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This is the author’s post-print version of an article published in the
IMA Journal
of Numerical Analysis, 33 (2)
White Rose Research Online URL for this paper:
http://eprints.whiterose.ac.uk/id/eprint/77779
Published article:
Ranner, T and Elliott, CM (2013)
Finite element analysis for a coupled bulk–
surface partial differential equation.
IMA Journal of Numerical Analysis, 33 (2).
377 - 402. ISSN 0272-4979
Finite element analysis for a coupled bulk-surface partial differential
equation
CHARLESM. ELLIOTT ANDTOMRANNER†
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
[Received on 17 January 2012; revised on 28 June 2012]
In this paper, we define a new finite-element method for numerically approximating the solution of a partial differential equation in a bulk region coupled to a surface partial differential equation posed on the boundary of the bulk domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and use piecewise polynomial boundary faces as an approximation of the surface. Two finite element spaces are defined, one in the bulk region and one on the surface, by taking the set of all continuous functions which are also piecewise polynomial on each bulk simplex or boundary face. We study this method in the context of a model elliptic problem; in particular, we look at well-posedness of the system using a variational formulation, derive perturbation estimates arising from domain approximation, and apply these to find optimal order error estimates. A numerical experiment is described which demonstrates the order of convergence.
Keywords: Surface finite elements; error analysis; bulk-surface elliptic equations
1. Introduction
Coupled bulk-surface partial differential equations arise in many applications; for example, they arise naturally in fluid dynamics and biological applications. This paper studies mathematically a finite ele-ment approach to a sample elliptic problem. The method is based on taking a polyhedral approximation of the domain. Given a sufficiently smooth boundary we go on to show error bounds of orderhk in theH1norm and orderhk+1in theL2norm, wherekis the polynomial degree in the underlying finite element space.
1.1 The coupled system
For a bounded domainΩ⊂RN(N=2,3) with boundaryΓ, we seek solutionsu:Ω→Randv:Γ→R
of the system
−∆u+u=f inΩ, (1.1a)
(αu−βv) +∂u
∂n =0 onΓ, (1.1b)
−∆Γv+v+
∂u
∂n =gonΓ. (1.1c)
Here we assume thatα andβ are given positive constants and f andgare known functions onΩ and
Γ respectively. We denote by∆Γ the Laplace–Beltrami operator onΓ and bynthe outward pointing normal toΓ.
1.2 Applications
In recent times there has been a great deal of attention paid to problems involving diffusion on a surface, for exampleDziuk & Elliott(2007b) and references therein. Of particular interest is cell biology, see, for example,Sbalzariniet al.(2006) andSchwartzet al.(2005). Indeed, cellular metabolism and signalling are mediated in part by trans-membrane receptors that can diffuse in the cell membrane; seeAlbertaet al.
(2002). There are also examples where this surface diffusion is coupled to diffusion in the bulk. For example, fluorescence loss in photobleaching (FLIP) where surface diffusion of a signalling molecule, G-protein RAC, cycles between the cytoplasm (bulk) and cell membrane (surface); see Novaket al.
(2007).
The coupling on the surface (1.1b,1.1c) has been used byNovaket al.(2007). It can be viewed as a linearisation of the more general equation
∂u
∂n
+L(u,v) =0
whereLu>0 andLv<0 which has been used inBooty & Siegel (2010); Kwon & Derby (2001);
Medvedev & Stuchebrukhov(2011);R¨atz & R¨oger(2012) for example. We leave the numerical analysis of more general couplings, the parabolic case, and evolving domains to future work.
1.3 Outline of paper
The paper is laid out as follows. In the second section we will derive a variational form for the equations. The third section looks at existence, uniqueness and regularity of variational solutions. The fourth section focuses on the approximations we make to the geometry of the problem. In the fifth section we develop the finite element method and in the sixth section we will look for error bounds for this method. In the final section we will show some numerical results.
2. Derivation of variational form
2.1 Surface properties
Throughout we will use the notation fromDeckelnicket al.(2005). We will assume thatΓ is a compact (N−1)dimensional hypersurface without boundary and thatΓisC2, so there exists a distance function
d:RN→Rdefined by
d(x) =
−inf{|x−y|:y∈Γ} ifx∈Ω,
0 ifx∈Γ,
inf{|x−y|:y∈Γ} ifx∈/Ω.
Since|∇d(x)| ≡1 in a neighbourhood aboutΓ, we can define the normal toΓ for almost everyx∈Γ
by
n(x) =∇d(x).
It follows that there exists a narrow bandU={x∈RN:|d(x)|<δ
Γ}aboutΓ, such thatd∈C2(U), for which we can also define the normal projectionx7→p(x)fromUontoΓ given by the unique solution
of
This is possible by the assumptions above; see, for example,Hildebrant(1982). Note thatp(x)is the closest point toxonΓ, so pis also the closest point operator. Since this decomposition is unique, we can extendnto a vector field on all ofUso thatn(x) =n(p(x)).
For a functionξ:Γ →Rwe define its surface gradient to be
∇Γξ:=∇ξ−(∇ξ·n)n,
where∇ξ denotes the gradient with respect to ambient coordinates of an arbitrary extension toU of
ξ. Alternatively, we can denote this relation as ∇Γξ =P∇ξ wherePis an N×N tensor given by Pi j=δi j−ninj. Notice thatPis symmetric. The Laplace–Beltrami operator is given by the surface divergence of the surface gradient, that is :
∆Γξ :=∇Γ·∇Γξ.
We denote byH =∇Γ·nthe mean curvature ofΓ. For facts about tangential gradients seeGilbarg &
Trudinger(1983), chapter 16.
We denote by dothe(N−1)dimensional surface measure onΓ. The formula for integration by parts onΓ is given by
Z
Γ
(∇Γ)iξdo=− Z
Γ
ξHnido.
