Uniform error estimates for general semilinear turning point problems on layer-adapted meshes

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by  Simon Becher

Uniform error estimates for general semilinear turning point
problems on layer-adapted meshes
Simon Becher∗
Abstract
7
1 We consider a singularly perturbed semilinear boundary value problem of a general form
0
thatallowsvarioustypesofturningpoints. Asolutiondecompositionisderivedthatseparates
2
thepotentialexponentialboundarylayerterms. Theproblemisdiscretizedusinghigherorder

n finiteelementsonsuitableconstructedlayer-adaptedmeshes. Finally,errorestimatesuniform
a with respect to thesingular perturbation parameter ε are proven in theenergy norm.
J
AMS subject classification (2010): 65L11, 65L50, 65L60,65L70.
3
2 Key words: singular perturbation, turning point, layer-adapted meshes, higher order, finite ele-
ments, uniform estimates.

]
A
N 1 Introduction
.
h
Let us consider a singularly perturbed semilinear boundary value problem of the type
t
a
m εu (x)+b(x)u(x)+f(x,u(x))=0, for x I :=[a,a],
′′ ′
− ∈ (1.1)
[ u(a)=ν , u(a)=ν ,
+
−
1
where 0 < ε 1 and b,f are supposed to be sufficiently smooth. Furthermore, we assume that
v ≪
3 there is a continuous function c such that
2
3 ∂uf(x,u)≥c(x)≥γ >0, for all (x,u)∈I ×R, c− 12b′ (x)≥γ˜ >0, for all x∈I. (1.2)
6
0 Apoint x¯ I is calledturning pointofthe problemifb((cid:0)x¯)=0 a(cid:1)ndforeveryneighborhoodU of x¯
. ∈
1 thereisapointx U I suchthatb(x)=0. Note thatthe assumptionsin(1.2)onb,c,andf are
∈ ∩ 6
0 very weak and especially allow an arbitrary number, location, and multiplicity of turning points.
7
But, since these functions are independent of ε, the turning points are also independent of ε. So,
1
we exclude the situation that an inner turning point moves to the boundary when ε goes to zero.
:
v As result of the general setting of problem (1.1) with (1.2), we have to be aware of many
i
X (possibly different) layers. One way to treat these layers and to enable uniform estimates is the
use of suitable layer-adaptedmeshes. This approachwas used by Liseikin in [4, Theorem 7.4.2]to
r
a provethe uniform first orderconvergenceofa simple upwind scheme for the consideredsemilinear
problem.
In this paper higher order finite elements shall be analyzed instead. For some special cases of
problem (1.1) it is already known that optimal order uniform error estimates can be proven on
layer-adapted meshes in an ε-weighted energy norm, see [8] for linear problems without turning
points or [1, 2] for linear problems with a single simple attractive interior turning point. But, is it
possibletoprovesuchestimatesinourgeneralsetting also? Andhowcouldsuitable layer-adapted
meshes look like?
We shallanswerboth questions inthe followingsections. But firstmoreinformationaboutthe
behaviorofthesolution,especiallyabouttheappearanceandtypesoflayers,isneeded. Therefore,
aprioriestimatesandasolutiondecompositionaregiveninSection2togetherwithsomecomments
onthelinearversionoftheconsideredproblem. Itshowsthatexponentialboundarylayers,interior
∗Institute of Numerical Mathematics, Technical University of Dresden, Dresden D-01062, Germany. e-mail:
[email protected]
1
cusp-type layers, and certain power-type boundary layers could occur. S-type meshes [6] and the
piecewiseequidistantmeshesproposedbySunandStynes[9],respectively,haveprovedtheirworth
in handling the first two classes of layers. Furthermore, it will turn out that the latter grids can
be adopted to the power-type boundary layers by simply adjusting a parameter. So at the end of
Section 4 we are able to give a convenient mesh construction strategy for the general problem.
The discretization of the problem is presented in Section 3 along with some first notes on
the estimation of the error. Then Section 5 is devoted to the completing proof of a uniform
error estimate for higher order finite elements. Several (new) difficulties have to be managed, for
example:
The semilinearity of the problem.
•
The different techniques for the different layers have to be combined.
•
In general the mesh outside the exponential boundary layer region is not quasi uniform and
•
so inverse inequalities, typically used in the analysis of S-type meshes, have to be handled
with additional care.
Exponential boundary layers of width (√εlog1/√ε), known from reaction-diffusion prob-
• O
lems, may also occur when b 0. Thus, the convection term has to be estimated for such
6≡
layers also.
Thereasoninginthecaseofacusp-typelayerhastobetransferredtothecaseofapower-type
•
boundary layer.
Finally, we will have a look at some examples of linear problems with different layersin Section 6.
