Adapted Numerical Methods for the Poisson Equation with $L^2$ Boundary Data in NonConvex Domains

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Adapted Numerical Methods for the Poisson Equation with L ² Boundary Data in NonConvex Domains

Thomas Apel, Serge Nicaise, Johannes Pfefferer

To cite this version:

Thomas Apel, Serge Nicaise, Johannes Pfefferer. Adapted Numerical Methods for the Poisson Equa-

tion with L

²

Boundary Data in NonConvex Domains. SIAM Journal on Numerical Analysis, Society

for Industrial and Applied Mathematics, 2017, 55 (4), pp.1937-1957. �10.1137/16m1062077�. �hal-

01957588�

ADAPTED NUMERICAL METHODS FOR THE POISSON EQUATION WITH L

²

BOUNDARY DATA IN NONCONVEX

DOMAINS

^∗

THOMAS APEL^†, SERGE NICAISE^‡,AND JOHANNES PFEFFERER^§

Abstract. The very weak solution of the Poisson equation withL²boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges in theL²(Ω)-norm with order 1/2 in convex domains but has a reduced convergence order in nonconvex domains although the solution remains to be contained inH^1/2(Ω). The reason is a singularity in the dual problem. In this paper we propose and analyze, as a remedy, both a standard finite element method with mesh grading and a dual variant of the singular complement method. The error order 1/2 is retained in both cases, also with nonconvex domains. Numerical experiments confirm the theoretical results.

Key words. elliptic boundary value problem, very weak formulation, finite element method, mesh grading, singular complement method, discretization error estimate

AMS subject classifications. 65N30, 65N15 DOI. 10.1137/16M1062077

1. Introduction. In this paper we consider the boundary value problem

−∆y = f in Ω, y = u on Γ = ∂Ω,

(1)

with right-hand side f ∈ H

⁻¹

(Ω) and boundary data u ∈ L

²

(Γ). We assume Ω ⊂ R

²

to be a bounded polygonal domain with boundary Γ. Such problems arise in optimal control when the Dirichlet boundary control is considered in L

²

(Γ); see for example [22, 24, 28].

For boundary data u ∈ L

²

(Γ) we cannot expect a weak solution y ∈ H

¹

(Ω).

Therefore we define a very weak solution by the method of transposition which goes back at least to Lions and Magenes [27, Chapter 2, section 6]: Find

y ∈ L

²

(Ω) : (y, ∆v)

Ω

= (u, ∂

n

v)

Γ

− (f, v)

Ω

∀v ∈ V (2)

with (w, v)

_G

:= R

G

wv denoting the L

²

(G) scalar product or an appropriate duality product. In our previous paper [4] we showed that the appropriate space V for the test functions is

V := H

_∆¹

(Ω) ∩ H

₀¹

(Ω) with H

_∆¹

(Ω) := {v ∈ H

¹

(Ω) : ∆v ∈ L

²

(Ω)}.

(3)

Note that from Theorems 4.4.3.7 and 1.4.5.3 of [25] the embedding V , → H

^3/2+ε

(Ω) for 0 < ε < ε

₀

follows with ε

₀

depending on the maximal interior angle of the domain Ω.

∗Received by the editors February 17, 2016; accepted for publication (in revised form) March 9, 2017; published electronically August 17, 2017. This paper is an extension of our previous technical report, arXiv:/505.00414 [math.NA], 2015 [2].

http://www.siam.org/journals/sinum/55-4/M106207.html

Funding:The work of the authors was partially supported by Deutsche Forschungsgemeinschaft, IGDK 1754.

†Institut f¨ur Mathematik und Bauinformatik, Universit¨at der Bundeswehr M¨unchen, D-85579 Neubiberg, Germany (thomas.apel@unibw.de).

‡LAMAV, Institut des Sciences et Techniques de Valenciennes, Universit´e de Valenciennes et du Hainaut Cambr´esis, B.P. 311, 59313 Valenciennes Cedex, France (snicaise@univ-valenciennes.fr).

§Lehrstuhl f¨ur Optimalsteuerung, Technische Universit¨at M¨unchen, D-85748 Garching bei M¨unchen, Germany (pfefferer@ma.tum.de).