This gives us a surface Green’s formula, for a surface without boundary,
Z
Γ
(−∆Γy)ξdo= Z
Γ
∇Γy·∇Γξdo. (2.2)
2.2 Variational form
We take functionsη,ξ in a suitable space of test functions, multiply (1.1a) byη and (1.1c) byξ, and integrate by parts to get
Z
Ω
∇u·∇η+uηdx−
Z
Γ
η∂u ∂ndo=
Z
Ω
fηdx, (2.3a)
Z
Γ
∇Γv·∇Γξ+vξdo+ Z
Γ
∂u
∂nξdo= Z
Γ
gξdo. (2.3b)
The boundary condition (1.1b) gives us that
−
Z
Γ
η∂u ∂ndo
=
Z
Γ
(αu−βv)ηdo and
Z
Γ
∂u
∂nξdo
=−
Z
Γ
(αu−βv)ξdo. (2.4)
We substitute these into (2.3) to get
Z
Ω
∇u·∇η+uηdx+
Z
Γ
(αu−βv)ηdo=
Z
Ω
fηdx (2.5a)
Z
Γ
∇Γv·∇Γξ+vξdo− Z
Γ
(αu−βv)ξdo=
Z
Γ
gηdo. (2.5b)
We now take a weighted sum of (2.5a) and (2.5b) to obtain the variational form
α
Z
Ω
∇u·∇η+uηdx+β
Z
Γ
∇Γv·∇Γξ+vξ
do
+
Z
Γ
(αu−βv)(α η−β ξ)do=α
Z
Ω
fηdx+β
Z
Γ
To help with notation later, we will writea (u,v),(η,ξ)for the left hand side of this equation and l (η,ξ)for the right hand side.
We will test this variational form over the spaceH1(Ω)×H1(Γ)which we define to be
H1(Ω)×H1(Γ):=(η,ξ)η∈H1(Ω),ξ ∈H1(Γ) . (2.7)
We equip this space with the inner product
h(η1,ξ1),(η2,ξ2)iH1(Ω)×H1(Γ)=hη1,η2iH1(Ω)+hξ1,ξ2iH1(Γ), (2.8)
and induced norm given by
k(η,ξ)kH1(Ω)×H1(Γ)= kηk2H1(Ω)+kξk2H1(Γ)
12
. (2.9)
One may define higher order spaces ifΓ is more regular: to defineHl(Ω)×Hl(Γ), we requireΓ to
beCj,κ withl
6 j+κandκ=0,1. SeeWloka(1987) for details of how to define the surface Sobolev
spaces.
Hence the variational formulation of the problem is to find(u,v)∈H1(Ω)×H1(Γ)such that
a (u,v),(η,ξ)=l (η,ξ)for all(η,ξ)∈H1(Ω)×H1(Γ). (2.10)
3. Existence, uniqueness and regularity
In this section, we apply the usual Lax–Milgram techniques (Evans,1998) to the variational form de-veloped in section2in order to find a unique solution to (2.10). Following this, we split the equations to show regularity with respect to the bulk and surface variables independently. To apply these techniques we must show thatais bounded and coercive andlis bounded overH1(Ω)×H1(Γ).
To see thatais bounded, notice that
a (w,y),(η,ξ)
6αkwkH1(Ω)kηkH1(Ω)+βkykH1(Γ)kξkH1(Γ)
+
Z
Γ
(αw−βy)(α η−β ξ)do
6√2 max{α,β} k(w,y)kH1(Ω)×H1(Γ)k(η,ξ)kH1(Ω)×H1(Γ) (3.1)
+2c2Tmax{α,β}2k(w,y)kH1(Ω)×H1(Γ)k(η,ξ)kH1(Ω)×H1(Γ)
6ck(w,y)kH1(Ω)×H1(Γ)k(η,ξ)kH1(Ω)×H1(Γ).
Here,cT is the constant from the Trace Theorem, seeEvans(1998). Coercivity ofais immediate since we have
a (η,ξ),(η,ξ)=αkηk2H1(Ω)+βkξk2H1(Γ)+kα η−β ξk2L2(Γ) (3.2) >√2 min{α,β} k(η,ξ)k2H1(Ω)×H1(Γ).
THEOREM3.1 (Existence and uniqueness) Given f∈H−1(Ω),g∈H−1(Γ), andα,β>0, there exists a unique pair(u,v)∈H1(Ω)×H1(Γ)such that
a (u,v),(η,ξ)=l (η,ξ)for all(η,ξ)∈H1(Ω)×H1(Γ). (3.3)
Furthermore, ifΓ isC3, we can achieve bounds in theH2-norms by consider restricting the bilinear
forms by settingηandξ =0 in turn.
Forη=0, we get
β
Z
Γ
∇Γv·∇Γξ+vξdo+ Z
Γ
β2vξdo=β
Z
Γ
gξdo+
Z
Γ
α βuξdo. (3.4)
This is exactly the variational form of the equation
−β ∆Γv+ (β+β 2)v=
βg+α βuonΓ. (3.5)
Hence by surface elliptic theory (Aubin,1982), ifΓ isC3, we have thatv∈H2(Γ). Since, by the Trace
Theorem,u∈L2(Γ), we have the bound:
kvkH2(Γ)6c kgkL2(Γ)+kvkL2(Γ)+kukH1(Ω)
. (3.6)
Forξ =0, we get
α
Z
Ω
∇u·∇η+uηdx+
Z
Γ
α2uηdo=α
Z
Ω
fηdx+
Z
Γ
α βvηdo. (3.7)
This equation arises as the variational form of the equations
−α ∆u+αu=αf inΩ (3.8a)
∂u
∂n+αu=βvonΓ (3.8b)
By regularity theory of elliptic problems with Robin boundary data (seeGilbarg & Trudinger(1983);
Ladyzhenskaia & Uraltseva(1968)), ifΓ isC3, we have the following result:
kukH2(Ω)6c
kfkL2(Ω)+kvkH1/2(Γ)
. (3.9)
THEOREM3.2 (Regularity) IfΓ isC3, f ∈L2(Ω),g∈L2(Γ)andα,β >0 and(u,v)solve the
varia-tional problem (2.6), then
u∈H2(Ω)andv∈H2(Γ),
and
k(u,v)kH2(Ω)×H2(Γ)6c kfkL2(Ω)+kgkL2(Γ). (3.10)
4. Domain perturbation and estimates
4.1 Domain approximation
The first step we take in discretising the system (1.1) is to takekth order approximationsΩh(k)andΓh(k)
to define the triangulation of our bulk domain and use results ofDziuk(1988);Dziuk & Elliott(2007b) andDemlow(2009) to make estimates about the perturbation of the boundary of this domain. To prove the results in this section, we will assumeΓ isCk+1. The higher order surface finite element spaces, used here, are described inHeine(2005).
Let ˇΩhbe a polyhedral approximation ofΩ and ˇΓh=∂Ωˇh. We suppose that the faces of ˇΓh are
(N−1)-simplices whose vertices lie onΓ so that ˇΓhis an interpolant ofΓ. We take a quasi-uniform triangulation ˇThof ˇΩh(Brenner & Scott,2002) consisting of closed simplices, either triangles inR2or
tetrahedra inR3.
We defineh=max{diam(T):T∈Thˇ}and assume thathis sufficiently small so that ˇΓh⊆Uso that for allx∈Γˇh, there exists a unique pointp=p(x)∈Γ defined by (2.1). Finally, we assume that for each T ∈Thˇ,T∩Γˇhhas at most one face ofT.