Notation: Throughoutthe paperlet C denote apositivegenericconstantindependent of εand
the number ofmeshintervalsN. ForS Rwe use the commonSobolev spacesWk, (S), Hk(S),
∞
⊂
H1(S), and Lp(S). The spaces of continuously and Lipschitz-continuously differentiable functions
0
willbewrittenasCk(S)andCk,1(S),respectively,andusedforS R2also. Furthermore,weshall
⊂
denote the L2-norm by , the H1-semi norm by , the (essential) supremum by ,
k·k0,S |·|1,S k·k ,S
andthe Lp-normby . If S is the whole interval,it willbe omitted to shortenthe notat∞ion.
k·kLp(S)
2 A priori estimates and solution decomposition
A priori estimates for the solution of problem (1.1) with (1.2) can be found, e.g., in [4, pp. 73,
74, 95, 96]. We denote by := x¯ ,x¯ ,... the set of all points in (a,a) with b(x¯ ) = 0 and
0 1 2 j
M { }
b(x¯ ) < 0, j = 1,2,..., i.e., all interior turning points, where b changes its sign from +1 to 1.
′ j
−
Note that is always finite, see [4, p. 73].
0
M
Theorem 2.1 (A priori estimates, cf. [4])
Let q N and suppose that b Cq(I) and f,f Cq(I R). Then we have for k = 0,...,q and
u
∈ ∈ ∈ ×
x I
∈
u(k)(x) C 1+φ (x a),k,b(a),b(a),ε +φ (a x),k, b(a),b(a),ε
a ′ a ′
≤ − − −
(cid:18)
(cid:12)(cid:12) (cid:12)(cid:12) (cid:0) (cid:1) (cid:0)+ ε1/2+ x x¯ λj(cid:1)−k , (2.1a)
j
| − |
Xj (cid:16) (cid:17) (cid:19)
where 0<λ <c(x¯ )/b(x¯ ) while
j j ′ j
| |
ε keax/ε, a<0,
−
ελ/2 ε1/2+x −λ−k, a=0, b>0,
φx¯(x,k,a,b,ε)=εε−1k//22(cid:0)e+−x√(λc−(x¯k)(cid:1)++b)εx/ε√1ε/,2+x −k−2, aa==00,, b0<≤0−,kb<c(x¯) (2.1b)
with 0<λ<c(x¯)/b(x¯).(cid:0)0, (cid:1) (cid:0) (cid:1) a>0,
′
| |
2
Using the standard inequality 1+x ex we observe that for β >0 and x˜,ε˜>0
≤
k k k k
k βx˜ k k
ε˜ ke βx/ε˜= e βx/ε˜ eβ(x˜ x)/ε˜ C(x˜) when x x˜. (2.2)
− − − −
βx˜ kε˜ ≤ βx˜ ≤ βx˜ ≤ ≥
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Thus awayfromthe locationof boundaryand interiorlayersthe solutionu andits derivativescan
be bounded by a constant. Typically we will use (2.2) with ε˜ ε,√ε .
∈{ }
We want derive a decomposition of the solution u = S +E that separates the potential ex-
ponential boundary layer terms E. Unlike for non turning point problems S is not simply the
“smooth” part but may consists of power layer terms also.
Let a,a denote the parts of the boundary where usually an exponential layer occurs.
exp
B ⊆{ }
That is
1, x¯=a,
exp := x¯ a,a :b(x¯) n(x¯)>0 or (b(x¯)=b′(x¯)=0) where n(x¯):= −
B ∈{ } · (1, x¯=a.
(cid:8) (cid:9)
In an analogous manner let a,a , defined by
pow
B ⊆{ }
:= x¯ a,a :b(x¯)=0, b(x¯)=0 ,
pow ′
B ∈{ } 6
denote the parts of the boundary wh(cid:8)ere usually a power-type layer o(cid:9)ccurs.
For x¯ a,a we define the minimal distance δ to other (possible) locations of layers by
x¯
∈{ }
δ =min x¯ y :y ( a,a x¯ ) x¯ .
x¯ j
| − | ∈ { }\{ } ∪ j{ }
n [ o
=M0
Moreover,for x¯ a,a we define δ¯ 0 (typical width of a po|ssi{bzle e}xponential layer at x¯) by
∈{ } x∗¯ ≥
ε log(1/ε), b(x¯) n(x¯)>0,
b(x¯) ·
δ¯x∗¯ =0|√,√c(εx¯|)log(1/√ε), bo(tx¯h)er=wi0s,e. b′(x¯)=0,
Now, we can prove the following solution decomposition.
