sup |||Θ|||2,h =1?A h (Θ,Φ)≥β2|||Φ|||2,h ?Φ∈V h .Hence the required result. 4.1.Existence and Local Uniqueness Results
Consider the nonlinear operator μ:V h ?→V h de?ned by
?A h (μ(Θ),Φ)=L h (Φ)+B h (Πh Ψ,Θ,Φ)+B h (Θ,Πh Ψ,Φ)?B h (Θ,Θ,Φ)?Φ∈V h .(4.3)A use of Lemma 4.1leads to the fact that the mapping μis well-de?ned and continuous.Also,any ?xed point of μis a solution of (3.2)and vice-versa.Hence,in order to show the existence of a solution to (3.2),we will prove that the mapping μhas a ?xed point.As a ?rst step to this,de?ne B R (Πh Ψ):= Φ∈V h :|||Φ?Πh Ψ|||2,h ≤R .
Theorem 4.2.(Mapping of ball to ball)For a su?ciently small choice of h ,there exists a positive constant R (h )such that for any Θ∈V h ,
|||Θ?Πh Ψ|||2,h ≤R (h )?|||μ(Θ)?Πh Ψ|||2,h ≤R (h ).
That is,μmaps the ball B R (h )(Πh Ψ)to itself.
Proof.Since the bilinear form ?A h (·,·)is nonsingular,from Lemma 4.1,there exists ˉΦ∈V h such that ˉΦ 2,h
=1and β4
|||μ(Θ)?Πh Ψ|||2,h ≤?A h (μ(Θ)?Πh Ψ,ˉΦ)Let E h ˉΦ
be an enrichment of ˉΦ(see Lemma 3.4).A use of (4.2),(4.3)and (2.2)yields ?A h (μ(Θ)?Πh Ψ,ˉΦ)=?A h (μ(Θ),ˉΦ)??A h (Πh Ψ,ˉΦ)
=L h (ˉΦ)+B h (Πh Ψ,Θ,ˉΦ)+B h (Θ,Πh Ψ,ˉΦ)?B h (Θ,Θ,ˉΦ)?A h (Πh Ψ,ˉΦ)?2B h (Πh Ψ,Πh Ψ,ˉΦ)=L h (ˉΦ
?E h ˉΦ)+ A h (Ψ,E h ˉΦ)?A h (Πh Ψ,ˉΦ) + B h (Ψ,Ψ,E h ˉΦ)?B h (Πh Ψ,Πh Ψ,ˉΦ) +B h (Πh Ψ?Θ,Θ?Πh Ψ,ˉΦ)=:T 1+T 2+T 3+T 4.(4.4)
Now we estimate {T i }4i =1.T 1can be estimated using Lemma 3.4and the continuity of L h .Using Lemma 3.6,continuity of A h (·,·)and Lemma 3.3,we obtain
T 2≤|A h (Ψ,E h ˉΦ)?A h (Πh Ψ,ˉΦ)|≤|A h (Ψ,E h ˉΦ?ˉΦ)|+|A h (Ψ?Πh Ψ,ˉΦ)|≤Ch α|||Ψ|||2+α.
A use of Lemmas 3.7,3.4,3.3and (3.18)leads to
T 3≤|B h (Ψ,Ψ,E h ˉΦ)
?B h (Πh Ψ,Πh Ψ,ˉΦ)|≤|B h (Ψ,Ψ,E h ˉΦ?ˉΦ)?B h (Πh Ψ?Ψ,Πh Ψ,Φ)?B h (Ψ,Πh Ψ?Ψ,ˉΦ)|≤Ch |||Ψ|||2+α|||Ψ|||2 E h ˉΦ 2,h +Ch α|||Ψ|||2+α|||Ψ|||2 ˉΦ 2,h
≤Ch α
|||Ψ|||2|||Ψ|||2+α.
Finally,T 4is estimated using (3.18)as
T 4≤|B h (Πh Ψ?Θ,Θ?Πh Ψ,ˉΦ)|≤C |||Θ?Πh Ψ|||22,h .
12TITLE WILL BE SET BY THE PUBLISHER
A substitution of the estimates derived for T 1,T 2,T 3and T 4in (4.4)and an appropriate grouping of the terms yields
|||μ(Θ)?Πh Ψ|||2,h ≤C 1 h α+|||Θ?Πh Ψ|||22,h (4.5)
for some positive constants C 1independent of h but dependent on |||Ψ|||2+α.A choice of h ≤h 3,where
h 3= 14C 21 1α,yields 4C 21h α≤1.Since |||Θ?Πh Ψ|||2,h ≤R (h ),for h ≤h 3,a choice of R (h ):=2C 1h α
leads to
|||μ(Θ)?Πh Ψ|||2,h ≤C 1h α 1+4C 21h α ≤R (h )
This completes the proof. Theorem 4.3.(Existence)For su?ciently small h ,there exists a solution Ψh of the discrete problem (3.2)that satis?es |||Ψh ?Πh Ψ|||2,h ≤R (h ),for some positive constant R (h )depending on h .Proof.Lemma 4.2leads to the fact that μmaps the ball B R (h )(Πh Ψ)to itself.Therefore,an application of Schauder ?xed point theorem [18]yields that the mapping μhas a ?xed point,say Ψh .Hence,Ψh is an approximate solution of (3.2)which satis?es |||Ψh ?Πh Ψ|||2,h ≤R (h ). Theorem 4.4.(Contraction result)For Θ1,Θ2∈B R (h )(Πh Ψ)with R (h )as de?ned in Theorem 4.2,the following contraction result holds true:
|||μ(Θ1)?μ(Θ2)|||2,h ≤Ch α|||Θ1?Θ2|||2,h ,
(4.6)
for some positive constant C independent of h .
Proof.For Θ1,Θ2∈B R (h )(Πh Ψ),let μ(Θi ),i =1,2be the solutions of:
?A h (μ(Θi ),Φ)=L h (Φ)+B h (Πh Ψ,Θi ,Φ)+B h (Θi ,Πh Ψ,Φ)?B h (Θi ,Θi ,Φ)?Φ∈V h .(4.7)The nonsingularity of ?A h (·,·)yields a ˉΦwith ˉΦ 2,h
=1.With (4.7)and (3.18),we obtain β|||μ(Θ1)?μ(Θ2)|||2,h ≤?A h (μ(Θ1)?μ(Θ2),ˉΦ)=B h (Πh Ψ,Θ1?Θ2,ˉΦ)
+B h (Θ1?Θ2,Πh Ψ,ˉΦ)+B h (Θ2,Θ2,ˉΦ)?B h (Θ1,Θ1,ˉΦ)=B h (Θ2?Θ1,Θ1?Πh Ψ,ˉΦ)+B h (Θ2?Πh Ψ,Θ2?Θ1,ˉΦ)≤C |||Θ2?Θ1|||2,h |||Θ1?Πh Ψ|||2,h +|||Θ2?Πh Ψ|||2,h .
Since Θ1,Θ2∈B R (h )(Πh Ψ),for a choice of R (h )as in the proof of Theorem 4.2,for su?ciently small h ,we obtain
|||μ(Θ1)?μ(Θ2)|||2,h ≤Ch α|||Θ2?Θ1|||2,h ,
(4.8)for some positive constant C independent of h .This completes the proof. Remark 4.5.(Local uniqueness)Let Ψbe an isolated solution of (2.2).For su?ciently small choice of h ,Theorem 4.4establishes the local uniqueness of the solution of (3.2).
TITLE WILL BE SET BY THE PUBLISHER13 4.2.Error Estimates
In this subsection,the error estimates in the broken energy and H1norms are established. Theorem4.6.(Energy norm estimate)LetΨandΨh be the solutions of(2.2)and(3.2)respectively. Under the assumption thatΨis an isolated solution,for su?ciently small h,it holds
|||Ψ?Ψh|||
2,h
≤Chα,(4.9)
whereα∈(1
2
,1]is the index of elliptic regularity.