4.1.1 Exact triangulation. In order to define our computational domains, we first define an exact triangulation ofΩ. For each simplexT∈Thˇ, we define an affine functionFT:RN→RN which maps
the unitN-simplex ˆT ontoT (mapping the vertices of ˆT onto the vertices ofT) which we write as
FT(xˆ) =ATxˆ+bT. (4.1)
We say that a closed setTeis a curvedN-simplex if there exists aC1-mappingFTethat maps ˆT ontoTe that is of the form
FTe=FT+ΦT, (4.2)
whereFT is the affine map from (4.1) andΦT is aC1-mapping from ˆT toRN satisfying
CT :=sup ˆ x∈Tˆ
DΦT(xˆ)A−T1
6C<1. (4.3)
From this definition we immediately have the following results:
PROPOSITION4.1 If the mapFTeexists, then it is aC1-diffeomorphism from ˆT ontoTeand satisfies
sup ˆ x∈Tˆ
|DFTe(xˆ)|6(1+CT)|AT|,
sup x∈Te
D(FTe)−1(x)
6(1−CT)−1
A−T1
,
(1−CT)N|detAT|6|detDFTe(xˆ)|6(1+CT)N|detDFT| for all ˆx∈Tˆ.
There are several ways of defining such aΦT given in the literature. Zlamal (1973, 1974) and
Scott(1973) considered problems with finite element spaces defined over curved spaces. Scott gives an explicit construction of an exact triangulation in two dimensions which was generalised byLenoir
(1986). Here, in this paper, we use a construction based on Dubois (1990) which uses the normal projection (2.1). We will adopt the notation ofB¨ansch & Deckelnick(1999);Deckelnicket al.(2009).
Bearing in mind our assumptions on the triangulation, eachT∈Thˇ is either an internal simplex, with at most one node on ˇΓh, in which case we setΦT =0; orT has more than one node on the boundary. We denote byl the number of nodes ofT that lie in ˇΓhand denote byψ1, . . . ,ψN+1the vertices ofT, ordered so thatψ1, . . . ,ψllie on ˇΓh. For each pointx∈T, we define barycentric coordinates by
x=
N+1
∑
j=1
x y
ψ3
ψ4 x
~ p(y)
ψ2
ψ1
FIG. 1: An illustration of the construction of the exact triangulation ofΩ. The pointxis mapped onto
y∈τ (the simplex spanned byψ1,ψ2,ψ3) and then toex=F
e
T(x)by using the closest point projection (2.1) ofy.
and write ˆx= (λ1, . . . ,λN)for the coordinates in ˆT. We next introduce
λ∗=λ∗(xˆ) =
l
∑
j=1
λj σˆ ={xˆ∈Tˆ:λ∗(xˆ) =0}.
In three dimensions this falls into the following cases:
1. T∩Γˇ
his an edge of a tetrahedron (l=2), then ˆσ is the inverse image of the edge spanned by
ψ3,ψ4underFT.
2. T∩Γˇ
his a face of a tetrahedron (l=3), then ˆσis the pointFT−1(ψ4). For ˆx∈/σˆ, we denote the projection ofxontoτbyy=y(xˆ)∈τby
y=
l
∑
j=1
λj
λ∗ψj∈τ.
Then using the normal projectionp(y)∈Γ ofygiven by (2.1) and we defineΦT by
ΦT(xˆ) =
(
(λ∗)k+2(p(y)−y) if ˆx∈/σˆ 0 if ˆx∈σˆ.
(4.4)
We now follow a sequence of Lemmas fromBernardi(1989) to show thatΦT satisfies (4.3).
LEMMA4.1 The mappingyis of classCk+1on ˆT\σˆ and satisfies
kDmxˆykL∞(Tˆ\σˆ)6
ch
Proof. SeeBernardi(1989), Lemma 6.2.
LEMMA4.2 The mappingp(y)is of classCk+1on ˆT\σˆ and we have the bound
kDmxˆ(p(y)−y)kL∞(Tˆ\σˆ)6
ch2
(λ∗)m. (4.6)
Proof. We remark, usingBernardi(1989) equation (2.9),
kDmxˆ(p(y)−y)kL∞(Tˆ\σˆ)6c
m
∑
r=1
Dry(p(y)−y)
L∞(τ)
m
∏
q=1
Dqxˆy
iq L∞(Tˆ\σˆ)
!
wherei= (i1, . . . ,im)∈Nmwith
m
∑
q=1
iq=r and
m
∑
q=1
q iq=m.
We notice that p(y) =yify=ψjfor any 06 j6l, soy|τ can be seen as a linear interpolant of p(y) onτ. Hence, from our geometric assumptions onΓ (Dziuk,1988),Dry(p(y)−y)
L∞(τ) 6ch
2−r for 06r62.
Using (4.5) we see, ifm62
kDmxˆ(p(y)−y)kL∞(Tˆ\σˆ)6c
m
∑
r=1
h2−rh(∑mq=1iq)(λ∗)−(∑mq=1q iq)6 ch 2
(λ∗)m,
and ifm>2,
kDmxˆ(p(y)−y)kL∞(Tˆ\σˆ)6c
2
∑
r=1
h2−rh(∑mq=1iq)(λ∗)−(∑mq=1qiq)+ m
∑
r=3
h(∑mq=1iq)(λ∗)−(∑mq=1qiq)
!
6 ch 2
(λ∗)m.
PROPOSITION4.2 The mappingΦT(xˆ) = (λ∗)k+2(p(y)−y)is of classCk+1on ˆT and satisfies
kDmΦTkL∞(Tˆ)6ch2 for 06m6k+1. (4.7)
Furthermore,ΦT satisfies (4.3).
Proof. Using the Leibniz formula, we have for any ˆxin ˆT\σˆ,
DmΦT(xˆ) =Dmxˆ (λ
∗)k+2(p(y)−y)) = m
∑
r=0
m r
(k+2). . .(k+3−r)(λ∗)k+2−r(Dxˆλ∗)rDmxˆ−r(p(y)−y),
so that applying (4.6)
D
m ˆ x (λ
∗)k+2(p(y)−y)
L∞(Tˆ\σˆ)6 c
m
∑
r=0
(λ∗)k+2−r ch
2
(λ∗)m−r 6ch
FIG. 2: A plot of two sections of triangulations. The left shows three tetrahedra in ˇThand the rights shows the corresponding three tetrahedra inTe
h. The surface is shown by spots on both sides. The red and yellow tetrahedra (left and right in each image) share a face with the boundary (l=3) and the blue tetrahedron (center in each image) shares an edge with the boundary (l=2). This means that the red and yellow curved tetrahedron have four curved faces and the blue tetrahedron has two curved faces.