1937

In particular this ensures ∂

n

v ∈ L

²

(Γ) for v ∈ V such that the formulation (2) is well defined. We proved the existence of a unique solution y ∈ L

²

(Ω) for u ∈ L

²

(Γ) and f ∈ H

⁻¹

(Ω), and that the solution is even in H

^1/2

(Ω). The method of transposition is used in different variants also in [24, 9, 15, 14, 22, 28].

Consider now the discretization of the boundary value problem. Let T

h

be a quasi-uniform family of conforming finite element meshes, and introduce the finite element spaces

Y

h

:= {v

h

∈ H

¹

(Ω) : v

h

|

T

∈ P

1

∀T ∈ T

h

}, Y

0h

:= Y

h

∩ H

₀¹

(Ω), Y

_h^∂

:= Y

h

|

∂Ω

. Since the boundary datum u is in general not contained in Y

_h^∂

we have to approximate it by u

^h

∈ Y

_h^∂

, e. g., by using L

²

(Γ)-projection or quasi-interpolation. In this way, the boundary datum is even regularized since u

^h

∈ H

^1/2

(Γ). Hence we can consider a regularized (weak) solution in Y

_∗^h

:= {v ∈ H

¹

(Ω) : v|

_Γ

= u

^h

},

y

^h

∈ Y

_∗^h

: (∇y

^h

, ∇v)

_Ω

= (f, v)

_Ω

∀v ∈ H

₀¹

(Ω).

(4)

The finite element solution y

h

is now searched for in Y

_∗h

:= Y

_∗^h

∩ Y

h

: Find y

h

∈ Y

_∗h

: (∇y

h

, ∇v

h

)

Ω

= (f, v

h

)

Ω

∀v

h

∈ Y

0h

.

(5)

The same discretization was derived previously by Berggren [9] from a different point of view. In [4] we showed that the discretization error estimate

ky − y

h

k

L²(Ω)

≤ Ch

^s

h

^1/2

kf k

_H⁻¹_(Ω)

+ kuk

L²(Γ)

holds for s = 1/2 if the domain is convex; this is a slight improvement of the result of Berggren, and the convex case is completely treated. In the case of nonconvex domains this convergence order is reduced although the very weak solution y is also in H

^1/2

(Ω); the finite element method does not lead to the best approximation in L

²

(Ω). In order to describe the result we assume for simplicity that Ω has only one corner with interior angle ω ∈ (π, 2π). We proved in [4] the convergence order s = λ− 1/2 − ε, where λ := π/ω and ε > 0 arbitrarily small, and showed by numerical experiments that the order of almost λ − 1/2 is sharp. Note that s → 0 for ω → 2π.

This is the state of the art for this kind of problem, and our aim is to devise methods to retain the convergence order s = 1/2 in the nonconvex case.

In order to explain the reduction in the convergence order and our first remedy, let us first mention that we have to modify the Aubin–Nitsche method to derive L

²

(Ω)- error estimates. The first reason is that our problem has no weak solution, only the dual problem,

v

_z

∈ V : (ϕ, ∆v

_z

)

_Ω

= (z, ϕ)

_Ω

∀ϕ ∈ L

²

(Ω), (6)

has. The second reason is that the solution y has inhomogeneous Dirichlet data such

that an estimate of the L

²

(Γ)-interpolation error of ∂

_n

v

_z

is needed. The H

¹

(Ω)-error

of a standard finite element method is of order one in convex domains but reduces

to s = λ − ε in the case of nonconvex domains; moreover, the order of the L

²

(Γ)-

interpolation error of ∂

n

v

z

reduces from 1/2 to λ − 1/2 − ε. It has been known for a

long time that locally refined (graded) meshes and augmenting of the finite element

space by singular functions are appropriate to retain the optimal convergence order

for such problems; see, e. g., [8, 11, 17, 29, 31, 33]. We use these strategies in this

paper.

The novelty is that the adapted methods act now implicitly and occur essentially in the analysis for the dual problem. This sounds particularly simple in the case of mesh grading. However, the convergence proof in [4] contains not only interpolation error estimates for the dual solution and its normal derivative (which are improved now) but also the application of an inverse inequality which gives a too pessimistic result if used unchanged in the case of graded meshes. We prove in section 2 a sharp result by using a weighted norm in intermediate steps. Note we suggest a strong mesh grading with grading parameter µ → 0 (the parameter is explained in section 2) for ω → 2π because of the interpolation error estimate of ∂

n

v

z

; the numerical tests show that weaker grading is not sufficient.