Proof.A use of triangle inequality yields
|||Ψ?Ψh|||
2,h ≤|||Ψ?ΠhΨ|||
2,h
+|||ΠhΨ?Ψh|||
2,h
.(4.10)
For su?ciently small h,Theorem4.3leads to
|||ΠhΨ?Ψh|||
2,h
≤Chα.(4.11) Now,Lemma3.3,(4.11)and(4.10)establish the required estimate. Theorem4.7.(H1estimate)LetΨandΨh be the solutions of(2.2)and(3.2)respectively.Assume thatΨis an isolated solution.Then,for su?ciently small h,it holds
|||Ψ?Ψh|||
1,h
≤Ch2α,(4.12)
whereα∈(1
2
,1]is the index of elliptic regularity.
Proof.A use of triangle inequality yields
|||Ψ?Ψh|||
1,h ≤|||Ψ?ΠhΨ|||
1,h
+|||ΠhΨ?Ψh|||
1,h
≤|||Ψ?ΠhΨ|||
1,h
+|||ρ?E hρ|||
1,h
+|||E hρ|||
1
,(4.13)
whereρ=ΠhΨ?Ψh.A choice of Q=??E hρandΦ=E hρin the dual problem(3.22)and a use of (2.2),(3.2)leads to
(?E hρ,?E hρ)=A h(E hρ,ζ)=A h(E hρ?ρ,ζ)+A h(ρ,ζ)
=A h(E hρ?ρ,ζ)+B h(Ψ,E hρ?ρ,ζ)+B h(E hρ?ρ,Ψ,ζ)
+A h(ΠhΨ?Ψ,ζ)+A h(Ψ?Ψh,ζ?Πhζ)+A h(Ψ,Πhζ?ζ)+L h(ζ?Πhζ)
+(B h(Ψ,ΠhΨ?Ψh,ζ)+B h(ΠhΨ?Ψh,Ψ,ζ)?B h(Ψ,Ψ,ζ)+B h(Ψh,Ψh,Πhζ))
=:
8
i=1
T i.(4.14)
T1is estimated using Lemma3.6and(4.11).T4and T6are estimated using Lemma3.6.T5is estimated using continuity of A h(·,·),Lemma3.3and Theorem4.6.The term T7is estimated using continuity of L h and Lemma3.3.T2is estimated using Remark3.8,Lemma3.4and(4.11)as
T2≤|B h(Ψ,E hρ?ρ,ζ)|≤C|||Ψ|||
2+α|||E hρ?ρ|||
1,h
|||ζ|||
2
≤Ch1+α|||Ψ|||
2+α
|||ζ|||
2
.(4.15)
T3is estimated using Remark3.10,Lemma3.4,(4.15)and(4.11)as
T3≤|B h(E hρ?ρ,Ψ,ζ)|≤|B h(Ψ,E hρ?ρ,ζ)|+Ch|||E hρ?ρ|||
2,h |||Ψ|||
2+α
|||ζ|||
2
≤Ch1+α|||Ψ|||
2+α
|||ζ|||
2
.
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Finally,a use of Remarks3.8,3.10,Lemmas3.3,3.7,Theorem4.6and(3.18)yields an estimate for T8as
T8=B h(Ψ,ΠhΨ?Ψh,ζ)+B h(ΠhΨ?Ψh,Ψ,ζ)?B h(Ψ,Ψ,ζ)+B h(Ψh,Ψh,Πhζ) =B h(Ψ,ΠhΨ?Ψ,ζ)+B h(ΠhΨ?Ψ,Ψ,ζ)
+B h(Ψ,Ψ?Ψh,ζ)+B h(Ψ?Ψh,Ψ,ζ)?B h(Ψ,Ψ,ζ)+B h(Ψh,Ψh,Πhζ)
=B h(Ψ,ΠhΨ?Ψ,ζ)+B h(ΠhΨ?Ψ,Ψ,ζ)
+B h(Ψ?Ψh,Ψ?Ψh,ζ)+B h(Ψh?Ψ,Ψh,Πhζ?ζ)+B h(Ψ,Ψh,Πhζ?ζ)
≤Ch2α(|||ζ|||
2+|||ζ|||
2+α
).
A combination of the estimates T1to T8and a priori bounds(3.23)for the linearized dual problem yields
(?E hρ,?E hρ)≤Ch2α|||??E hρ|||
?1≤Ch2α|||E hρ|||
1
=?|||E hρ|||
1
≤Ch2α.(4.16)
A use of Lemmas3.3,3.4,(4.11)and the last statement of(4.16)in(4.13)completes the proof.
4.3.Convergence of the Newton’s Method
In this subsection,we de?ne a working procedure to?nd an approximation for the discrete solution Ψh.The discrete solutionΨh of(3.2)is characterized by the?xed point of(4.3).This depends on the unknownΠhΨand hence the approximate solution for(3.2)is computed using Newton’s method in implementation.The iterates of the Newton’s method are de?ned by
A h(Ψn h,Φ)+
B h(Ψn?1
h ,Ψn h,Φ)+B h(Ψn h,Ψn?1
h
,Φ)=B h(Ψn?1
h
,Ψn?1
h
,Φ)+L h(Φ)?Φ∈V h.(4.17)
Now we establish that these iterates in fact converge quadratically to the solution of(3.2). Theorem4.8.(Convergence of Newton’s method)LetΨbe an isolated solution of(2.2)and letΨh
solve(3.2).There existsρ>0,independent of h,such that for any initial guessΨ0
h which satis?es
Ψ0h?Ψh
2,h
≤ρ,|||Ψn h?Ψh|||
2,h
≤
ρ
2n
holds true.That is,the iterates of the Newton’s method
de?ned in(4.17)are well de?ned and converge quadratically toΨh.
Proof.From Lemma4.1,there existsδ>0such that for each Z h∈V h satisfying|||Z h?ΠhΨ|||
2,h ≤δ,
the form
A h(Θ,Φ)+
B h(Z h,Θ,Φ)+B h(Θ,Z h,Φ)(4.18) is non singular in V h×V h.From(4.11),for su?ciently small h,|||ΠhΨ?Ψh|||
2,h
≤Chα.Thus h can
be chosen su?ciently small so that|||ΠhΨ?Ψh|||
2,h ≤δ
2
.De?ne
ρ:=min
δ
2
,
β
16C b
(4.19)
whereβand C b are respectively the coercivity constant of A h(·,·)and boundedness constant of B h(·,·,·)
(see(3.7)).Assume that the initial guessΨ0
h satis?es
Ψh?Ψ0
h
2,h
≤ρ.Then
ΠhΨ?Ψ0h
2,h
≤|||ΠhΨ?Ψh|||
2,h
+
Ψh?Ψ0h
2,h
≤δ
Since(4.18)is nonsingular,the?rst iterateΨ1
h of the Newton’s method in(4.17)is well de?ned for the
initial guessΨ0
h .Using the nonsingularity of(4.18),there existsˉΦ∈V h such that
ˉ
Φ
2,h
=1which
satis?es
β8
Ψ1h?Ψh
2,h
≤A h(Ψ1h?Ψh,ˉΦ)+B h(Ψ0h,Ψ1h?Ψh,ˉΦ)+B h(Ψ1h?Ψh,Ψ0h,ˉΦ).