The mappingΦTis is of classCk+1on ˆT\σˆ with derivatives of order less than or equal tok+1 tending to zero when ˆxtends to a point in ˆσ. Hence, it can be extended to aCk+1-mapping on ˆT (Gilbarg & Trudinger,1983) which satisfies (4.7).
Since
∂xˆl ∂xj
6c/h(Ciarlet & Raviart,1972a, page 239), we know that
A−T1 =
c h.
This result together with (4.7) shows
CT 6sup
ˆ x∈Tˆ
|DΦT(xˆ)|
A−T1
6ch,
henceΦT satisfies (4.3) forhsmall enough.
REMARK 4.1 Note that we could have chosenΦT(xˆ) =λ∗(p(y)−y). However this function is not
C1(T), and the interpolation theory ofBernardi (1989) would be unavailable. Our construction is a combination of ideas fromLenoir(1986) andDubois(1990).
We will call the exact triangulation, defined byFTeabove,The. Notice that under this construction, simplices inThe, which have more than one vertex on the boundary, can have more than one curved face. See Figure2for an example.
4.1.2 Computational domain. We can now define our computational domainsΩh(k)andΓh(k). LetT∈
ˇ
Thandφ1k, . . . ,φnkk be a Lagrangian basis of degreekon ˆT corresponding to the nodal points ˆx1, . . . ,xˆnk.
Then for ˆx∈Tˆ, we can define a parametrisation of a polynomial simplexT(k)by
FT(k)(xˆ) =
nk
∑
j=1
We can carry out this procedure for each simplexT∈Thˇ. Since the basis functions{φkj}are unisolvent,
FT(k)is also a diffeomorphism. We defineΩh(k)as the union of elementsTh(k)given by
T(k):={FT(k)(xˆ): ˆx∈Tˆ} T(k)
h :={T
(k)|T ∈Thˇ}. (4.8)
ThenΓh(k)is the boundary of the domainΩh(k)with the triangulationTh(k)
Γh(k)
. This construction admits
quasi-uniform triangulationsTh(k) and Th(k)
Γh(k)
forΩh(k) andΓh(k) respectively. Notice that, like the
exact simplices inThe, the simplcies inTh(k)can have curved (polynomial) faces.
4.2 Bulk estimates
We define a functionGh: Ω
(k)
h →Ω locally byGh|T(k) :=FTe◦(F
(k)
T )
−1for eachT(k)∈T(k)
h . This is a homeomorphism, which when restricted to interior simplices (those with at most one vertex on the boundary) is the identity.
We use the notationDGhfor the gradient ofGhwhere(DGh)i j=∂∂xj(Gh)i, andDGht for its
trans-pose. Also, we will writeDG−h1forD(G−h1) = (DGh)−1. We denote byJh|T the absolute value of the determinant ofDGh|T.
We denote byBhthe union of elements inTh(k)which have more than one vertex on the boundary
Γh(k)andB`hthe associated exact elements inThe. Notice thatBhis the region whereGhis different from the identity.
Let us use the notation that for a fixed ˆx∈Tˆ, we denoteF(k)
T (xˆ) =x, then one may write that
Gh(x) =FTe((F
(k)
T )
−1(x)) =Fe
T(xˆ) =x+ (FTe(xˆ)−F
(k)
T (xˆ)). (4.9)
LEMMA4.3 IfΓisCk+1thenGh|T∈Ck+1(T(k))for eachT(k)∈Th(k)and we have thatkGhkWk+1,∞(T(k))
is bounded independently ofh.
Proof. Using (4.9), we can writeGhas
Gh(x) =FT(xˆ) +ΦT(xˆ).
Sincex7→xˆis smooth,Ghis the sum of an affine function and aCk+1function, soGhis of classCk+1 onT(k). To achieve the bound independently ofh, we use (4.3).
PROPOSITION4.3 (Geometric bulk estimates) LetT ∈Th(k) be a boundary simplex (one which has
more than one vertex on the boundaryΓh(k)), andTe the associated exact triangle inThe. Under the
assumption thatThis quasi-uniform, for sufficiently smallh, we have that
DGht|T−Id
L∞(T)6ch
k, (4.10a)
kJh|T−1kL∞(T)6chk. (4.10b)
Proof. We will bound
∂
∂xj
(Gh)i−δi j
which will show the estimates above.
We start by taking thexjderivative ofGhto get
∂
∂xj
(Gh)i=
∑
l∂(FT(k))−1(x)l
∂xj
∂ FTe(xˆ)i ∂xˆl
,
where we have used the substitutionFT(k)(xˆ) =x. We note that this means
∑
l
∂(FT(k))−1(x) l
∂xj
∂ FT(k)(xˆ)i ∂xˆl
=∂(F
(k)
T )
−1(x)
∂xj
=δi j
Hence
∂
∂xj
(Gh)i−δi j=
∑
l∂(FT(k))−1(x) l
∂xj
∂
∂xˆl
FTe(xˆ)−FT(k)(xˆ)i.
It is classical (Ciarlet & Raviart,1972a, Lemma 7, page 238) that
∂ (FT(k))−1(xˆ)
l
∂xj
=
∂xˆl
∂xj
6 c h,
and from standard interpolation theory, we see that
∂
∂xˆl
FTe(xˆ)−FT(k)(xˆ)
i 6 c D
k+1 ˆ x (F
e T)
L∞(Tˆ).
However we may use the fact thatDmxˆ+1xj
6chm (Ciarlet & Raviart,1972a, page 239) and change
coordinates to see
D
k+1 ˆ x (F
e T)
L∞(Tˆ)6ch
k+1 (F
e T◦(F
(k)
T )
−1)
Wk+1,∞(T(k))=ch
k+1kG
hkWk+1,∞(T(k)).
From Lemma4.3, we knowkGhkWk+1,∞(T(k))is bounded independently ofh, this shows that
∂
∂xj
(Gh)i−δi j
6
chk.
We can now lift a function defined onΩh(k)onto a function defined onΩ.
DEFINITION4.4 For a functionηh:Ω
(k)
h →R, we define its liftηh`:Ω→Rby
ηh`:=ηh◦G−h1.
For a functionη:Ω →R, we can also define an inverse liftη−`:Ωh(k)→Rby
η−`:=η◦Gh.