The basic idea of the dual singular function method (see [11]), or the singular complement method (see [17]), is to augment the approximation space for the solution by one (or more, if necessary) singular function of type r

^λ

sin(λθ) and the space of test functions by a dual function of type r

^−λ

sin(λθ), where r, θ are polar coordinates at the concave corner. In this paper we do it the other way round and compute an approximate solution

z

h

∈ Y

h

⊕ Span{r

^−λ

sin(λθ)}

such that the error estimate

ky − z

_h

k

_L2(Ω)

≤ Ch

^1/2

h

^1/2

kf k

_H−1(Ω)

+ kuk

_L2(Γ)

can be shown. Note that the original singular complement method augments the stan- dard finite element space with a function which is part of the representation of the solution. Here, we complement the finite element space with r

^−λ

sin(λθ) 6∈ H

^1/2

(Ω), and, although y ∈ H

^1/2

(Ω), this has an effect on the approximation order in the L

²

(Ω)-norm. This makes the method different from the original singular comple- ment method, [17], and we call it the dual singular complement method. Numerical experiments in section 4 confirm the theoretical results.

Finally in this introduction, we would like to note that higher order finite elements are not useful here since the solution has low regularity. The extension of our methods to three-dimensional domains should be possible in the case of mesh grading (at considerable technical expenses in the analysis) but is not straightforward in the case of the dual singular complement method since the space V \ H

²

(Ω) is in general not finite dimensional; see [18] for the Fourier singular complement method to treat special domains. Curved boundaries could be treated at the price of using nonaffine finite elements; see, e. g., [10, 12, 22].

2. Graded meshes. Recall from the introduction that Ω ⊂ R

²

is a bounded polygonal domain with boundary Γ, and we consider here the case that Ω has exactly one corner (called singular corner ) with interior angle ω ∈ (π, 2π). The convex case was already treated in [4] and the case of more than one nonconvex corner can be treated similarly since corner singularities are local phenomena.

Without loss of generality we can assume that the singular corner is located at the origin of the coordinate system, and that one boundary edge is contained in the positive x

1

-axis. We recall from [25, Theorem 4.4.3.7] or [26, sections 1.5, 2.3, and 2.4] that the weak solution of the boundary value problem (1) with f ∈ L

²

(Ω) and u = 0 is not contained in H

²

(Ω) but in

H

_∆¹

(Ω) ∩ H

₀¹

(Ω) =

H

²

(Ω) ∩ H

₀¹

(Ω)

⊕ Span{ξ(r) r

^λ

sin(λθ)},

(7)

ξ being a cutoff function, while r and θ denote polar coordinates at the singular corner.

Let the finite element mesh T

h

= {T } be graded with the mesh grading parameter µ ∈ (0, 1], i. e., the element size h

T

= diam T and the distance r

T

of the element T to the singular corner are related by

c

1

h

^1/µ

≤ h

T

≤ c

2

h

^1/µ

for r

T

= 0, c

₁

hr

_T^1−µ

≤ h

_T

≤ c

₂

hr

_T^1−µ

for r

_T

> 0.

(8)

This type of graded mesh was investigated before in [8, 29, 31, 32]; see also the overview and background information in [5, section 2.3] and [1, section 7]. Define the finite element spaces

Y

h

= {v

h

∈ H

¹

(Ω) : v

h

|

T

∈ P

1

∀T ∈ T

h

}, Y

0h

= Y

h

∩ H

₀¹

(Ω), Y

_h^∂

= Y

h

|

∂Ω

, (9)

and let the regularized boundary datum u

^h

∈ Y

_h^∂

⊂ H

^1/2

(Γ) be defined by the L

²

(Γ)- projection Π

h

u or by the Carstensen interpolant C

h

u; see [13]. To define the latter let N

Γ

be the set of nodes of the triangulation on the boundary, and set

C

_h

u = X

x∈NΓ

π

_x

(u)λ

_x

with π

_x

(u) = R

ω_x

uλ

x

R

ω_x

λ

x

= (u, λ

x

)

ω_x

(1, λ

x

)

ωx

,

where λ

x

is the standard hat function related to x and ω

x

= supp λ

x

⊂ Γ. As already outlined in [4], the advantages of the interpolant in comparison to the L

²

-projection are its local definition and the property

u ∈ [a, b] ⇒ C

_h

u ∈ [a, b];

see [21]; a disadvantage may be that C

h

u

h

6= u

h

for piecewise linear u

h

. With these regularized boundary data we then define the regularized weak solution y

^h

∈ Y

_∗^h

:=

{v ∈ H

¹

(Ω) : v|

Γ

= u

^h

} by (4).