TITLE WILL BE SET BY THE PUBLISHER 15
A use of (4.17),(3.2),(3.18)yields A h (Ψ1h ?Ψh ,ˉΦ)+
B h (Ψ0h ,Ψ1h ?Ψh ,ˉΦ)+B h (Ψ1h ?Ψh ,Ψ0h ,ˉΦ)
=B h (Ψ0h ,Ψ0h ,ˉΦ)
+L h (ˉΦ)?A h (Ψh ,ˉΦ)?B h (Ψ0h ,Ψh ,ˉΦ)?B h (Ψh ,Ψ0h ,ˉΦ)=B h (Ψ0h ,Ψ0h ,ˉΦ)+B h (Ψh ,Ψh ,ˉΦ)?B h (Ψ0h ,Ψh ,ˉΦ)?B h (Ψh ,Ψ0h ,ˉΦ)=B h (Ψ0h ?Ψh ,Ψ0h ?Ψh ,ˉΦ)≤C b Ψ0h
?Ψh 22,h .(4.20)Hence, Ψ1h ?Ψh 2,h ≤8C b β Ψ0h ?Ψh 22,h
.Since Ψ0h ?Ψh ≤ρ≤β16C b ,we obtain Ψ1h ?Ψh 2,h ≤12 Ψ0h ?Ψh 2,h ≤ρ2
.(4.21)Since Ψ1h
?Ψh 2,h ≤ρ,the form (4.18)is nonsingular for Z h =Ψ1h .Continuing the process,we obtain |||Ψn h ?Ψh |||2,h ≤ρ2
n .(4.22)Moreover,proceeding as in the proof of the estimate (4.20),it can be shown that
Ψn +1h ?Ψh 2,h ≤(8C b /β)|||Ψn h ?Ψh |||22,h .(4.23)
This establishes that the Newton’s method converges quadratically to Ψh .This completes the proof.
Remark 4.9.(Local uniqueness)The local uniqueness of solution of (3.2)also follows from Theo-rem 4.8.We observe that the de?nition of ρin (4.19)does not depend on h .From Theorem 4.8,it is clear that for any initial guess Ψ0h which lies in the ball of radius ρwith center at Ψh ,the sequence gen-erated by (4.17)will converge uniquely to Ψh .In particular,if we choose the initial guess Ψ0h =Πh Ψ,then the sequence generated by the iterates of the Newton’s method will also converge to Ψh which shows the local uniqueness of the solution Ψh .
5.Numerical Experiments
In this section,two numerical experiments that justify the theoretical results are presented.The implementations have been carried out in MATLAB.The results illustrate the order of convergence obtained for the numerical solution of (1.1)-(1.2)computed using the Morley ?nite element scheme.For a detailed description of construction of basis functions for the Morley element,see Ming &Xu [21].We implement the Newton’s method de?ned in (4.17)to solve the discrete problem (3.2).
Example 5.1.In the ?rst example,we choose the right hand side load functions such that the exact solution is given by
u (x,y )=x 2(1?x )2y 2(1?y )2;v (x,y )=sin 2(πx )sin 2(πy )
on the unit square.The initial triangulation is chosen as shown in Figure 2(A).In the uniform red-re?nement process,each triangle T is divided into four similar triangles [1]as in Figure 2(B).
Let the mesh parameter at the N -th level be denoted by h N and the computational error by e N .The experimental order of convergence at the N -th level is de?ned by
αN :=log (e N ?1/e N )/log (h N ?1/h N )=log (e N ?1/e N )/log (2).
Tables 1and 2show the errors and experimental convergence rates for the variables u h and v h .In Figures 3-4,the convergence history of the errors in broken energy,H 1and L 2norms are illustrated.
16TITLE WILL BE SET BY THE PUBLISHER
(a)(b)
Figure 2.Initial triangulation T 0and its red re?nement T 1#unknowns
|u ?u h |2,h Order |u ?u h |1,h Order u ?u h L 2Order 25
0.874685E-1-0.102155E-1-0.386068E-2-113
0.405787E-1 1.10800.257318E-2 1.98910.919743E-3 2.0695481
0.209921E-10.95080.732470E-3 1.81270.248134E-3 1.89011985
0.106209E-10.98290.191118E-3 1.93830.636227E-4 1.96358065
0.532754E-20.99530.483404E-4 1.98310.160158E-4 1.9900325130.266595E-20.99880.121213E-4 1.99560.401107E-5 1.9974
Table 1.Errors and convergence rates of u h in broken H 2,H 1and L 2norms #unknowns
|v ?v h |2,h Order |v ?v h |1,h Order v ?v h L 2Order 25
19.245671- 2.140613E-0-0.770876E-0-113
9.5043699 1.01780.569979E-0 1.90900.177898E-0 2.1154481
5.05492090.91090.161737E-0 1.81720.482777E-1 1.88161985
2.57589390.97260.421546E-1 1.93980.123930E-1 1.96188065
1.29449290.99260.106618E-1 1.98320.312076E-2 1.9895325130.64808480.99810.267351E-2 1.99560.781643E-3 1.9973
Table 2.Errors and convergence rates of v h in broken H 2,H 1and L 2norms
The computational order of convergences in broken H 2,H 1norms are quasi-optimal and verify the theoretical results obtained in Theorems 4.6and 4.7for α=1.The order of convergence with respect to L 2norm is sub-optimal justifying the results in [17]that using a lower order ?nite element method,the order of convergence in L 2norm cannot be improved than that of the H 1norm.
Example 5.2.Consider the L-shaped domain ?=(?1,1)2\([0,1)×(?1,0])(see Figure 5).Choose the right hand functions such that the exact singular solution [16]in polar coordinates is given by
u (r,θ)=(r 2cos 2θ?1)2(r 2sin 2θ?1)2r 1+αg α,ω(θ);
v (r,θ)=u (r,θ),where ω:=3π
2and α:=0.5444837367is a non-characteristic root of sin 2(αω)=α2sin 2(ω)with
g α,ω(θ)=
1α?1sin (α?1)ω ?1α+1sin (α+1)ω cos (α?1)θ ?cos (α+1)θ ? 1α?1sin (α?1)θ ?1α+1sin (α+1)θ cos (α?1)ω ?cos (α+1)ω .
TITLE WILL BE SET BY THE PUBLISHER17
Figure3.Convergence history of displacement for Example1
Figure4.Convergence history of Airy stress for Example1
Tables3and4show the errors and experimental convergence rates for the variables u h and v h.The domain being non-convex,we do not obtain linear and quadratic order of convergences in broken energy and H1norms for displacement and Airy stress functions.
18TITLE WILL BE SET BY THE PUBLISHER
Figure5.L-shaped domain and its initial triangulation
#unknowns|u?u h|2,h Order|u?u h|1,h Order u?u h L2Order
1729.209171- 6.363539E-0- 2.769499E-0-
8114.130192 1.0476 1.682747E-0 1.91900.693436E-0 1.9977
3537.56513000.90130.491659E-0 1.77500.200814E-0 1.7879
1473 3.96201260.93310.146551E-0 1.74620.583024E-1 1.7842
6017 2.08411410.92670.487106E-1 1.58910.179703E-1 1.6979
24321 1.12525340.88910.187772E-1 1.37520.613474E-2 1.5505 Table3.Errors and the experimental convergence rates for u h in broken H2,H1
and L2norms for L-shaped domain
#unknowns|v?v h|2,h Order|v?v h|1,h Order v?v h L2Order
1724.759835- 4.932699E-0- 2.069151E-0-
8115.2932700.6951 1.779132E-0 1.47120.727981E-0 1.5070
3537.85093220.96190.483823E-0 1.87860.199644E-0 1.8664
1473 4.05312690.95380.137278E-0 1.81730.557622E-1 1.8400
6017 2.12199880.93360.439086E-1 1.64450.165699E-1 1.7507
24321 1.14219380.89360.165883E-1 1.40430.545066E-2 1.6040 Table4.Errors and the experimental convergence rates for v h in broken H2,H1and
L2norms for L-shaped domain
6.Conclusions&Perspectives
In this work,an attempt has been made to obtain approximate solutions for the clamped von K′a rm′a n equations de?ned on polygonal domains using nonconforming Morley elements.Error esti-mates in broken energy and H1norms are established for su?ciently small discretization parameters. Numerical results that substantiate the theoretical results are obtained.A future area of interest would be derivation of reliable a posteriori error estimates that drive the adaptive mesh re?nements.