We also have equivalence of norms via this lifting process:
PROPOSITION4.5 Letηh:Ω
(k)
h →Rand letηh`: Ω→Rbe its lift. Then there exists constantsc1,c2 independent ofhsuch that
c1 η ` h
L2(Ω)6kηhkL2(Ω(k) h )6
c2 η ` h
L2(Ω) (4.11a) c1 ∇η ` h
L2(Ω)6k∇ηhkL2(Ω(k) h )6
c2 ∇η ` h
L2(Ω). (4.11b) Proof. We can write integrals overΩh(k)in the following way
Z
Ωh(k)
ηh(x)dx= Z
Ω
ηh`(y) 1
J`
h(y) dy,
and the gradient onΩh(k)as
∇xηh(x) =DGht(y)∇yηh`(y).
The results follows simply from applying the previous proposition. In the following error analysis, we will require the following narrow band trace inequality.
LEMMA4.4 LetNδ ⊆Ube the band of widthδ<δΓ given by
Nδ={x∈Ω :−δ<d(x)<0}.
It holds that forη∈H1(Ω)
kηkL2(N
δ)6cδ
1 2kηk
H1(Ω). (4.12)
Proof. First, we may assume thatη∈C1(Ω), since the more general result will follow by a density
argument. Note thatd∈C2(Nδ)and|∇d| ≡1 onNδ. We can apply the coarea formula to integrals overNδ as follows:
Z
Nδ
η(y)2dy=
Z
Nδ
η(y)2|∇d(y)|dy
=
Z 0
−δ Z
Γs
η2|Γsdods.
HereΓsdenotes theC2hypersurface which is the inverse image ofsunderd, namelyΓs={x∈Nδ : d(x) =s}. Next, we wish to apply a trace inequality type argument to bound the right hand side of this equation. We follow the proof of the trace inequality (Theorem 1.5.1.10) fromGrisvard(2011). Let the vector fieldD: ¯Ω →RN be an extension of∇dof classC1on ¯Ω, equal to∇d onNδ, with the bound
kDkC1(Ω¯)6ckdkC2(N
δ). SettingΩs={x∈Ω :d(x)<s}, see Figure3, we have that
Z
Ωs
∇(η2)·Ddx=2 Z
Ωs
η∇η·Ddx.
On the other hand applying Green’s theorem, using the notationnsfor the normal toΓs, we obtain
Z
Ωs
∇(η2)·Ddx=
Z
Γs
η2D·nsdo− Z
Ωs
Γ
Γs
Ω Ω
s
FIG. 3: A cartoon of the setup ofΩs(yellow) andΓslying insideΩ (red).
SinceD·ns=1 onΓs, combining these two equations we have that
Z
Γs
η2D·nsdo=2 Z
Ωs
η∇η·Ddx+
Z
Ωs
η2∇·Ddx,
which means that
Z
Γs
η2do62 max ¯ Ωs
|D|
Z
Ωs
|η| |∇η|dx+max ¯ Ωs
|∇·D|
Z
Ωs
η2dx.
Since we have thatΩs⊆Ω, applying a Young’s inequality gives
Z
Γs
η2do6ckDkC1(Ω¯) Z
Ω
∇η2
+η2dx.
Hence we have that
Z
Nδ
η2dy6cδkηk2H1(Ω). (4.13)
4.3 Surface estimates
We have the following geometric estimates for the surfaceΓh. They follow sinceΓhcan be viewed as an interpolant ofΓ. Details can be found inDziuk(1988);Dziuk & Elliott(2007a,b);Demlow(2009).
PROPOSITION4.6 (Geometric surface estimates) Under the above assumptions onΓ andΓhwe have that
kdk
L∞(Γ(k)
h )6 chk+1.
Letµhbe the quotient of the measures on the surface and the approximate surface, so that do=µhdoh. Then we have the the estimate
sup
Γh(k)
|1−µh|6chk+1. (4.14)
LetPandPhdenote the projections onto the tangent spaces ofΓ andΓhrespectively. We introduce the notation
Qh= 1 µh
(Id−dH)PPhP(Id−dH) (4.15)
then we have the estimate that
p(x)
ν(x) x h
p(x) ν(x)
x Γ
Γ
FIG. 4: A section of a surface triangulation with normal lifts shown inR3
A proof can be found inDziuk(1988); Dziuk & Elliott(2007a) for the linear case and Demlow
(2009) for higher orders.
We use the closest point operator (2.1) to define the lift and inverse lift of surface functions.
DEFINITION4.7 Givenξh:Γh(k)→R, we define its lift, denoted byξh`:Γ →R, by
ξh`(p(x)):=ξh(x).
Similarly for a functionξ:Γ →R, we define its inverse lift, writtenξ−`:Γh(k)→R, by
ξ−`(x):=ξ(p(x)).
It can be shown that the following norms are equivalent via this lifting process.
PROPOSITION4.8 Letξh:Γh(k)→Rand letξh`:Γ →Rbe its lift. Then there exists constantsc1,c2 independent ofhsuch that
c1
ξ
`
h
L2(Γ)6kξhkL2(Γ(k)
h )6 c2
ξ
`
h
L2(Γ) (4.17a)
c1
∇Γξ
`
h
L2(Γ)6
∇Γhξh
L2(Γ(k) h )6
c2
∇Γξ
`
h
L2(Γ). (4.17b)
A proof is given inDziuk(1988);Dziuk & Elliott(2007a) fork=1 andDemlow(2009) for any k>1.
5. Finite Element Method
5.1 Isoparametric finite element spaces
We use this section to define the finite element spacesVhandShthat our finite element method will be based on. We recall that the computational domainsΩhandΓhare defined elementwise by a parametri-sationFT(k): ˆT →T(k)⊂Ωh(k)as in (4.8). In both the bulk and surface cases, we define the finite element functions to be continuous functions which are piecewise polynomials of degreekwith respect to the barycentric coordinates of the reference element in dimensionN andN−1. An important part of the construction is that the trace of a function onΓh(k)inVhlies inSh.
More precisely for the bulk finite element functions:
Vh={ηh∈C(Ω
(k)
h ):ηh|T=ηˆh◦(F
(k)
T )
−1with ˆ
ηh∈Pk(Tˆ)for allT ∈Th}
For the surface finite element functions, we introduce
Sh={ξh∈C(Γ
(k)
h ):ξh|τ=ξˆh◦(F
(k T )
−1with ˆ
ξh∈Pk(τ)ˆ for allT∈Thwithτ=T∩Γh6=/0}.
We have used the notation ˆτ= (FT(k))−1(τ)for the face of the reference element ˆT corresponding toτ, andPk(ω)for the space of polynomials of degreekonω.
From now on we will assumekis fixed and writeΩh,Γh,ThforΩh(k),Γh(k),Th(k)without ambiguity.