Lemma 2.1. If the mesh is graded with parameter µ < 2λ − 1 the effect of the regularization of the boundary datum can be estimated by

ky − y

^h

k

_L2(Ω)

≤ ch

^1/2

kuk

_L2(Γ)

. Proof. In view of

ky − y

^h

k

_L2(Ω)

= sup

z∈L²(Ω),z6=0

(y − y

^h

, z)

_Ω

kzk

L²(Ω)

(10)

we have to estimate (y − y

^h

, z)

Ω

. To this end, let z ∈ L

²

(Ω) be an arbitrary function and let v

z

∈ V be defined by (6). Since the weak regularized solution y

^h

∈ Y

_∗^h

:=

{v ∈ H

¹

(Ω) : v|

Γ

= u

^h

} defined by (4) is also a very weak solution, (y

^h

, ∆v)

Ω

= (u

^h

, ∂

n

v)

Γ

− (f, v)

Ω

∀v ∈ V, (11)

we get with (2) and (6)

(y − y

^h

, z)

_Ω

= (u − u

^h

, ∂

_n

v

_z

)

_Γ

.

(12)

If u

^h

is the L

²

(Γ)-projection Π

h

u of u we can continue with

(u − u

^h

, ∂

_n

v

_z

)

_Γ

= (u − u

^h

, ∂

_n

v

_z

− Π

_h

(∂

_n

v

_z

))

_Γ

= (u, ∂

_n

v

_z

− Π

_h

(∂

_n

v

_z

))

_Γ

≤ kuk

L²(Γ)

k∂

n

v

z

− Π

h

(∂

n

v

z

)k

L²(Γ)

≤ kuk

_L2(Γ)

k∂

n

v

z

− C

h

(∂

n

v

z

)k

_L2(Γ)

= kuk

L²(Γ)

X

x∈NΓ

∂

n

v

z

− π

x

(∂

n

v

z

) λ

x

_L2(Γ)

≤ ckuk

_L2(Γ)



 X

x∈NΓ

k∂

_n

v

_z

− π

_x

(∂

_n

v

_z

)k

²_L2(ω_x)





1/2

.

If u

^h

is the Carstensen interpolant of u, there holds (u − C

h

u, ∂

n

v

z

)

Γ

=



 X

x∈N_Γ

(u − π

x

u)λ

x

, ∂

n

v

z





Γ

= X

x∈N_Γ

(u − π

x

(u), (∂

n

v

z

)λ

x

)

Γ

= X

x∈NΓ

(u − π

x

(u), (∂

n

v

z

− π

x

(∂

n

v

z

))λ

x

)

Γ

≤ X

x∈N_Γ

kuk

_L2(ωx)

k∂

n

v

z

− π

x

(∂

n

v

z

)k

_L2(ωx)

≤ ckuk

L²(Γ)



 X

x∈NΓ

k∂

n

v

z

− π

x

(∂

n

v

z

)k

²_L2(ωx)





1/2

,

i. e., in both cases we have to estimate P

x∈NΓ

k∂

n

v

z

− π

x

(∂

n

v

z

)k

²_L2(ωx)

. To this end we notice that

v

z

∈ V =

H

²

(Ω) ∩ H

₀¹

(Ω)

⊕ Span{ξ(r) r

^λ

sin(λθ)}

and, consequently,

∂

n

v

z

∈ V

Γ

=





N

Y

j=1

H

₀₀^1/2

(Γ

j

)



 ⊕ Span{ξ(r) r

^λ−1

};

see [4, Remark 2.2] or [25, Theorem 1.5.2.8]. This means that we can split ∂

n

v

z

= αξ(r) r

^λ−1

+ P

N

j=1

w

j

with w

j

∈ H

₀₀^1/2

(Γ

j

) and

|α| +

N

X

j=1

kw

j

k

_H1/2

00 (Γ_j)

=: k∂

n

v

z

k

V_Γ

≤ ckv

z

k

V

:= k∆v

z

k

_L2(Ω)

= kzk

_L2(Ω)