Acknowledgments:The authors would like to sincerely thank Professors S.C.Brenner and Li-yeng Sung for their suggestions on extension of the results to non-convex polygonal domains and to Dr.Thirupathi Gudi for his comments.The?rst author would also like to thank National Board for Higher Mathematics(NBHM) for the?nancial support towards the research work.
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有机化学规律总结 一．有机物组成和结构的规律 １．在烃类中，烷烃CnH2n+2随分子中碳原子的增多，其含碳量增大；炔烃、二烯烃、苯的同系物随着碳原子增加，其含碳量减少；烯烃、环烷烃的含碳量为常数（８５．７１％）２．一个特定的烃分子中有多少种结构的氢原子，一般来说其一卤代物就有多少种同分异构体． ３．最简式相同的有机物，不论以何种比例混合，其元素的质量分数为常数．例：m克葡萄糖，n克甲醛，x克乙酸，y克甲酸甲酯，混合后，求混合物碳的质量分数． ４．烃及烃的含氧衍生物中，氢原子个数一定为偶数． ６．常见有机物中最简式同为"CH"的有乙炔、苯、苯乙烯；同为“CH2”的为单烯烃和环烷烃； 7、烃类的熔、沸点变化规律 （1）有机物一般为分子晶体，在有机物同系物中，随碳原子数增加，相对分子质量增大，分子间作用力增大，熔、沸点逐渐升高。如：气态烃：CxHy x≤4 （2）分子式相同的烃，支链越多，熔、沸点越低。如沸点： 正戊烷（36.07℃）>异戊烷（27.9℃）>新戊烷（9.5℃） （3）苯的同系物，熔沸点。邻位>间位>对位, 如沸点： 邻二甲苯（144.4℃）>间二甲苯(139.1℃）>对二甲苯(138.4℃） 二．有机物燃烧规律 １．有机物燃烧通式：CxHy+(x+y/4)O2 xCO2+y/2H2O 2、烃（CXHY）完全燃烧前后物质的量的变化 （1）当Y=4时，反应前后物质的量相等。若同温同压下，100℃以上时，反应前后体积不变。 （2）当Y〈4时，燃烧后生成物分子数小于反应物分子总数。 （3）当Y〉4时，燃烧后生成物分子数大于反应物分子总数。 3、若分子为CnH2n、CnH2nO 或CnH2nO2时，其完全燃烧时生成CO2和H2O的物质的量之比为1：1。 4、若有机物在足量的氧气里完全燃烧,其所耗氧气物质的量与燃烧后生成CO2的物质的量相当,则有机物分子组成中氢与氧原子数比为2:1. 5、等质量的烃完全燃烧时,耗氧量决定于氢元素的含量,它越高,耗氧量越高,如甲烷耗氧量最高。 6、最简式相同的有机物,不论以何种比例混合，只要混合物的质量一定,它们完全燃烧后生成CO2总量为常数。 7.不同的有机物完全燃烧时,若生成CO2和H2O的质量比相同,则它们分子中碳与氢的原子个数也相同。 8、烃CXHY燃烧后体积变化规律 （1）当100℃以上，水为气态，若烃中H数或混合烃中平均H数小于4时，体积减小，1体积烃减少（1-Y/4）体积，若有a体积烃，则减少（1-Y/4）a 即△V=（1-Y/4）a 若烃中H数或混合烃中平均H数等于4时，△V=0。 若烃中H数或混合烃中平均H数大于4时，气体体积增大。1体积烃增大（Y/4-1）体积，若有a体积烃增大（Y/4-1）a体积，即 △V=（Y/4-1）a (2)当生成的水被浓H2SO4吸收或水蒸汽冷却为液体时。气体的体积会减少，每体积烃减少（1+Y/4)体积，若有a体积烃，则减少（1+Y/4）a体积即：△V=（1+Y/4）a 三、有机物的水溶性和密度大小规律
“十二五”规划12th Five-Year Plan 《关于禁止发展、生产和储存细菌（生物）及毒素武器和销毁此种武器的公约》（于1972年4月10日在伦敦、莫斯科和华盛顿签订）Prohibition of the Development, Production and Stockpiling of Bacteriological (Biological) and Toxin Weapons and on Their Destruction, (signed at London, Moscow and Washington on 10 April 1972) 《关于禁止在战争中使用窒息性、毒性或其他气体和细菌作战方法的议定书》（1925年《日内瓦议定书》）的原则和目标the principles and objectives of the Protocol for the Prohibition of the Use in War of Asphyxiating, Poisonous or Other Gases, and of Bacteriological Methods of Warfare, signed at Geneva on 17 June 1925 (the Geneva Protocol of 1925) 《联合国宪章》的宗旨和原则purposes and principles of the Charter of the United Nations 《维也纳外交关系公约》Vienna Convention on Diplomatic Relations ?根据本公约进行规定活动的视察组成员应免纳外交代表根据《维也纳外交关系公约》第34条所免纳的一切捐税。The members of the inspection team carrying out prescribed activities pursuant to this Convention shall be accorded the exemption from dues and taxes accorded to diplomatic agents pursuant to Article 34 of the Vienna Convention on Diplomatic Relations. 1, 3-二（2-氯乙硫基）正丙烷1,3-Bis (2-chloroethylthio)-n-propane 1, 4-二（2-氯乙硫基）正丁烷1,4-Bis (2-chloroethylthio)-n-butane 1, 5-二（2-氯乙硫基）正戊烷1,5-Bis (2-chloroethylthio)-n-pentane 1925年《日内瓦议定书》规定的义务obligations assumed under the Geneva Protocol of 1925 2,2-二苯基-2-羟基乙酸2,2-Diphenyl-2-hydroxyacetic acid 2-氯乙基氯甲基硫醚2-Chloroethylchloromethylsulfide atmospheric storage tank BZ：二苯乙醇酸-3-奎宁环酯BZ: 3-Quinuclidinyl benzilate DCS(分散控制系统、集散控制系统) distributed control system DF：甲基膦酰二氟DF: Methylphosphonyldifluoride flash drum HN1：N, N-二（2-氯乙基）乙胺HN1: Bis(2-chloroethyl)ethylamine HN2：N, N-二（2-氯乙基）甲胺HN2: Bis(2-chloroethyl)methylamine HN3：三（2-氯乙基）胺HN3: Tris(2-chloroethyl)amine O形环O-ring PFIB: 1,1,3,3,3 - 五氟-2-三氟甲基-1-丙烯PFIB:1,3,3,3-Pentafluoro-2-(trifluoromethyl)-1-propene pump (for firefighting water) QL：甲基亚膦酸乙基-2-二异丙氨基乙酯QL O-Ethyl O-2-diisopropylaminoethyl methylphosphonite slide valve U-V组合坡口combination U and V groove U形螺栓U-bolt U形弯管“U” bend VX（甲基硫代膦酸乙基-S-2-二异丙氨基乙酯）VX (O-Ethyl -2-diisopropylaminoethyl methyl phosphonothiolate) V形坡口V groove V型混合机V-type mixer X形坡口double V groove Y型粗滤器Y-type strainer γ射线照相Gamma radiography 安（培）Ampere (A) 安瓶ampule 安全safety 安全标志safety mark 安全等级safety class 安全防护装置safety protection device 安全连锁safety interlock (SI) 安全喷淋洗眼器safe spray eye wash 安全设施safety device 安全完整性等级safety integrity level (SIL) 安全卫生safety & health 安全系数margin of safety; safety coefficient; safety factor; coefficient of safety 安全仪表系统safety instrumented system (SIS) 安全照明safety lighting 安慰剂dummy drug ?