Theorem 2.2 (Solution decomposition)
Let q N and suppose that b Cq(I) and f,f Cq(I R). Then u has the representation
u
∈ ∈ ∈ ×
u=S+E with E =E +E , where for all k =0,...,q and x I
a a
∈
|S(k)(x)|≤C 1+φSa (x−a),k,b(a),b′(a),ε +φaS (a−x),k,−b(a),b′(a),ε
(cid:18)
(cid:0) (cid:1) +(cid:0) ε1/2+ x x¯ λj−(cid:1)k , (2.3)
j
| − |
Xj (cid:16) (cid:17) (cid:19)
with 0<λ <c(x¯ )/b(x¯ ) and
j j ′ j
| |
E(k)(x) CφE (x a),k,b(a),b(a),ε , E(k)(x) CφE (a x),k, b(a),b(a),ε . (2.4)
| a |≤ a − ′ | a |≤ a − − ′
Here φS and φE are(cid:0)given by (cid:1) (cid:0) (cid:1)
x¯ x¯
ελ/2 ε1/2+x −λ−k, a=0, b>0,
φSx¯(x,k,a,b,ε)= ε1/2(cid:0)+x λ−k(cid:1)+ε ε1/2+x −k−2, a=0, b<0, (2.5)
0, otherwise
(cid:0) (cid:1) (cid:0) (cid:1)
with 0<λ<c(x¯)/b(x¯) and 
′
| |
ε keax/ε, a<0,
−
φEx¯(x,k,a,b,ε)=ε−k/2e−√c(x¯)x/√ε, a=b=0, (2.6)
0, otherwise.

3
Proof: Setδ =min qδ¯ ,δ /2 forallx¯ a,a . Forconvenienceleta =a+δ anda =a δ .
x∗¯ { x∗¯ x¯ } ∈{ } ∗ a∗ ∗ − a∗
The construction of δ yields
x∗¯
εq, b(x¯) n(x¯)>0,
·
φEx¯ δx∗¯,0,−b(x¯)·n(x¯),b′(x¯),ε ≥φEx¯ qδ¯x∗¯,0,−b(x¯)·n(x¯),b′(x¯),ε =εq/2, b(x¯)=b′(x¯)=0,
0, otherwise.
(cid:0) (cid:1) (cid:0) (cid:1)
(2.7)

This and (2.2), respectively, gives that up to the qth derivative the exponential boundary layer
terms can be bounded by a constant inside the interval [a ,a ]. Note that because of δ δ /2
∗ ∗ x∗¯ ≤ x¯
this interval is not empty and, moreover,we have
φS (x a),k,b(a),b(a),ε +φS (a x),k, b(a),b(a),ε + ε1/2+ x x¯ λj−k C
a − ′ a − − ′ | − j| ≤
(cid:0) (cid:1) (cid:0) (cid:1) Xj (cid:16) (cid:17)
(2.8)
for all x I [a ,a ], where the constant C may depend on min δ :x¯ .
∗ ∗ x¯ exp
∈ \ { ∈B }
In order to prove the decomposition,we adaptan idea from [7, p. 23,24]. Set S(x):=u(x) for
x [a ,a ]. From (2.1), (2.2), and (2.7) we get that S satisfies (2.3) with (2.5) on [a ,a ]. Then
∗ ∗ ∗ ∗
∈
S can be extended to a smooth function (i.e., S Cq) defined on I that satisfies the requested
∈
bound on the whole interval.
We nowdefineE :=u S. Obviously,wehaveE =0on[a ,a ]. Combining (2.1), (2.2),(2.7),
∗ ∗
−
and (2.8) we get for x I [a ,a ] that
∗ ∗
∈ \
E(q)(x) u(q)(x) + S(q)(x)
≤
(cid:12) (cid:12) C(cid:12) 1+φ(cid:12)E (cid:12)(x a),(cid:12)q,b(a),b(a),ε +φE (a x),q, b(a),b(a),ε
(cid:12) (cid:12)≤(cid:12) (cid:12)a (cid:12) − (cid:12) ′ a − − ′
(cid:16)φE (x(cid:0) a),q,b(a),b(a),ε , (cid:1) for x(cid:0)<a , (cid:1)(cid:17)
C a − ′ ∗
≤ (φEa(cid:0)(a−x),q,−b(a),b′(a),(cid:1)ε , for a∗ <x.
Integrating E(k) for k =q,q(cid:0) 1,...,1 we inductively(cid:1)gain (recalling that E =0 on [a ,a ])
∗ ∗
−
x x
E(k 1)(x) E(k)(s)ds C φE (x a),k,b(a),b(a),ε ds
− ≤(cid:12)Za∗ (cid:12)≤ Za∗ a − ′
(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)≤CφEa (x−(cid:0) a),k−1,b(a),b′(a),ε(cid:1), for x<a∗
and analogously (cid:0) (cid:1)
E(k 1)(x) CφE (a x),k 1, b(a),b(a),ε , for a <x.