. In the remaining part of the proof we show for j = 1, . . . , N,



 X

x∈NΓ

kw

j

− π

x

w

j

k

²_L2(ωx)





1/2

≤ ch

^1/2

kw

j

k

_H1/2 00 (Γj)

, (13)



 X

x∈NΓ

kξ(r) r

^λ−1

− π

x

(ξ(r) r

^λ−1

)k

²_L2(ωx)





1/2

≤ ch

^1/2

,

(14)

to conclude P

x∈NΓ

k∂

n

v

z

− π

x

(∂

n

v

z

)k

²_L2(ω_x)

^1/2

≤ ch

^1/2

kzk

L²(Ω)

and, hence, (u − u

^h

, ∂

_n

v

_z

)

_Γ

≤ ch

^1/2

kuk

_L2(Γ)

kzk

_L2(Ω)

which, together with (10) and (12), finishes the proof.

We extend w

j

to the whole boundary Γ by zero on Γ \ Γ

j

and start with the estimate

kw

j

− π

s

w

j

k

_L2(ωx)

≤ ch

^s_x

ks

j

k

_Hs(ωx)

, s = 0, 1, x ∈ N

Γ

. (15)

This estimate follows for s = 0 from the definition of π

x

. For s = 1 it follows from a Bramble–Hilbert-type argument if x is not a corner of Ω. In the case of a corner point x we use instead the zero boundary condition of w

_j

on one end of ω

_x

. Adding these estimates and using that H

₀₀^1/2

(Γ

j

) is an interpolation space of L

²

(Γ

j

) and H

₀¹

(Γ

j

) we obtain (13). Note that the local element size h

x

is bounded by h from above.

Denote by N

Γ,reg

⊂ N

Γ

the set of nodes where ω

x

does not contain the singular corner. Let r

x

be the distance of x ∈ N

Γ,reg

to the set of corners of Ω, and note that the local mesh size satisfies both h

x

≤ chr

^1−µ_x

and h

x

≤ cr

x

. One can estimate by using (15) with s = 1,

X

x∈NΓ,reg

kξ(r) r

^λ−1

− π

x

(ξ(r) r

^λ−1

)k

²_L2(ω_x)

≤ c X

x∈NΓ,reg

h

²_x

kr

^λ−2

k

²_L2(ω_x)

≤ ch X

x∈NΓ,reg

r

^1−µ_x

r

x

kr

^λ−2

k

²_L2(ω_x)

≤ ch

Z

diamΩ 0

r

^{2−µ+2(λ−2)}

dr = ch

for µ < 2λ−1. For the three nodes x ∈ N

Γ

\N

Γ,reg

we cannot use the H

¹

(ω

x

)-regularity of r

^λ−1

but, by using the stability of π

x

, the properties of ξ(·), and h

x

∼ h

^1/µ

there holds

kξ(r) r

^λ−1

− π

x

(ξ(r) r

^λ−1

)k

_L2(ωx)

≤ ckr

^λ−1

k

_L2(ωx)

∼ h

^λ−1/2_x

∼ h

^{(λ−1/2)/µ}

≤ ch

^1/2

for µ < 2λ − 1. Note that we computed the norm in the middle step. This finishes the proof.

We consider now a lifting ˜ B

_h

u

^h

∈ Y

_∗h

:= Y

_∗^h

∩ Y

_h

defined by the nodal values as follows:

( ˜ B

h

u

^h

)(x) =

( u

^h

(x) for all nodes x ∈ Γ, 0 for all nodes x ∈ Ω.

(16)

The function y

^h

and its finite element approximation y

h

∈ Y

_∗h

are now defined by y

^h

= y

_f

+ ˜ B

_h

u

^h

+ ˜ y

^h₀

as well as y

_h

= y

_{f h}

+ ˜ B

_h

u

^h

+ ˜ y

_0h

,

(17)

where y

f

, y ˜

₀^h

∈ H

₀¹

(Ω) and y

f h

, y ˜

0h

∈ Y

0h

satisfy (∇y

f

, ∇v)

Ω

= (f, v)

Ω

∀v ∈ H

₀¹

(Ω), (18)

(∇y

f h

, ∇v

h

)

_Ω

= (f, v

_h

)

_Ω

∀v

h

∈ Y

_0h

, (19)

(∇˜ y

^h₀

, ∇v)

Ω

= −(∇( ˜ B

h

u

^h

), ∇v)

Ω

∀v ∈ H

₀¹

(Ω), (20)

(∇ y ˜

0h

, ∇v

h

)

Ω

= −(∇( ˜ B

h

u

^h

), ∇v

h

)

Ω

∀v

h

∈ Y

0h

. (21)

In order to estimate ky

^h

− y

h

k

_L2(Ω)

we estimate ky

f

− y

f h

k

_L2(Ω)

and k˜ y

₀^h

− y ˜

0h

k

_L2(Ω)

.