他们对第二组用了安慰剂（无任何作用的药）。For the second group they used dummy drug. 安装erection; installation 氨氮ammonia-nitrogen (NH4+-N) 氨分离器ammonia separator 氨分缩器ammonia partial condenser 氨解ammonolysis 氨冷冻剂ammonia refrigerant 氨冷凝器ammonia condenser 氨冷器ammonia chiller 氨气ammonia gas 氨收集器ammonia accumulator 氨树脂amino resin 氨吸收制冷ammonia absorption refrigeration 鞍座saddle ?固定鞍座fixed saddle ?滑动鞍座sliding saddle 按国际惯例in line (accordance) with internationally accepted practices; compatible with established international practices; in accordance with international practice 胺吸磷：硫代磷酸二乙基-S-2-二乙氨基乙酯及相应烷基化盐或质子化盐Amiton: O,O-Diethyl S-[2-(diethylamino)ethyl] Phosphorothiolate and corresponding alkylated or protonated salts 奥氏体不锈钢Austenitic stainless steel 板框压滤器plate-and-fram filter press 板式输送机slat type conveyor 半成品semi-manufactured goods; semi- finished products; semi-finished goods semi-processed goods 半径radius (R) 半模拟盘（屏）semi-graphic panel 半贫液semi-lean solution 半衰期half-life; half-life period 半自动程序控制器semi-automatic sequence controller 1
基础化学常用英语词汇汇总
基础化学常用英语词汇380条 1. The Ideal-Gas Equation 理想气体状态方程 2. Partial Pressures 分压 3. Real Gases: Deviation from Ideal Behavior 真实气体：对理想气体行为的偏离 4. The van der Waals Equation 范德华方程 5. System and Surroundings 系统与环境 6. State and State Functions 状态与状态函数 7. Process 过程 8. Phase 相 9. The First Law of Thermodynamics 热力学第一定律 10. Heat and Work 热与功 11. Endothermic and Exothermic Processes 吸热与发热过程 12. Enthalpies of Reactions 反应热 13. Hess’s Law 盖斯定律 14. Enthalpies of Formation 生成焓 15. Reaction Rates 反应速率 16. Reaction Order 反应级数 17. Rate Constants 速率常数 18. Activation Energy 活化能 19. The Arrhenius Equation 阿累尼乌斯方程 20. Reaction Mechanisms 反应机理 21. Homogeneous Catalysis 均相催化剂 22. Heterogeneous Catalysis 非均相催化剂 23. Enzymes 酶 24. The Equilibrium Constant 平衡常数 25. the Direction of Reaction 反应方向 26. Le Chatelier’s Principle 列·沙特列原理
SA T II 化学词汇表 Part 1 foundation chemistry 基础化学 Chapter 1 acid 酸 apparatus 仪器，装置 aqueous solution水溶液 arrangement of electrons 电子排列 assumption 假设 atom 原子 atomic mass原子量 atomic number 原子序数 atomic radius原子半径 atomic structure 原子结构 be composed of 由……组成 bombard ment 撞击 boundary 界限 cathode rays 阴极射线 cathode-ray os cilloscope (C.R.O) 阴极电子示波器ceramic 陶器 charge-clouds 电子云 charge-to-mass ratio(e/m) 质荷比 chemical behaviour 化学行为 chemical property化学性质 clockwise顺时针方向的 compound 化合物 configuration 构型 copper 铜 correspond to 相似 corrosive 腐蚀 d-block elements d 区元素 deflect 使偏向，使转向 derive from 源于 deuterium (D)氘 diffuse mixture 扩散混合物 distance effect 距离效应? distil 蒸馏 distinguish 区别 distribution 分布 doubly charged(2+) ion正二价离子 dye 染料 effect of electric current in solutions 电流在溶液里的影响electrical charge电荷 electrical field 电场
有机化学规律方法总结 第一：有机化学中的方法规律 1.有机物同分异构体的书写方法 〖碳链异构的书写方法〗以己烷（ ）为例，共五种同分异构体（氢原子省略） （1）先直链、一条线 （2）摘一碳、挂中间、往边移、不到端 （3）摘两碳、二甲基、同邻间、不重复、要写全 如果碳链更长，还可以摘两碳、三碳，先甲基，后乙基…… 〖取代基位置异构的书写方法〗 1、对称法（等效氢法） a、同一碳原子上的氢原子是等效的； b、同一碳原子所连甲基上的氢原子是等效的； c、处于镜面对称位置上的氢原子是等效的 2、换元法 详解：同分异构体书写规律：遵循对称性、有序性原则，一般按照下列顺序书写：官能团类型异构；碳链异构；官能团或取代基位置异构；立体异构（较少涉及）口诀：主链长到短，支链整到散，位置心到边，排布对邻间 2.有机物类型异构大全
3.常见有机物的分离提纯方法
4.常见有机物的检验与鉴别
第二：有机化学知识点总结 1．需水浴加热的反应有：（1）、银镜反应（2）、乙酸乙酯的水解（3）苯的硝化（4）糖的水解（5）、酚醛树脂的制取（6）固体溶解度的测定凡是在不高于100℃的条件下反应，均可用水浴加热，其优点：温度变化平稳，不会大起大落，有利于反应的进行。 2．需用温度计的实验有：（1）、实验室制乙烯（170℃）（2）、蒸馏（3）、固体溶解度的测定（4）、乙酸乙酯的水解（70－80℃）（5）、中和热的测定（6）制硝基苯（50－60℃） 〔说明〕：（1）凡需要准确控制温度者均需用温度计。（2）注意温度计水银球的位置。 3．能与Na反应的有机物有：醇、酚、羧酸等——凡含羟基的化合物。