− ≤ a − − − ′ ∗
Since E =0 on [(cid:12)a ,a ]= (cid:12)by const(cid:0)ruction, we simply have to s(cid:1)et
(cid:12) ∗ ∗ (cid:12)
6 ∅
E, on [a,a ], E, on [a ,a],
∗ ∗
E = and E =
a a
(0, otherwise, (0, otherwise,
respectively, to get the two terms E and E . Thus the statement is proven. (cid:3)
a a
Remark 2.3 (Comments on the linear version of the problem)
Consider the linear version of problem (1.1) where f(x,u(x)) is replaced by c(x)u(x) f(x), i.e.,
−
Lu:= εu′′(x)+b(x)u′(x)+c(x)u(x)=f(x), for x I :=[a,a],
− ∈ (2.9)
u(a)=ν , u(a)=ν ,
+
−
with 0<ε 1, problem data b,c,f sufficiently smooth, and suppose that
≪
for all x I : b(x)=0 = c(x)>0, (2.10a)
∈ ⇒
for all x I : b(x)=0 = c 1b (x)>0. (2.10b)
∈ ⇒ − 2 ′
(cid:0) (cid:1)
4
Then we may assume without loss of generality (for ε sufficiently small) that (1.2) holds, i.e.,
c(x) γ >0, c 1b (x) γ˜ >0, for all x I.
≥ − 2 ′ ≥ ∈
(cid:0) (cid:1)
This can always be achieved by a suitable problem transformation, see Appendix A for a proof
of this statement. Such transformations are widely known in the case that b(x) =0 and was also
6
studied in [9] in the case of a single interior turning point. But to the authors knowledge this
statement is new in this general setting.
Moreover,wewanttonote thatforthe linearproblema solutiondecompositionsimilarto that
of Theorem 2.2 can also be derived using boundary layer corrections. The interested reader is
referred to Appendix B for more details.
♣
3 FEM discretization
In order to fix the notation we want to present the discretization of the problem by higher order
finite elements now. We will consider homogeneous Dirichlet boundary conditions ν = ν = 0
+
−
only. Note that these can be easily ensured by modifying the nonlinear term f(x,u). Indeed, set
u˜(x)=((a x)ν +(x a)ν )/(a a). Then u+u˜ solves (1.1) when u itself solves
+
− − − −
εu′′(x)+b(x)u′(x)+f˜(x,u(x))=0, for x I :=[a,a], u(a)=u(a)=0,
− ∈
where f˜(x,u) = f(x,u+u˜(x)) εu˜ (x)+b(x)u˜ (x) for u R. Obviously, the assumption (1.2)
′′ ′
also holds for f˜. − ∈
For v,w V :=H1((a,a)) we set
∈ 0
B (v,w):=(εv ,w )+(bv ,w)+(f(,v),w).
ε ′ ′ ′
·
Then we obtain the following weak formulation of (1.1) with ν =ν =0:
+
−
Find u V such that
∈
B (u,v)=0, for all v V. (3.1)
ε
∈
By a = x < ... < x < ... < x = a an arbitrary mesh is given on the interval [a,a]. Let
0 i N
h :=x x denote the mesh interval lengths. We define the trial and test space VN by
i i i 1
− −
VN := v C([a,a]):v P ((x ,x )) i, v( 1)=v(1)=0 V
∈ |(xi−1,xi) ∈ k i−1 i ∀ − ⊂
(cid:8) (cid:9)
where k 1. The space P ((x ,x )) comprises all polynomials up to order k over (x ,x ). The
k a b a b
≥
discrete problem arises from replacing V in (3.1) by the finite dimensional subspace VN:
Find uN VN such that
∈
B uN,vN =0, for all vN VN. (3.2)
ε
∈
Let vI denote the standard L(cid:0)agrangi(cid:1)an interpolant into VN of v V. As interpolation points
∈
we choosethe meshpoints and k 1 (arbitrary)inner points per interval. For example uniformor
−
Gauß-Lobatto points could be used.
Assuming v Wk+1, ((x ,x )), the standard interpolation theory leads to the error esti-
∞ i 1 i
∈ −
mates: For all j =0,...,k+1
(cid:13)(v−vI)(j)(cid:13)∞,(xi−1,xi) ≤Chki+1−j(cid:13)v(k+1)(cid:13)∞,(xi−1,xi) (3.3)
(cid:13) (cid:13) (cid:13) (cid:13)
and (cid:13) (cid:13) (cid:13) (cid:13)
(v vI)(j) C v(j) . (3.4)
(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)∞,(xi−1,xi) ≤ (cid:13)(cid:13) (cid:13)(cid:13)∞,(xi−1,xi)
5
The constant C depends on the location of the inner interpolation points. Furthermore, for all
vN VN the inverse inequality
∈
(vN)′ Lp((xi−1,xi)) ≤Ch−i 1 vN Lp((xi−1,xi)) (3.5)
holds for p [1, ]. (cid:13) (cid:13) (cid:13) (cid:13)
∈ ∞ (cid:13) (cid:13) (cid:13) (cid:13)
Now we have a closer look on the nonlinear term. The following identity will be exploited
severaltimes later on. For a function g =g(x,v) with g,g C(I R) we have
v
∈ ×
1
g(x,v ) g(x,v )= ∂ g(x,v +s(v v ))ds (v v ). (3.6)
1 2 v 2 1 2 1 2
− − −
(cid:20)Z0 (cid:21)
Especially, choosing g(x,v)=f(x,v) 1bv and noting that
− 2 ′
∂ g(x,v)=∂ f(x,v) 1b(x) c(x) 1b(x) γ˜ >0
v u − 2 ′ ≥ − 2 ′ ≥
by (1.2), we obtain for v ,v V and shortly v =v v
1 2 1 2
∈ −
Bε(v1,v) Bε(v2,v)=ε(v′,v′)+(bv′,v)+(f(,v1) f(,v2),v)
− · − ·
=ε v 2+ [f(,v ) 1bv ] [f(,v ) 1bv ],v
| |1 · 1 − 2 ′ 1 − · 2 − 2 ′ 2 (3.7)
≥ε|v|21+(cid:0)(c− 21b′)v,v (cid:1)
ε v 2+γ(cid:0)˜ v 2. (cid:1)
≥ | |1 k k0
Therefore,for the analysisof the problemanappropriatedweightedenergynorm is givenby
|||·|||ε
1/2
v := ε v 2+γ˜ v 2 .