Lemma 2.2. If the mesh is graded with parameter µ < λ the error in approximat- ing y

f

satisfies

ky

f

− y

_{f h}

k

L²(Ω)

≤ chkf k

H⁻¹(Ω)

.

Note that the condition µ < λ is weaker than the condition µ < 2λ−1 from Lemma 2.1 since λ < 1.

Proof. As in the proof of Lemma 2.1, let z ∈ L

²

(Ω) be an arbitrary function, let v

z

∈ V be defined via (6), and let v

zh

∈ Y

0h

be the Ritz projection of v

z

. By the definitions (18) and (19) and using the Galerkin orthogonality we get

(y

_f

− y

_{f h}

, z)

_Ω

= (∇(y

_f

− y

_{f h}

), ∇v

_z

)

_Ω

= (∇(y

_f

− y

_{f h}

), ∇(v

_z

− v

_zh

))

_Ω

= (∇y

f

, ∇(v

z

− v

zh

))

Ω

≤ k∇y

f

k

L²(Ω)

k∇(v

z

− v

zh

)k

L²(Ω)

. By using standard a priori estimates (see, e.g., [7, Theorem 3.2]), we obtain with grading µ < λ the bounds k∇y

f

k

_L2(Ω)

≤ kf k

_H−1(Ω)

, k∇(v

z

−v

zh

)k

_L2(Ω)

≤ chkzk

_L2(Ω)

, and, hence, with

ky

_f

− y

_{f h}

k

_L2(Ω)

= sup

z∈L²(Ω),z6=0

(y

f

− y

f h

, z)

Ω

kzk

L²(Ω)

,

the assertion of the lemma.

In the proof of Lemma 2.4 we will employ a regularity result which is proved in [4, section II.C]. Reducing notation for the price of a slightly weaker statement we have the folllowing lemma.

Lemma 2.3. If ω > π then the very weak solution y from (2) satisfies kr

^−β

yk

L²(Ω)

≤ c

kf k

H⁻¹(Ω)

+ kuk

L²(Γ)

for all β ∈ 1 − λ, 1

2 i .

Proof. The statement is proved in [4, Lemma 2.8]. Concerning the assumptions on the regularity of the data, note that f and u are from bigger spaces there if β ≤

¹₂

; see [4, Remark 2.7]. Concerning the definition of the solution y in [4, (2.15)] note that the test space there contains V , which is seen by using the splitting (7), and since the solutions of both formulations are unique they must be equal.

In order to estimate k˜ y

₀^h

− y ˜

0h

k

L²(Ω)

, we divide the domain Ω into subsets Ω

J

, i.e.,

Ω =

I

[

J=0

Ω

J

,

where Ω

_J

:= {x ∈ Ω : d

_J+1

≤ |x| ≤ d

_J

} for J = 1, . . . , I − 1, Ω

_I

:= {x ∈ Ω : |x| ≤ d

_I

}, and Ω

₀

:= Ω\ S

I

J=1

Ω

_J

. The radii d

_J

are set to 2

^−J

and the index I is chosen such that

d

I

= 2

^−I

= c

I

h

^1/µ

(22)

with a constant c

I

> 1 exactly specified later on. In addition we define the extended domains Ω

⁰_J

and Ω

⁰⁰_J

by

Ω

⁰_J

:= Ω

J−1

∪ Ω

_J

∪ Ω

_J+1

and Ω

⁰⁰_J

:= Ω

⁰_J−1

∪ Ω

⁰_J

∪ Ω

⁰_J+1

,

respectively, with the obvious modifications for J = 0, 1 and J = I − 1, I .