LESSON11NEUTRALIZATION–ACIDSREACTWITHBASES 中和——酸碱反应 生词 neutralization [,nju:tr?lai'zei??n,li'z-] n.[化学]中和；[化学]中和作用；中立状态 Drano Drano是一个排水管清洁剂的品牌，主要成分是氢氧化钠。 《绯闻女孩》-YouarenotusingBlairassexualDrano. lye [lai] n.碱液vt.用碱液洗涤；sodalye：氢氧化钠 oven ['?v?n] n.炉，灶；烤炉，烤箱 Sodiumchloride chloride ['kl?:raid] n.氯化物 add [?d] vi.加； vinegar ['viniɡ?] n.醋 hydrochloric [,haidr?u'kl?rik] adj.氯化氢的，盐酸的 chapter ['t??pt?] n.章; recognize ['rek?ɡnaiz] vt.认出，识别；承认vi.确认，承认；具结 shortcut ['??:tk?t] n.捷径；被切短的东西；快捷键 -desktopshortcut：桌面快捷方式；桌面捷径 beneath [bi'ni:θ]prep.在…之下adv.在下方 literally ['lit?r?li] adv.照字面地；逐字地 coefficient [,k?ui'fi??nt] n.[数]系数；率；协同因素adj.合作的；共同作用的calcium ['k?lsi?m] n.[化学]钙 electriccharge n.电荷（等于charge，electricity）；电费 assimpleamanneraspossible attachedto 附属于；系于；爱慕 incontactwith 接触；与…有联系
英语化学词汇的组词规律 1．化学元素（Chemical elements） A—音译transliteration 锂lithium 镁magnesium 硒selenium 氦helium 锰manganese 钡barium 氟fluorine 钛titanium 铀uranium 氖neon 镍nickel 钙calcium B—意译transcribe or free translation 氧铝溴 氢hydrogen 硅silicon 钾potassium 碳carbon 磷phosphorus 汞mercury 氮nitrogen 硫sulphur(sulfur) 钠sodium C—意音兼译translation by meaning and pronunciation 碘iodine 氯chlorine 砷arsenic 2．化学基团chemical groups 各种元素element的原子atom组合成分子molecule，按性质可以把分子分为不同的原子团radicle，分子中这些具有相对独特性质的原子团，称为官能团functional group，或范围更广一点，称作化学基团chemical group. propylene 丁/四butyl butane butanol butylene butyne butanone 戊/五pentyl pentane pentanol pentylene戊二烯 pentene 戊烯 pentyne pentanone 己/六hexyl hexane hexanol hexylene hexyne hexanone 庚/七heptyl heptane heptanol heptylene heptyne heptanone 辛/八octyl octane octanol octylene octyne octanone 壬/九nonyl nonane nonanol nonylene nonyne nonanone 癸/十decatyl decane decanol decylene decyne decanone 说明：-ol表示醇羟基，-an与烷有关，表示是饱和状态，-en与烯有关，有不饱和的意义，相应的饱和状态的羟基即醇-anol，不饱和状态的羟基即酚-enol。在结构较为复杂的醇类化合物中，常常只用-ol；酮-one是相当于用氧o替换了烯-ene中的一个碳，二者均为不饱和结构。 ether(醚)，ester(酯)，只要在化学基后加上这两个词，就获得了两组英文词。例如：甲醚methyl ether，甲酯methyl ester。 3．化合物类别及化学官能团Classes of compounds and functional groups 化合物类别 烃hydrocarbon 酚hydroxybenzene/phenol 胺amine 烷alkane 醚ether 亚胺imine 烯alkene 酮ketone 腈nitrile 炔alkyne 醛aldehyde 亚砜sulfoxide 苯benzene 酸acid 砜sulfone 醇alcohol 酐anhydride 碱alkali/base 官能团 烷基alkyl 羟基hydroxyl 羰基carbonyl
有机化学知识要点总结 一、有机化学基础知识归纳 1、常温下为气体的有机物： ①烃：分子中碳原子数n≤4（特例：）,一般：n≤16为液态，n＞16为固态。 ②烃的衍生物：甲醛、一氯甲烷。 2、烃的同系物中，随分子中碳原子数的增加，熔、沸点逐渐_ _____，密度增大。同分异构 体中，支链越多，熔、沸点____________。 3、气味。无味—甲烷、乙炔（常因混有PH3、AsH3而带有臭味） 稍有气味—乙烯特殊气味—苯及同系物、萘、石油、苯酚刺激性—--甲醛、甲酸、乙酸、乙醛香味—----乙醇、低级酯 甜味—----乙二醇、丙三醇、蔗糖、葡萄糖苦杏仁味—硝基苯 4、密度比水大的液体有机物有：溴乙烷、溴苯、硝基苯、四氯化碳等。 5、密度比水小的液体有机物有：烃、苯及苯的同系物、大多数酯、一氯烷烃。 6、不溶于水的有机物有：烃、卤代烃、酯、淀粉、纤维素。 苯酚：常温时水溶性不大，但高于65℃时可以与水以任意比互溶。 可溶于水的物质：分子中碳原子数小于、等于3的低级醇、醛、酮、羧酸等 7、特殊的用途：甲苯、苯酚、甘油、纤维素能制备炸药；乙二醇可用作防冻液；甲醛的水溶 液可用来消毒、杀菌、浸制生物标本；葡萄糖或醛类物质可用于制镜业。 8、能与Na反应放出氢气的物质有：醇、酚、羧酸、葡萄糖、氨基酸、苯磺酸等含羟基的 化合物。 9、显酸性的有机物有：含有酚羟基和羧基的化合物。 10、能发生水解反应的物质有：卤代烃、酯（油脂）、二糖、多糖、蛋白质（肽）、盐。 11、能与NaOH溶液发生反应的有机物： （1）酚；（2）羧酸；（3）卤代烃(NaOH水溶液：水解；NaOH醇溶液：消去) （4）酯：（水解，不加热反应慢，加热反应快）；（5）蛋白质（水解） 12、遇石蕊试液显红色或与Na2C03、NaHC03溶液反应产生CO2：羧酸类。 13、与Na2CO3溶液反应但无CO2气体放出：酚； 14、常温下能溶解Cu(OH)2：羧酸； 15、既能与酸又能与碱反应的有机物：具有酸、碱双官能团的有机物（氨基酸、蛋白质等） 16、羧酸酸性强弱： 17、能发生银镜反应或与新制的Cu(OH)2悬浊液共热产生红色沉淀的物质有：醛、甲酸、 甲酸盐、甲酸酯、葡萄糖、麦芽糖等凡含醛基的物质。 18、能使高锰酸钾酸性溶液褪色的物质有： （1）含有碳碳双键、碳碳叁键的烃和烃的衍生物、苯的同系物 （2）含有羟基的化合物如醇和酚类物质
化学常用英语词汇 1．TheIdeal-GasEquation理想气体状态方程 2.PartialPressures分压 3.RealGases:DeviationfromIdealBehavior真实气体：对理想气体行为的偏离 4.ThevanderWaalsEquation范德华方程 5.SystemandSurroundings系统与环境 6.StateandStateFunctions状态与状态函数 7.Process过程 8.Phase相 9.TheFirstLawofThermodynamics热力学第一定律 10.HeatandWork热与功 11.EndothermicandExothermicProcesses吸热与发热过程 12.EnthalpiesofReactions反应热 13.Hess’sLaw盖斯定律 14.EnthalpiesofFormation生成焓 15.ReactionRates反应速率 16.ReactionOrder反应级数 17.RateConstants速率常数 18.ActivationEnergy活化能 19.TheArrheniusEquation阿累尼乌斯方程 20.ReactionMechanisms（机制）反应机理 21.HomogeneousCatalysi（s catalysis英[k?'t?l?s?s]n.催化作用）均相催化剂 22.HeterogeneousCatalysis非均相催化剂 23.Enzymes酶 24.TheEquilibriumConstant平衡常数 25.theDirectionofReaction反应方向 26.LeChatelier’sPrinciple列沙特列原理 27.EffectsofVolume,Pressure,TemperatureChangesandCatalys体ts积，压力，温度变化以及催化剂的影响