5.2 Description of the method
We define approximate data fh,ghusing the appropriate inverse lifts. That is
fh=f−`Jh gh=g−`µh. (5.1)
The approximate problem is then to find(uh,vh)∈Vh×Shsuch that
α
Z
Ωh
∇uh·∇ηh+uhηhdx+β Z
Γh
∇Γhvh·∇Γhξh+vhξhdoh
+
Z
Γh
(αuh−βvh)(α ηh−β ξh)doh=α Z
Ωh
fhηhdx+β Z
Γh
ghξhdoh, (5.2)
for all(ηh,ξh)∈Vh×Sh,
where∇Γhis the surface gradient onΓh.
REMARK5.1 This choice of fhandghis not fully practical for arbitrary(f,g)∈L2(Ω)×L2(Γ)as the right hand side integrals would need to be calculated via some numerical integration rule. We are not concerned in analysing such errors in this paper and will assume that it is possible to calculate these integrals exactly. For general results on numerical integration in the context of curved domains see
Ciarlet & Raviart(1972b);Barrett & Elliott(1987).
We introduce the bilinear and linear forms onVh×Sh:
ah (wh,yh),(ηh,ξh)
=α
Z
Ωh
∇wh·∇ηh+whηhdx
+β
Z
Γh
∇Γhyh·∇Γhξh+yhξhdoh,
+
Z
Γh
(αwh−βyh)(α ηh−β ξh)doh
lh (ηh,ξh)=α
Z
Ωh
fhηhdx+β Z
Γh
ghξhdoh,
so that we can write (5.2) as: find(uh,vh)∈Vh×Shsuch that
ah (uh,vh),(ηh,ξh)
=lh (ηh,ξh)
for all(ηh,ξh)∈Vh×Sh. (5.3)
THEOREM5.1 The finite element method defined in (5.2) has a unique solution(uh,vh)∈Vh×Shfor allhwhich satisfies the bound
k(uh,vh)kH1(Ωh)×H1(Γ
h)6ck(f,g)kL2(Ω)×L2(Γ). (5.4) Proof. It is clear that the equations have a unique solution sinceahis also coercive – this follows from the same reasoning as (3.2). To show the bound we use the coercivity ofah, the equivalence of norms shown in (4.17a,4.11a) and (4.14,4.10) to see forhsmall enough
k(uh,vh)kH1(Ω
h)×H1(Γh)6ck(fh,gh)kL2(Ωh)×L2(Γh) 6ck(f,g)kL2(Ω)×L2(Γ).
5.3 Lifted finite element spaces
In order to prove error bounds, we define the lifted finite element spaces that lifts of finite element functions live in. In particular this allows us to define(u`h,v`h), the lifts of the finite element solution, defined on the same domain as the solutions of the continuous problem. We define the lift of the finite element spaces as
Vh`={ηh`:ηh∈Vh} ⊆H1(Ω)
S`h={ξh`:ξh∈Sh} ⊆H1(Γ).
(5.5)
It is important to note that the trace onΓ of functions inVh`live inS`h.
PROPOSITION5.2 (Approximation property) For the lifted finite element spacesVh`,Sh`defined above there exists an interpolation operatorIh:Hk+1(Ω)×Hk+1(Γ)→Vh`×S`hsuch that for 26m6k+1,
k(w,y)−Ih(w,y)kL2(Ω)×L2(Γ)+hk(w,y)−Ih(w,y)kH1(Ω)×H1(Γ)6chmk(w,y)kHm(Ω)×Hm(Γ) (5.6)
Proof. We start by defining the interpolation operatoreIh:H2(Ω)×H2(Γ)→Vh×Sh so that(w,y) andIeh(w,y)agree at the nodes ofΩh andΓh. We use both lifts to define Ih(w,y) =
e
Ih(w,y)
`
. The error bounds follow from given interpolation theory; seeBernardi(1989), Corollary 4.1, for the bulk
andDemlow(2009) for the surface.
Using the fact that
∇(w`h) =∇(wh◦G−h1) =DG
−t h (∇wh)`, (writingDG−htfor(DGh−1)t) and fromDziuk(1988),
(Ph(Id−dH))∇Γ(y
`
h) = (∇Γhyh)
`,
we have that
ah (wh,yh),(ηh,ξh)
=α
Z
Ω
DGht∇w`h·DGht∇ηh`+w`hηh`1
Jh`dx
+β
Z
Γ Q`
h∇Γy`h·∇Γξh`+y`hξh` 1
µh`do
+
Z
Γ
(αw`h−βy`h)(α ηh`−β ξh`) 1 µh`do
=:a`h (w`h,yh`),(ηh`,ξh`).
for all(wh,yh),(ηh,ξh)∈Vh×Shwith lifts(w`h,y`h),(ηh`,ξh`)∈Vh`×S`h. Whereas for the right hand side, we immediately have thatlh (ηh,ξh)
=l (ηh`,ξh`)since Z
Ωh
fhηhdx= Z
Ωh
(f−`Jh)ηhdx= Z
Ω
(f−`Jh)`ηh`1
Jh`dx= Z
Ω
f Jh`ηh`1
Jh`dx= Z
Ω fηh`dx,
and
Z
Γh
ghξhdoh= Z
Γh
(g−`µh)ξhdoh= Z
Γ
(g−`µh)`ξh` 1
µh`
do=
Z
Γ
gµh`ξh` 1 µh`
do=
Z
Γ gξh`do.
Hence we may rewrite (5.3) as: Find(u`h,vh`)∈Vh`×S`h
a`h (u`h,vh`),(ηh`,ξh`)
=l (ηh`,ξh`)
for all(ηh`,ξh`)∈Vh`×S`h. (5.7)
We will make use of the fact thata`hnow make sense for all function pairs inH1(Ω)×H1(Γ)in the following.
6. Error analysis
In this section, we wish to compare the error of the solutions(u,v)of the exact problem (1.1) to the solutions(uh,vh)of the approximate problem (5.2) defined in section5.
One of the problems we have to overcome is the fact that the two problems are posed over different domains. However the lift operators we have defined will help us.