||| |||ε | |1 k k0
(cid:16) (cid:17)
Notethat(3.7)impliestheuniquenessoftheweakandthediscretesolution. Inthecaseofalinear
problem B (, ) is a bilinear form which is uniformly coercive over H1((a,a)) H1((a,a)) with
ε · · 0 × 0
respect to due to (3.7).
|||·|||ε
As usual, to estimate the discretization error a splitting is used, that is
u uN =(u uI)+(uI uN). (3.8)
− − −
Combining (3.7) and the problem formulations (3.1) and (3.2), we conclude
uN uI 2 B (uN,vN) B (uI,vN)=B (u,vN) B (uI,vN) (3.9)
− ε ≤ ε − ε ε − ε
where vN = uN(cid:12)(cid:12)(cid:12) uI V(cid:12)(cid:12)(cid:12)N V. Unlike for linear problems [1, 8] the right hand side of (3.9)
(cid:12)(cid:12)(cid:12)− ∈ (cid:12)(cid:12)(cid:12) ⊂
contains nonlinear terms in u and uI. So the nonlinearity needs additional consideration.
Fortunately, from inverse monotonicity properties, see [4, p. 74], we have for the solution u of
problem (1.1) that
u max ν , ν , f(,0)/c() .
+
k k∞ ≤ | −| | | k · · k∞
Hence, with (3.4) we get (cid:8) (cid:9)
u + uI C˜,
k k ≤
∞ ∞
where C˜ > 0 is independent of ε and depends(cid:13)on t(cid:13)he problem data and the location of the inner
(cid:13) (cid:13)
interpolation points only. Furthermore, for f C(I R) we can define
u
∈ ×
C˜ :=max ∂ f(x,v) :(x,v) I [ C˜,C˜] .
f u
| | ∈ × −
Therefore, using (3.6) with g(x,v)=(cid:8)f(x,v) yields (cid:9)
1
f x,u(x) f x,uI(x) ∂ f x,uI(x)+s u(x) uI(x) ds u(x) uI(x) (3.10)
u
− ≤ − −
(cid:20)Z0 (cid:21)
(cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) (cid:12) (cid:0) (cid:0) (cid:1)(cid:1)(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) ≤C˜f (cid:12) (cid:12) (cid:12)
which simply reduces the differe|nce of the nonlinea{rzterms to a basic int}erpolation error. Thus,
the nonlinearity raises no further difficulties in the error estimation.
Under the above assumptions the existence of unique solutions for (3.1) and (3.2) is clear. For
details we refer to [3, Section 3.5].
6
4 Layer-adapted meshes
Inthissectionwewanttodescribehowalayer-adaptedmeshforproblem(1.1)with(1.2)couldbe
constructed, see Section 4.3. Before some layer-adapted meshes are presented each of which can
be used to capture a single layer-type. In order to describe these meshes we shall always consider
the interval [0,1] and assume that the layer is located at zero.
4.1 S-type meshes for exponential layers
S-typemeshesareverypopularinthecasesofexponentiallayers. Theyarecharacterizedbyavery
fine mesh in the layer region and a coarse mesh away from the layer. Dependent on the detailed
structure of the exponential layer term φ the transition point τ from the fine to the coarse part
E
is defined by
ε˜ ε, if φ(j)(x) Cε je βx/ε, 0 j k+1,
τ =ρ lnN, ε˜= E ≤ − − ≤ ≤ (4.1)
β (√ε, if (cid:12)(cid:12)φ(Ej)(x)(cid:12)(cid:12)≤Cε−j/2e−βx/√ε, 0≤j ≤k+1,
(cid:12) (cid:12)
with ρ>0. This definition yields φE(τ) (cid:12) CN−ρ(cid:12).