Atoms，Elements and Ions atom 原子 element 元素 ion 离子 anion 阴离子 cation 阳离子 electron 电子 neutral 中性的 proton 质子 atomic nucleus 原子核 nucleon 核子 nuclide symbol 核素符号 nuclide 核素 isotope 同位素 alpha particle 粒子 alpha ray 射线 beta particle 粒子 anode 阳级，正极 cathode 阴极 atomic mass unit 原子质量单位 element symbol 元素符号 atomic number 原子序数 atomic weight 原子量 mass number 质量数 atomic theory 原子理论 brownian motion 布朗运动 cathode ray 阴极射线 chemical change 化学变化，化学反应 compound 化合物 deuterium 氘 electric charge 电荷 heavy water 重水 isotopic abundance 同位素的丰度 natural abundance 自然丰度 isotopic mass 同位素质量 law of conservation of mass 物质守恒定律，物质不灭定律 law of definite proportions 确定化定律 law of multiple proportions 整比定律 mass spectrometer 质谱仪 mass spectrometry 质谱学，质谱测量，质谱分析 mass spectrum 质谱（复数形式：mass spectra） nuclear binding energy 核能量 radioactivity 放射能 X–ray spectrum X射线光谱（复数形式：X-ray spectra）
表2-1 中英文对照
CC
表8-1 中英文对照 常用玻璃仪器英文表示 adapter 接液管广口瓶air condenser 空气冷凝管beaker 烧杯boiling flask 烧瓶 boiling flask-3-neck 三口烧瓶burette clamp 滴定管夹 burette stand 滴定架台Busher funnel 布氏漏斗Claisen distilling head 减压蒸馏头 condenser-Allihn type 球型冷凝管 condenser-west tube 直型冷凝管
crucible tongs 坩埚钳 crucible with cover 带盖的坩埚distilling head 蒸馏头 distilling tube 蒸馏管 Erlenmeyer flask 锥型瓶evaporating dish (porcelain) 瓷蒸发皿filter flask(suction flask) 抽滤瓶florence flask 平底烧瓶fractionating column 分馏柱 Geiser burette (stopcock) 酸氏滴定管graduated cylinder 量筒 Hirsch funnel 赫氏漏斗 long-stem funnel 长颈漏斗medicine dropper 滴管 Mohr burette for use with pinchcock 碱氏滴定管 Mohr measuring pipette 量液管mortar 研钵 pestle 研杵 pinch clamp 弹簧节流夹 plastic squeeze bottle 塑料洗瓶reducing bush 大变小转换接头rubber pipette bulb 吸耳球 screw clamp 螺旋夹 separatory funnel 分液漏斗stemless funnel 无颈漏斗 test tube holder 试管夹 test tube 试管 Thiele melting point tube 提勒熔点管transfer pipette 移液管 tripod 三角架 volumetric flask 容量瓶 watch glass 表皿 wide-mouth bottle 广口瓶 【分享】有机物英语单词后缀表 有机物英语单词后缀表 -acetal 缩醇 acid 酸 -al 醛 alcohol 醇 -aldehyde 醛 -aldechydic acid 醛酸
Bunsen burner 本生灯 product 化学反应产物 flask 烧瓶 apparatus 设备 PH indicator PH值指示剂,氢离子(浓度的)负指数指示剂 matrass 卵形瓶 litmus 石蕊 litmus paper 石蕊试纸 graduate, graduated flask 量筒,量杯 reagent 试剂 test tube 试管 burette 滴定管 retort 曲颈甑 still 蒸馏釜 cupel 烤钵 crucible pot, melting pot 坩埚 pipette 吸液管 filter 滤管 stirring rod 搅拌棒 element 元素 body 物体 compound 化合物 atom 原子 gram atom 克原子 atomic weight 原子量 atomic number 原子数 atomic mass 原子质量 molecule 分子 electrolyte 电解质 ion 离子 anion 阴离子 cation 阳离子 electron 电子 isotope 同位素 isomer 同分异物现象 polymer 聚合物 symbol 复合 radical 基 structural formula 分子式 valence, valency 价 monovalent 单价 bivalent 二价 halogen 成盐元素bond 原子的聚合mixture 混合combination 合成作用compound 合成物alloy 合金 metal 金属metalloid 非金属Actinium(Ac) 锕Aluminium(Al) 铝Americium(Am) 镅Antimony(Sb) 锑Argon(Ar) 氩Arsenic(As) 砷Astatine(At) 砹Barium(Ba) 钡Berkelium(Bk) 锫Beryllium(Be) 铍Bismuth(Bi) 铋Boron(B) 硼Bromine(Br) 溴Cadmium(Cd) 镉Caesium(Cs) 铯Calcium(Ca) 钙Californium(Cf) 锎Carbon(C) 碳Cerium(Ce) 铈Chlorine(Cl) 氯Chromium(Cr) 铬Cobalt(Co) 钴Copper(Cu) 铜Curium(Cm) 锔Dysprosium(Dy) 镝Einsteinium(Es) 锿Erbium(Er) 铒Europium(Eu) 铕Fermium(Fm) 镄Fluorine(F) 氟Francium(Fr) 钫Gadolinium(Gd) 钆Gallium(Ga) 镓Germanium(Ge) 锗Gold(Au) 金Hafnium(Hf) 铪Helium(He) 氦
有机化学必背44条规律 1．聚集状态 (1)烃：碳原子数小于或等于4的烃都是气体。(新戊烷是气体，沸点：9.5℃) 烷烃：C1～C4为气体，C5～C16为液体，C17以上为固体。 烯烃：C2～C4为气体，C5～C18为液体，C19以上为固体。 苯的同系物多数为液体，和苯一样有特殊的香味，其蒸气有毒。但对二甲苯为固体。 环丙烷、环丁烷为气体，环戊烷为液体，高级同系物为固体。 (2)烃的衍生物：CH3Cl、CH2＝CHCl、C2H5Cl、HCHO、CH3Br等为气体。 饱和一元醇中，C1～C4为酒味液体，C17以上为固体。 饱和一元羧酸中，C1～C3为具有强烈酸味和刺激性的流动液体，C4～C9为具有无色无臭的油状液体，C10以上为石蜡状固体。 硝基化合物中，一硝基化合物为高沸点液体，其余为结晶固体。 酚类、饱和高级脂肪酸、二元羧酸、芳香酸、脂肪、糖类、氨基酸及萘等为固体。 不饱和脂肪酸(如油酸)为液体。 2．一些有机物的沸点： 3．溶解性 (1)难溶于水：烃类、卤代烃、酯、硝基化合物、高级脂肪酸、多糖、高分子化合物等。 (2)溶于水：低级醇、醛、羧酸、单糖、二糖、氨基酸、丙酮、某些蛋白质溶于水。 (3)与水互溶：乙炔、乙醚微溶于水。常温下，苯酚微溶于水，70℃以上与水
互溶。 (4)难溶于水、且比水轻：烷烃、烯烃、炔烃、C n H n+1Cl、汽油、乙醚、苯(0.8765 g / cm3)、甲苯(0.8669 g / cm3)、二甲苯、环己烷、乙酸乙酯(0.9003 g / cm3)、油酸及低级酯类、油。 (5)难溶于水、且比水重：溴甲烷(1.6755 g / cm3)、溴乙烷(1.4604 g / cm3)、溴苯(1.495 g / cm3)、CCl4、硝基苯、多氯代烃、溴代烃等。 4．最简式相同的 （1）CH：乙炔(C2H2)、(C4H4)、苯(C6H6)、立方烷(C8H8) CH2：烯烃(C2H4)、环烷烃(C n H2n) CH2O：甲醛(CH2O)。乙酸和甲酸甲酯(C2H4O2)、乳酸(C3H6O3)、葡萄糖和果糖(C6H10O6) C6H10O5(葡萄糖单元)：淀粉和纤维素[(C6H10O5)n] C∶H＝1∶1的有：乙炔、苯、苯乙烯、苯酚等； C∶H＝1∶2的有：甲醛、乙酸、甲酸甲酯、葡萄糖、果糖、单烯烃、环烷烃等； C∶H＝1∶4的有：甲烷、甲醇、尿素等。 （2）最简式相同的，所含元素的百分含量不变。最简式相同的有机物，无论多少种，以何种比例混合，混合物中元素质量比值相同。要注意：①含有n个碳原子的饱和一元醛或酮与含有2n个碳原子的饱和一元羧酸和酯具有相同的最简式；②含有n个碳原子的炔烃与含有3n个碳原子的苯及其同系物具有相同的最简式。 （3）最简式相同的有机物，当组成混合物时，只要质量一定，无论以任何配比混合，完全燃烧后，生成CO2的量一定，耗O2量相同。 （4）等质量的最简式相同的化合物燃烧时耗氧量相同。 （5）具有相同的相对分子质量的有机物为：①含有n个碳原子的醇或醚与含有(n－1)个碳原子的同类型羧酸和酯。②含有n个碳原子的烷烃与含有(n－1)个碳原子的饱和一元醛或酮。此规律用于同分异构体的推断。 M得整数商由相对分子质量求有机物的分子式(设烃的相对分子质量为M)① 12 M的余数为0或和余数，商为可能的最大碳原子数，余数为最小氢原子数。② 12 碳原子数≥6时，将碳原子数依次减少一个，每减少一个碳原子即增加12个氢原子，直到饱和为止。 5．(单)烯烃的特点