Lemma 2.4. With σ := r + d

I

there holds the estimate

kσ

^(1−µ)/2

∇˜ y

₀^h

k

L²(Ω)

+ kσ

^(1−µ)/2

∇( ˜ B

h

u

^h

)k

L²(Ω)

≤ ch

^−1/2

kuk

L²(Γ)

. Proof. We start by rearranging terms, i.e.,

kσ

^(1−µ)/2

∇˜ y

^h₀

k

²_L2(Ω)

= Z

Ω

σ

^1−µ

∇˜ y

₀^h

· ∇˜ y

₀^h

= Z

Ω

∇ y ˜

^h₀

· ∇(˜ y

^h₀

σ

^1−µ

) − Z

Ω

˜

y

^h₀

∇˜ y

^h₀

· ∇σ

^1−µ

. (23)

For the first term in (23) we conclude according to (20) Z

Ω

∇ y ˜

^h₀

· ∇(˜ y

^h₀

σ

^1−µ

) = − Z

Ω

∇( ˜ B

_h

u

^h

) · ∇(˜ y

₀^h

σ

^1−µ

)

= − Z

Ω

σ

^1−µ

∇( ˜ B

h

u

^h

) · ∇˜ y

^h₀

− Z

Ω

˜

y

^h₀

∇( ˜ B

h

u

^h

) · ∇σ

^1−µ

≤ kσ

^(1−µ)/2

∇( ˜ B

h

u

^h

)k

_L2(Ω)

kσ

^(1−µ)/2

∇˜ y

₀^h

k

_L2(Ω)

+ kσ

^{(−1−µ)/2}

y ˜

₀^h

k

_L2(Ω)

, (24)

where we used the Cauchy–Schwarz inequality and

∇σ

^1−µ

= (1 − µ)σ

^−µ

(cos θ, sin θ)

^T

. (25)

Having in mind the decomposition of the domain in subdomains Ω

J

, an application of the Poincar´ e inequality yields for the latter term in (24)

kσ

^{(−1−µ)/2}

y ˜

₀^h

k

²_L2(Ω)

=

I

X

J=0

kσ

^{(−1−µ)/2}

y ˜

₀^h

k

_L2(ΩJ)

kσ

^{(−1−µ)/2}

y ˜

₀^h

k

_L2(ΩJ)

≤

I

X

J=0

d

^{(−1−µ)/2}_J

k y ˜

₀^h

k

L²(Ω_J)

kσ

^{(−1−µ)/2}

y ˜

^h₀

k

L²(Ω_J)

≤ c

I

X

J=0

d

^(1−µ)/2_J

k∇˜ y

₀^h

k

_L2(Ω⁰_J)

kσ

^{(−1−µ)/2}

y ˜

^h₀

k

_L2(Ω_J)

≤ ckσ

^(1−µ)/2

∇ y ˜

^h₀

k

L²(Ω)

kσ

^{(−1−µ)/2}

y ˜

₀^h

k

L²(Ω)

,

where we used d

J

∼ σ for x ∈ Ω

⁰_J

twice and the discrete Cauchy–Schwarz inequality.

Consequently, we get from (24) Z

Ω

∇ y ˜

^h₀

· ∇(˜ y

^h₀

σ

^1−µ

) ≤ ckσ

^(1−µ)/2

∇( ˜ B

h

u

^h

)k

_L2(Ω)

kσ

^(1−µ)/2

∇˜ y

₀^h

k

_L2(Ω)

. (26)

Similarly to the above steps, we get for the second term in (23) by means of (25)

Z

Ω

˜

y

^h₀

∇˜ y

^h₀

· ∇σ

^1−µ

≤ kσ

^(1−µ)/2

∇˜ y

^h₀

k

L²(Ω)

kσ

^{(−1−µ)/2}

y ˜

₀^h

k

L²(Ω)

(27)

≤ kσ

^(1−µ)/2

∇˜ y

₀^h

k

_L2(Ω)

kσ

^{(−1−µ)/2}

(˜ y

^h₀

+ ˜ B

_h

u

^h

)k

_L2(Ω)

+ kσ

^{(−1−µ)/2}

B ˜

_h

u

^h

k

_L2(Ω)

, such that we infer from (23), (26), and (27) that

kσ

^(1−µ)/2

∇˜ y

^h₀

k

_L2(Ω)

≤ c

kσ

^{(−1−µ)/2}

B ˜

_h

u

^h

k

_L2(Ω)

+ kσ

^(1−µ)/2

∇( ˜ B

_h

u

^h

)k

_L2(Ω)

+ kσ

^{(−1−µ)/2}

(˜ y

^h₀

+ ˜ B

h

u

^h

)k

_L2(Ω)

.