≤
Now, we set x =τ and define the mesh on the interval [0,x ] by a mesh generating func-
N/2 (cid:12) (cid:12) N/2
tion ϕ, where we suppose that ϕ (cid:12)is cont(cid:12)inuous, monotonically increasing, piecewise continuously
differentiable, and furthermore satisfies ϕ(0) = 0 and ϕ(1) = lnN. On the interval [x ,1] a
2 N/2
coarsemeshis generated. Often anequidistantpartition(with h CN 1) is chosen. Inthis case
i −
≤
we have
ρε˜ϕ(t ), t =i/N, i=0,1,...,N/2,
x = β i i
i (1 (1 τ)2(N−i), i=N/2+1,...,N.
− − N
Moreover,the mesh characterizing function ψ is defined by
βx
i
ϕ= lnψ ψ(t )=exp .
i
− ⇒ − ρε˜
(cid:18) (cid:18) (cid:19)(cid:19)
The further analysis is based on the assumption (typical for FEMs on S-type meshes)
maxϕ′ CN. (4.2)
≤
Therefore, by the construction of the mesh we get for 1 i N/2
≤ ≤
ε˜ ε˜
h =x x =ρ ϕ(t ) ϕ(t ) ρ N 1maxϕ Cε˜. (4.3)
i i i 1 i i 1 − ′
− − β − − ≤ β ≤
(cid:0) (cid:1)
We now prove some estimates for the interpolation error.
Lemma 4.1
Under the assumption (4.2) and for ρ k+1 we have
≥
φ φI C N 1max ψ k+1, φ φI CN ρ,
E − E ,[0,τ] ≤ − | ′| E − E ,[τ,1] ≤ −
∞ ∞
ε˜1/(cid:13)(cid:13)2 φE −φIE(cid:13)(cid:13) 1,[0,τ] ≤C(cid:0)N−1max|ψ′|(cid:1)k, ε˜1/(cid:13)(cid:13)2 φE −φIE(cid:13)(cid:13) 1,[τ,1] ≤CN−ρ,
(cid:12)φ φI (cid:12) Cε˜(cid:0)1/2(lnN)1/2 N(cid:1) 1max ψ k+1, (cid:12)φ φI (cid:12) CN ρ,
(cid:12) E − E (cid:12)0,[0,τ] ≤ − | ′| (cid:12) E − E (cid:12)0,[τ,1] ≤ −
(cid:13) (cid:13) (cid:0) (cid:1) (cid:13) (cid:13)
or if ρ(cid:13) k+3/2(cid:13) (cid:13) (cid:13)
≥
φ φI Cε˜1/2 N 1max ψ k+1.
E − E 0,[0,τ] ≤ − | ′|
(cid:13) (cid:13) (cid:0) (cid:1)
Here φI(cid:13) VN de(cid:13)notes a standard Lagrangian interpolant of φ (k+1 points per mesh interval).
E ∈ E
7
Proof: Theprooffollows[8,Section2]butisgeneralizedwithrespecttothedifferentwidthofthe
exponential layer. We first study the interval [0,τ], so let 1 i N/2. Applying some standard
≤ ≤
interpolation error estimates we obtain
φ φI Chk+1 φ(k+1) .
E − E ∞,[xi−1,xi] ≤ i (cid:13) E (cid:13)∞,[xi−1,xi]
(cid:13) (cid:13) (cid:13) (cid:13)
Together with (cid:13) (cid:13) (cid:13) (cid:13)
ε˜ ε˜
hi ρ N−1maxϕ′ ρ N−1max ψ′ eβxi/(ρε˜) (4.4)
≤ β ≤ β | |
and the general bound h Cε˜(from (4.3)) this yields for ρ k+1
i
≤ ≥
φE −φIE ∞,[xi−1,xi] ≤C N−1max|ψ′| k+1eβ((k+ρ1)xi−xi−1)/ε˜≤C N−1max|ψ′| k+1.
(cid:13) (cid:13) (cid:0) (cid:1) (cid:0) (cid:1)
Th(cid:13)e first bo(cid:13)und in the L2-norm follows from the maximum norm estimate since
φE −φIE 0,[0,τ] ≤τ1/2 φE −φIE ,[0,τ] ≤Cε˜1/2(lnN)1/2 N−1max|ψ′| k+1.
∞
(cid:13) (cid:13) (cid:13) (cid:13) (cid:0) (cid:1)
In the(cid:13)integral-b(cid:13)asednorms (w(cid:13)ith j =0(cid:13),1) we can also proceed as follows
(cid:12)φE −φIE(cid:12)21−j,[xi−1,xi] ≤Ch2i(k+j)Zxxi−i1(cid:16)φ(Ek+1)(cid:17)2 ≤Ch2i(k+j)ε˜−(2k+1)(cid:16)e−2βxi−1/ε˜−e−2βxi/ε˜(cid:17)
(cid:12) (cid:12) Cε˜−1+2j N−1max ψ′ 2(k+j)e2(k+j)βxi/(ρε˜)e−2βxi−1/2/ε˜sinhβhi.