☆常用： ppm: parts per million ppb: parts per billion pH: potential of hydrogen 1. 化合物的命名：规则：金属(或某些非金属)元素+阴离子名称 (1)MgCl2 magnesium [m?ɡ’ni:zj ?m] chloride (2)NaNO2 sodium nitrite [‘naitrait] (3)KNO3 potassium[p ?’t?si ?m] nitrate [‘naitreit] (4)硝酸 nitric acid (5)NaHCO3 sodium hydrogen carbonate 练习： ? FeBr2 ? (NH4)2SO4 ? NH4H2PO4
?KMnO4 ?亚硫酸 ?sulfurous acid ?H2S ?NO 2 有机物命名 ?Hydrocarbon ?{Aliphatic hydrocarbon; Aromatic Hydrocarbon} ?Aliphatic hydrocarbon (脂肪烃) ?{Alkane (烷); Alkene(烯); Alkyne(炔)} ?Alcohol 醇 ?Aldehyde 醛 ?Ketone [‘ki:t?un] 酮 ?Carboxylic acid 羧酸 ?Aromatic hydrocarbon(芳香烃) ?{benzene (苯) hydroxybenzene(酚) quinone(醌) 无机物中关于数字的写法 mono-, di-, tri-, tetra-, penta- hexa-, hepta-, octa-, nona-, deca- 一，二，三，四，五，六，七，八，九，十 有机物中关于数字的写法 meth-, eth-, prop-, but-, pent-, hex-, 甲乙丙丁戊已 hept-, oct-, non-, dec-, cyclo-, poly- 庚辛壬葵环聚 练习 ?甲烷乙炔 ?丙酮丁醇 ?戊烷己烯 ?庚醛辛烷 ?2-甲基壬酸 3,5-二乙基癸醇
普通化学术语中英文对照表原子atom 原子核nucleus 质子proton 中子neutron 电子electron(abbr.e) 离子ion[an] 阳离子cation 阴离子anion 分子molecular[mlekjl] 单质element 化合物compound 纯净物puresubstance/chemicalsubstance 混合物mixture 相对原子质量relativeatomicmass(abbr.Ar) 相对分子质量relativemolecularmass(abbr.Mr) 化学式量relativeformulamass 物质的量amountofsubstance 阿伏加德罗常数Avogadroconstant(abbr.NA) 摩尔mole 摩尔质量molarmass 实验式/最简式/经验式empiricalformula 气体摩尔体积molarvolume 阿伏加德罗定律Avogadro’sLaw 溶液solution 溶解dissolve 浓度concentration 质量/体积分数mass/volumeconcentration 浓缩concentrate 稀释dilute 蒸馏distillation 萃取extraction 化学反应方程式chemicalequation 反应物reactant 生成物product 固体solid 液体liquid 气体gaseous
溶液aqueous 配平balance 化学计量学stoichiometry 酸acid 碱 base/alkali(referstosolublebases)盐salt 酸式盐acidsalt 离子方程式ionicequation 结晶水waterofcrystallisation 水合的hydrated 无水的anhydrous 滴定titration 指示剂indicator 氧化数oxidationnumber 氧化还原反应redoxreaction 氧化反应oxidationreaction 还原反应reductionreaction 歧化反应disproportionationreaction 归中反应 comproportionation(symproportiona tion) reaction 氧化剂oxidisingagent 还原剂reducingagent 电子层(壳)(electron)shell 电离能ionisationenergy(abbr..) 第一，第二，第三等电离能 first,second,third,etc.ionisation energy(abbr.1st,2nd,3rd,etc..) 电子亲和能 electronaffinityenergy(abbr..) 屏蔽效应electronicshielding(screening) 原子轨道atomicorbital 电子云electroncloud 电子亚层sub-shell 能层layer 能级energylevel 电子排布electronconfiguration 电子云重叠shelloverlap 洪特规则Hund’srule 泡利原理Pauliexclusionprinciple
高中有机化学规律总结 高中化学 2011-04-05 19:26 一、综观近几年来的高考有机化学试题中有关有机物组成和结构部分的题型，其共同特点是通过题给某一有机物的化学式(或式量)，结合该有机物性质，对该有机物的结构进行发散性的思维和推理，从而考查“对微观结构的一定想象力”。为此，必须对有机物的化学式(或式量)具有一定的结构化处理的本领，才能从根本上提高自身的“空间想象能力”。 1．式量相等下的化学式的相互转化关系 一定式量的有机物若要保持式量不变，可采用以下方法 (1) 若少1个碳原子，则增加12个氢原子。 (2) 若少1个碳原子，4个氢原子，则增加1个氧原子。 (3) 若少4个碳原子，则增加3个氧原子。 2．有机物化学式结构化的处理方法 若用C n H m O z (m≤2n＋2，z≥0，m、n?N，z属非负整数)表示烃或烃的含氧衍生 O z(z≥0)相比较，若少于两个H原子，则相当于原有机物物，则可将其与C n H 2n+2 中有一个C＝C，不难发现，有机物C n H m O z分子结构中C＝C数目为个，然后以双键为基准进行以下处理 (1) 一个C＝C相当于一个环。 (2) 一个碳碳叁键相当于二个碳碳双键或一个碳碳双键和一个环。 (3) 一个苯环相当于四个碳碳双键或两个碳碳叁键或其它(见(2))。 (4) 一个羰基相当于一个碳碳双键。 二、有机物结构的推断是高考常见的题型，学习时要掌握以下规律 1．不饱和键数目的确定
(1) 有机物与H 2(或X 2 )完全加成时，若物质的量之比为1∶1，则该有机物 含有一个双键；1∶2时，则该有机物含有一个叁键或两个双键；1∶3时，则该有机物含有三个双键或一个苯环或其它等价形式。 (2) 由不饱和度确定有机物的大致结构 对于烃类物质C n H m，其不饱和度W＝ ① C＝CW＝1； ②CoCW＝2； ③环W＝1； ④苯W＝4； ⑤萘W＝7； ⑥复杂的环烃的不饱和度等于打开碳碳键形成开链化合物的数目。 2．符合一定碳、氢之比的有机物 C∶H＝1∶1的有乙炔、苯、苯乙烯、苯酚等； C∶H＝1∶2的有甲醛、乙酸、甲酸甲酯、葡萄糖、果糖、单烯烃、环烷烃等； C∶H＝1∶4的有甲烷、甲醇、尿素等。 近几年有关推测有机物结构的试题，有这样几种题型 1．根据加成及其衍变关系推断 这类题目的特点是通过有机物的性质推断其结构。解此类题目的依据是烃、醇、醛、羧酸、酯的化学性质，通过知识串联，综合推理，得出有机物的 结构简式。具体方法是① 以加成反应判断有机物结构，用H 2、Br 2 等的量确定 分子中不饱和键类型(双键或叁键)和数目；或以碳的四价及加成产物的结构确定不饱和键位置。② 根据有机物的衍变关系推断有机物的结构，要找出衍变关系中的突破口，然后逐层推导得出结论。 2．根据高聚物(或单体)确定单体(或高聚物) 这类题目在前几年高考题中经常出现，其解题依据是加聚反应和缩聚反应原理。方法是按照加聚反应或缩聚反应原理，由高分子的键节，采用逆向思维，反推单体的结构。由加聚反应得到的高分子求单体，只要知道这个高分子