In order to derive optimal order estimates fork>1, we must assume higher regularity of the smooth solution(u,v)of (2.10) and the surfaceΓ. We require(u,v)∈Hk+1(Ω)×Hk+1(Γ)which requiresΓ
THEOREM6.1 Let(u,v)∈Hk+1(Ω)×Hk+1(Γ)be the solution of the variational problem (2.10) and let(uh,vh)∈Vh×Shbe the solution of the finite element scheme given by (5.2). Denote byu`handv`h the lifts ofuhandvhrespectively. Then we have the following error bounds:
(u−u
`
h,v−v
`
h)
H1(Ω)×H1(Γ)6C1h
k, (6.1)
where
C1=c k(u,v)kHk+1(Ω)×Hk+1(Γ)+k(f,g)kL2(Ω)×L2(Γ),
and
(u−u
`
h,v−v
`
h)
L2(Ω)×L2(Γ)6C2h
k+1, (6.2)
where
C2=c k(u,v)kHk+1(Ω)×Hk+1(Γ)+k(f,g)kL2(Ω)×L2(Γ)
.
6.1 Geometric errors
Part of the error of the finite element method comes from the fact that there is a so-called ‘variational crime’, that is we are using different bilinear forms in the exact and approximate formulations and Vh6⊆H1(Ω)andSh6⊆H1(Γ). These errors come from the change in geometry of the computational domain.
LEMMA6.1 For(w,y),(η,ξ)∈Vh`×S`hwe have
a (w,y),(η,ξ)
−a`h (w,y),(η,ξ)
6chkkwkH1(B`
h)kηkH1(B`h)
+chk+1k(w,y)kH1(Ω×H1(Γ)k(η,ξ)kH1(Ω)×H1(Γ).
(6.3)
Proof. To prove this lemma we will split the formsaanda`hinto bulk, surface and cross terms. That is
a(Ω)(w,η) =
α
Z
Ω
∇w·∇η+wηdx
a(Γ)(y,
ξ) =β
Z
Γ
∇Γy·∇Γξ+yξdo
a(×) (w,y),(η,ξ)=
Z
Γ
(αw−βy)(α η−β ξ)do.
We definea(h·)`similarly.
Givenw,η∈Vh`, for the bulk term we see that
Z
Ωh
∇w−`·∇η`dx−
Z
Ω
∇w·∇ηdx
=A1+A2+A3,
where
A1=
Z
Ω
(DGht−Id)∇w·DGht∇η 1
Jh`dx,
A2=
Z
Ω
∇w·(DGht−Id)∇η 1
Jh`dx,
A3=
Z
Ω
∇w·∇η(1
Making use of the fact that
1
Jh`−1=0 and DG
t
h−Id=0, inΩ\B
`
h,
we actually have
A1=
Z
B`
h
(DGht−Id)∇w·DGht∇η1
Jh`dx,
A2=
Z
B`h
∇w·(DGht−Id)∇η 1
J`
h dx,
A3=
Z
B`h
∇w·∇η(1
Jh`−1)dx.
Using Proposition4.3, we see the three termsAjare bounded by
chkk∇wkL2(B`
h)
k∇ηkL2(Ω).
Similarly, Z Ωh
w−`η−`dx−
Z
Ω wηdx
= Z Ω wη(1
J`
h
−1)dx)
6
chkkwkL2(Ω)kηkL2(Ω).
Giveny,ξ ∈S`h, for the surface terms we see that, using Proposition4.6,
Z Γh
∇Γhy
−`·
∇Γhξ
−`do
h− Z
Γ
∇Γy·∇Γξdo
= Z Γ
(Id−µhQ` `h)∇Γy·∇Γξdo
6
chk+1k∇ΓykL2(Γ)k∇ΓξkL2(Γ),
and Z Γh
y−`ξ−`doh− Z
Γ yξdo
= Z Γ yξ( 1
µh`
−1)do
6
chk+1kykL2(Γ)kξkL2(Γ).
Using the previous result we also have that
Z Γh
(αw−`−βy−`)(α η−`−β ξ−`)doh− Z
Γ
(αw−βy)(α η−β ξ)do
= Z Γ
(αw−βy)(α η−β ξ)(1 µh`
−1)do
6chk+1k(w,y)kL2(Γ)×L2(Γ)k(η,ξ)kL2(Γ)×L2(Γ) 6chk+1k(w,y)kH1(Ω)×H1(Γ)k(η,ξ)kH1(Ω)×H1(Γ).
We remark briefly that sinceB`his contained inΩ we also have for functions(η,ξ)∈H1(Ω)×
H1(Γ)
a (w,y),(η,ξ)
−a`h (w,y),(η,ξ)
6chkk(w,y)kH1(Ω)×H1(Γ)k(η,ξ)kH1(Ω)×H1(Γ).
(6.4)
Finally, we remark we can use Lemma4.4for integrals overB`h.
LEMMA6.2 Forη∈H1(Ω),
kηkL2(B`
h)6ch 1 2kηk
H1(Ω). (6.5)
Proof. We may apply Lemma4.4to the domainNδ. We can chooseδ such thatδΓ >ch>δ>h>0, since the width ofB`his just one element. Hence
kηkL2(B`
h)6kηkL2(Nδ)6cδ
1 2kηk
H1(Ω)6ch 1 2kηk
H1(Ω).
6.2 Proof of error bounds
Let(u,v)∈Hk+1(Ω)×Hk+1(Γ)be the solution of the variational problem (2.6) and let(uh,vh)∈Vh×Sh be the solution of the finite element scheme given by (5.2). Denote byu`handv`hthe lifts ofuhandvh respectively. DefineFh:H1(Ω)×H1(Γ)→Rby
Fh (η,ξ):=a (u−uh`,v−v`h),(η,ξ). (6.6)
LEMMA6.3 If(η,ξ) = (ηh`,ξh`)∈Vh`×S`h, thenFhis bounded by
Fh (η
`
h,ξh`)
6ch
k (u
`
h,v`h)
H1(Ω)×H1(Γ)
(η
`
h,ξh`)
H1(Ω)×H1(Γ). (6.7)
If(η,ξ)∈H2(Ω)×H2(Γ)then, we can improve the bound onFhto
Fh η,ξ
6 chk+1 (u
`
h,v`h)
H1(Ω)×H1(Γ)+ch k
(u
`
h−u,v`h−v)
H1(Ω)×H1(Γ)
+chk+1k(u,v)kH2(Ω)×H2(Γ)k(η,ξ)kH2(Ω)×H2(Γ).
(6.8)
Proof. First, we notice that if(η,ξ) = (ηh`,ξh`)∈Vh`×S`h, using the fact that(u,v)satisfies (2.6) and (u`h,v`h)satisfies (5.7),Fhcan be written as
Fh (ηh`,ξh`)
=a (u−u`h,v−v`h),(ηh`,ξh`) =l (ηh`,ξh`)−a (uh`,v`h),(ηh`,ξh`) =l (ηh`,ξh`)−l (ηh`,ξh`)
−a (u`h,v`h),(ηh`,ξh`)−a`h (u`h,v`h),(ηh`,ξh`)
=−a (u`h,vh`),(ηh`,ξh`)−a`h (u`h,v`h),(ηh`,ξh`)
Applying the result from (6.4) gives (6.7).