≤ | | ε˜
(cid:0) (cid:1)
By (4.3) we have sinhβhi sinhCβ C and thus
ε˜ ≤ ≤
βh βh ti ti ψ ti
sinh i C i =C ϕ′ =C − ′ Ceβxi/(ρε˜) ( ψ′).
ε˜ ≤ ε˜ ψ ≤ −
Zti−1 Zti−1(cid:18) (cid:19) Zti−1
Combining
e2(k+j)βxi/(ρε˜)e−2βxi−1/2/ε˜eβxi/(ρε˜) =e2β((k+j+ρ1/2)xi−xi−1/2)/ε˜ C
≤
which holds for ρ k+j+1/2 and
≥
1/2
( ψ )=ψ(0) ψ(1/2) 1
′
− − ≤
Z0
we get
ε˜1 2j φ φI 2 C N 1max ψ 2(k+j).
− E − E 1 j,[0,τ] ≤ − | ′|
−
(cid:12) (cid:12) (cid:0) (cid:1)
The estimate in the interv(cid:12)al [τ,1] is(cid:12)based on the smallness of the exponential boundary layer
term. Using some stability properties of the interpolant, we conclude from (4.1)
φ φI φ + φI C φ C φ (τ) CN ρ.
E − E ,[τ,1] ≤k Ek ,[τ,1] E ,[τ,1] ≤ k Ek ,[τ,1] ≤ | E |≤ −
∞ ∞ ∞ ∞
(cid:13) (cid:13) (cid:13) (cid:13)
Immed(cid:13)iately, this(cid:13)implies (cid:13) (cid:13)
φE −φIE 0,[τ,1] ≤C φE −φIE ,[τ,1] ≤CN−ρ.
∞
(cid:13) (cid:13) (cid:13) (cid:13)
Using the stability/error(cid:13)estimate f(cid:13)or the inter(cid:13)polator w(cid:13)e also get
1
φE −φIE 21,[τ,1] ≤C|φE|21,[τ,1] ≤C ε˜−2e−2βx/ε˜≤Cε˜−1 e−2βτ/ε˜−e−2β/ε˜ ≤Cε˜−1N−2ρ
(cid:12) (cid:12) Zτ (cid:16) (cid:17)
whi(cid:12)ch compl(cid:12)etes the proof. (cid:3)
8
Remark 4.2
The estimates of Lemma 4.1 on the coarsepart[τ,1] are independent of the exact structure of the
mesh in this interval. The proof only uses the smallness of φ , i.e., φ (x) CN ρ for x τ.
E E −
≤ ≥ ♣
Lemma 4.3 (cid:12) (cid:12)
(cid:12) (cid:12)
Under the assumption (4.2) and for ρ k+1, ℓ 1 we have
≥ ≥
xℓ(φ φI ) C(2ε˜)ℓ 1 N 1max ψ k.
E − E ′ ,[0,τ] ≤ − − | ′|
∞
(cid:13) (cid:13) (cid:0) (cid:1)
Proof: Let 1 i (cid:13)N/2. Combin(cid:13)ing standard interpolation estimates, the bounds for φ ,
E
≤ ≤
and (4.4) we obtain
xℓ(φ φI )
E − E ′ ∞,[xi−1,xi]
(cid:13) (cid:13) h k
(cid:13) ≤C(xi−1+(cid:13)hi)ℓhki (cid:13)φ(Ek+1)(cid:13)∞,[xi−1,xi] ≤C2ℓ−1 xℓi−1+hℓi (cid:18) ε˜i(cid:19) ε˜−1e−βxi−1/ε˜
C(2ε˜)ℓ−1 N−1(cid:13)(cid:13)max ψ′(cid:13)(cid:13)k xi−1 ℓeβkxi/(ρε˜)(cid:0)+ N−1m(cid:1)ax ψ′ k+1eβ(k+1)xi/(ρε˜) e−βxi−1/ε˜.
≤ | | ε˜ | |
(cid:18)(cid:0) (cid:1) (cid:16) (cid:17) (cid:0) (cid:1) (cid:19)
From the well known inequality 1+x ex we conclude
≤
x ℓ ℓ(k+1) ℓ
i−1 e−βxi−1/ε˜ e−βkxi−1/((k+1)ε˜).
ε˜ ≤ β
(cid:16) (cid:17) (cid:18) (cid:19)
Hence, we have for ρ k+1
≥
x(φ φI )
E − E ′ ∞,[xi−1,xi]
(cid:13)(cid:13) C(2ε˜)ℓ(cid:13)(cid:13)−1 N−1max ψ′ keβ(kρxi−k+k1xi−1)/ε˜+ N−1max ψ′ k+1eβ(k+ρ1xi−xi−1)/ε˜
≤ | | | |
C(2ε˜)ℓ−1(cid:16)(cid:0)N−1max ψ′ (cid:1)k+ N−1max ψ′ k+1(cid:0), (cid:1) (cid:17)
≤ | | | |
(cid:16)(cid:0) (cid:1) (cid:0) (cid:1) (cid:17)
where x x =h Cε˜was used in the last inequality. (cid:3)
i i 1 i
− − ≤
4.2 Piecewise equidistant mesh for power-type layers
ThepiecewiseequidistantmeshpresentedinthissectionwasfirstintroducedbySunandStynes[9,
Section 5.1] to treat an interior cusp-type layer. We will see that it can be easily adapted such
that uniform estimates are possible for all kinds of power-type layers appearing in (2.5) as well.
This fact is based on the observation that all of these layer terms can be bounded as
λ j
φ(j)(x) C 1+ ε1/2+x − , 0 j k+1, (4.5)
S ≤ ≤ ≤
(cid:12) (cid:12) (cid:18) (cid:16) (cid:17) (cid:19)
(cid:12) (cid:12)
with λ 0. (cid:12) (cid:12)
≥
We shall assume in the following that λ [0,k + 1). This is the most difficult case since
∈
otherwise all crucial derivatives of φ could be bounded by a constant independent of ε which
S
would allow uniform estimates for the layer terms using standard methods on uniform meshes.
The mesh parameters are determined as in [1, Section 3]. For ε (0,1] and given positive
∈
integer N we set
σ =max ε(1−λ/(k+1))/2,N−(2k+1) (4.6)
(cid:8) (cid:9)
and
ln(σ)
= 1 , (4.7)
K − ln(10)
(cid:22) (cid:23)
where z denotes the largest integer less or equal to z.
⌊ ⌋
9
The piecewise equidistant mesh is constructed in two steps: First, we divide the interval (0,1]
into the + 1 subintervals (0,10 ], (10 ,10 +1],...,(10 1,1]. Afterwards each of these
−K −K −K −
K
subintervals is partitioned uniformly into N/( +1) parts, where for simplicity we assume that
⌊ K ⌋
N/( +1) =N/( +1). Thus, by construction we have
⌊ K ⌋ K
h =( +1)10 N 1, for x (0,10 ] (4.8)
i −K − i −K
K ∈
and
hi =9( +1)10−lN−1, for xi (10−l,10−l+1] and l =1,..., . (4.9)
K ∈ K
The properties of the logarithm together with (4.6) and (4.7) yield
ln(σ) 1 λ/(k+1) ln(ε) ln(N)
+1 2 2+min − | |,(2k+1) .
K ≤ − ln(10) ≤ 2 ln(10) ln(10)
(cid:26) (cid:27)
Thisensuresforsufficiently largeN (dependenton k)thatthe numberofsubintervals +1is less
K
than the number of mesh intervals N since
+1 ClnN. (4.10)
K ≤
Moreover,from (4.7) we easily see that
10−1σ 10−K <σ. (4.11)
≤
Note that by a simple modification of the construction one can guarantee that the mesh consists
of exactly N mesh intervals, see [9, Section 6]. Let Φ denote the associated mesh generating
N,λ
function which is continuous and piecewise linear.
The next lemma is taken from [1]. Note that the arguments used therein also works for the
(formal) choice λ=0.
Lemma 4.4 (see [1, Lemma 3.1])
Let j =0,1. The following inequalities hold
hki+1−j xi−1+ε1/2 λ−(k+1−j) ≤C (K+1)N−1 k+1−j, for xi ∈(10−K,1], (4.12)
hki+1−j(cid:16)xi−1+ε1/2(cid:17)λ−(k+1−j) ≤C((cid:0)i−1)−(k+1−(cid:1)j), for xi ∈(x1,10−K]. (4.13)
(cid:16) (cid:17)
If σ =ε(1 λ/(k+1))/2, then
−
hki+1−j xi−1+ε1/2 λ−(k+1−j) ≤C (K+1)N−1 k+1−j, for xi ∈(0,10−K]. (4.14)
In general, the(cid:16)mesh interv(cid:17)al length can be(cid:0)bounded by (cid:1)
h ( +1)N 1.
i −
≤ K
Furthermore, in the case of σ =N (2k+1), we have
−
x =h ( +1)N 2(k+1). (4.15)
1 1 −
≤ K
Usingthetechniquesof[1,Lemma3.2](alsoherethereasoningcanbeadoptedforthe(formal)
choiceλ=0),weobtainthe followinginterpolationerrorestimates onthe layer-adaptedpiecewise
equidistant mesh.
Lemma 4.5 (cf. [1, Lemma 3.2])
Let φ satisfy (4.5) and let φI VN be its interpolant on the piecewise equidistant mesh given
S S ∈
by (4.6) – (4.9). Then
φ φI C ( +1)N 1 k+1 (4.16)
S − S 0 ≤ K −
and (cid:13) (cid:13) (cid:0) (cid:1)
(cid:13) (cid:13)
φ φI + x(φ φI) C ( +1)N 1 k. (4.17)
S − S ε S − S ′ 0 ≤ K −
(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) (cid:13) (cid:13) (cid:0) (cid:1)
(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) (cid:13) (cid:13)
10