To show the second result, we assume (η,ξ)∈H2(Ω)×H2(Γ) and introduce the interpolant Ih(η,ξ)∈Vh`×S`hof(η,ξ), so that
Fh (η,ξ)
=a (u−u`h,v−v`h),(η,ξ)
=a (u−u`h,v−vh`),(η,ξ)−Ih(η,ξ)
+a (u−u`h,v−v`h),Ih(η,ξ)
.
Then, again we can use the fact that(u,v)satisfies (2.6) and(u`h,v`h)satisfies (5.7), so that
Fh (η,ξ)=a (u−u`h,v−v`h),(η,ξ)−Ih(η,ξ)+
a`h (u`h,v`h),Ih(η,ξ)−a (u`h,vh`),Ih(η,ξ)
.
Hence we have that
Fh (η,ξ)=a (u−u`h,v−v`h),(η,ξ)−Ih(η,ξ) +a`h (u`h,v`h),Ih(η,ξ)−(η,ξ)
−a (u`h,v`h),Ih(η,ξ)−(η,ξ)
+a`h (u`h−u,v`h−v),(η,ξ)−a (u`h−u,v`h−v),(η,ξ)
+a`h (u,v),(η,ξ)−a (u,v),(η,ξ)
.
(6.9)
We bound each of the terms on the right hand side of (6.9) in turn. For the first term we apply (6.1) together with the Approximation Property (Proposition5.2) to see
a (u−u
`
h,v−v`h),(η,ξ)−Ih(η,ξ)
6C1h
kchk(η,
ξ)kH2(Ω)×H2(Γ).
For the second term, we use the geometric bound (6.4), again, with the Approximation Property (Propo-sition5.2) to get
a
`
h (u`h,v`h),Ih(η,ξ)−(η,ξ)−a (u`h,v`h),Ih(η,ξ)−(η,ξ)
6chk
(u
`
h,v
`
h)
H1(Ω)×H1(Γ)chk(η,ξ)kH2(Ω)×H2(Γ).
A bound for the third term follows by applying the geometric bound (6.4)
a
`
h (u`h−u,v`h−v),(η,ξ)
−a (u`h−u,v`h−v),(η,ξ)
6chk
(u
`
h−u,v`h−v)
H1(Ω)×H1(Γ)k(η,ξ)kH1(Ω)×H1(Γ).
Finally, for the fourth term, we simply apply (6.3) followed by the result from Lemma6.2to see
a
`
h (u,v),(η,ξ)
−a (u,v),(η,ξ)
6chkkukH1(B`
h)kηkH1(B`h)+ch
k+1k(u,v)k
H1(Ω)×H1(Γ)k(η,ξ)kH1(Ω)×H1(Γ) 6chk+1k(u,v)kH2(Ω)×H2(Γ)k(η,ξ)kH2(Ω)×H2(Γ).
REMARK6.1 Notice that for(η,ξ) = (ηh,ξh)∈Vh×Shin the absence of domain perturbation then
Fh (ηh,ξh)
=0,
where this is simply Galerkin orthogonality, whereas, in the absence of the bulk equations then the bound would be orderhk+1,Demlow(2009).
Proof of Theorem6.1. The error estimate (6.1) follows simply by combining the approximation
property (Proposition5.2) with the bound onFhfrom above (6.7). We rewrite the error as
a (u−u`h,v−v`h),(u−u`h,v−v`h)
=a (u−u`h,v−v`h),(u,v)−Ih(u,v)
+a (u−u`h,v−v`h),Ih(u,v)−(u`h,v`h)
=a (u−u`h,v−v`h),(u,v)−Ih(u,v)
+Fh Ih(u,v)−(u`h,v`h)
.
The result follows from application of a Cauchy inequality and the coercivity of the bilinear form a (3.2). To show the given value of C1 we use (5.4) from Theorem5.1 and (4.17, 4.11) to bound
(u`h,v`h)
H1(Ω)×H1(Γ).
We will use an Aubin-Nitsche duality argument to show theL2bound. Forζ = (ζ1,ζ2)∈L2(Ω)× L2(Γ), we define the dual problem: Findzζ ∈H1(Ω)×H1(Γ)such that
a (η,ξ),zζ) =hζ,(η,ξ)iL2(Ω)×L2(Γ)for all(η,ξ)∈H1(Ω)×H1(Γ). (6.10)
Here,h(w,y),(η,ξ)i ∈L2(Ω)×L2(Γ)denotes the sum of theL2inner products betweenwandη on
Ω andyandξ onΓ. Similar to Theorem3.2, one can show the following regularity result for the dual problem:
zζ
H2(Ω)×H2(Γ)6ckζkL2(Ω)×L2(Γ). (6.11) We write the error,
e= (u−u`h,v−vh`)∈L2(Ω)×L2(Γ),
as the data for the dual problem and test with(η,ξ) =eso that
kek2L2(Ω)×L2(Γ)=a(e,ze) =Fh(ze).
Hence, using (6.8) combined with theH1error bound (6.1) the dual regularity result (6.11), we have
kek2L2(Ω)×L2(Γ)=Fh(ze)6C2hk+1kekL2(Ω)×L2(Γ),
withC2as in the statement of the theorem.
7. Numerical Results
We have implemented the above finite element method using the ALBERTA finite element toolbox (Schmidtet al.,2005).
The data was chosen, withα=β =1, so that the exact solution was,
u(x1,x2,x3) =βexp(−x1(x1−1)x2(x2−1))
FIG. 5: Plot of the solution of the finite element scheme ath≈.2,k=2, along the planex=yinΩh, with mesh shown, (left) and the surfaceΓh(right).
We calculate the right hand side by setting(fh,gh) =Ieh(f,g). We ran two simulations: one withk=1, one withk=2. We present the error calculated after solving the matrix system at each mesh size in Tables1–4. A plot of the solution is provided in Figure5.
Acknowledgements
The work of C. M. Elliott was supported by the UK Engineering and Physical Sciences Research Council EPSRC Grant EP/G010404 and the work of T. Ranner was supported by a EPRSRC PhD studentship. The authors also thank the Centre for Scientific Computing at the University of Warwick for supplying valuable computation time.
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TABLE1. Error table for the case k=1- bulk errors,ku−uhk.
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TABLE4. Error table for the case k=2- surface errors,kv−vhk.
h L2error eoc H1error eoc
1.000000e+00 1.538024e-01 - 1.258018e+00
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