(28)

Due to the definition of ˜ B

h

and the definition of the element size h

T

in the case of graded meshes we easily obtain by means of the norm equivalence in finite dimensional spaces that

kσ

^{(−1−µ)/2}

B ˜

_h

u

^h

k

L²(Ω)

+ kσ

^(1−µ)/2

∇( ˜ B

_h

u

^h

)k

L²(Ω)

≤ ch

^−1/2

ku

^h

k

L²(Γ)

≤ ch

^−1/2

kuk

_L2(Γ)

, (29)

where we employed the stability of u

^h

in L

²

(Γ) in the last step. Having in mind the definition (22) of d

I

and applying Lemma 2.3 with β =

¹₂

for the solution ˜ y

^h₀

+ ˜ B

h

u

^h

of the homogeneous equation with boundary datum u

^h

we conclude that

kσ

^{(−1−µ)/2}

(˜ y

^h₀

+ ˜ B

_h

u

^h

)k

_L2(Ω)

≤ d

^−µ/2_I

kσ

^−1/2

(˜ y

^h₀

+ ˜ B

_h

u

^h

)k

_L2(Ω)

≤ ch

^−1/2

kr

^−1/2

(˜ y

^h₀

+ ˜ B

h

u

^h

)k

_L2(Ω)

≤ ch

^−1/2

ku

^h

k

_L2(Γ)

≤ ch

^−1/2

kuk

_L2(Γ)

, (30)

where we used again the stability of u

^h

. The estimates (28), (29), and (30) end the proof.

Lemma 2.5. Let σ := r + d

I

and µ ∈ (0, 2λ − 1). Then there is the estimate kσ

^{−(1−µ)/2}

(˜ y

₀^h

− y ˜

0h

)k

L²(Ω)

≤ ch

^1/2

kuk

L²(Γ)

.

Proof. Let v ∈ H

₀¹

(Ω) be the weak solution of

−∆v = σ

^−(1−µ)

(˜ y

₀^h

− y ˜

_0h

) in Ω, v = 0 on Γ,

which, according to Theorem 2.15 of [20], has the regularity v ∈ V

_(1−µ)/2^2,2

(Ω) (as µ < 2λ − 1) and hence

¹₂

(1 − µ) > 1 − λ) and satisfies the a priori estimate

|v|

_V2,2

(1−µ)/2(Ω)

≤ ckσ

^−(1−µ)

(˜ y

₀^h

− y ˜

_0h

)k

_V0,2

(1−µ)/2(Ω)

≤ ckσ

^{−(1−µ)/2}

(˜ y

₀^h

− y ˜

_0h

)k

_L2(Ω)

, (31)

where we use the weighted Sobolev space V

_β^k,2

(Ω) := {v ∈ D

⁰

: kvk

_Vk,2

β (Ω)

< ∞} with kvk

²

V_β^k,2(Ω)

:=

k

X

j=1

|v|

²_Vj,2

β−k+j(Ω)

, |v|

_Vj,2

β (Ω)

:= kr

^β

∇

^j

vk

_L2(Ω)

. Then we obtain by using integration by parts and the Galerkin orthogonality

kσ

^{−(1−µ)/2}

(˜ y

₀^h

− y ˜

0h

)k

²_L2(Ω)

= (˜ y

^h₀

− y ˜

0h

, −∆v)

Ω

= (∇(˜ y

₀^h

− y ˜

0h

), ∇(v − I

h

v))

Ω

≤

I

X

J=0

k∇(˜ y

^h₀

− y ˜

0h

)k

_L2(ΩJ)

k∇(v − I

h

v)k

_L2(ΩJ)

, (32)

where I

h

is the Lagrange interpolant.

By employing standard interpolation error estimates on graded meshes we obtain for any µ ∈ (0, 1]

k∇(v − I

h

v)k

_L2(ΩJ)

≤ chd

^(1−µ)/2_J

|v|

_V2,2

(1−µ)/2(Ω⁰_J)

, (33)

where the constant c is independent of c

I

; see, e.g., [6, Lemma 3.7] or [30, Lemma

3.58]. In fact, the constant is essentially the one appearing in the local, elementwise

interpolation error estimate. Note that this kind of independence will be crucial when applying a kickback argument further below.

Local finite element error estimates from [23, Theorem 3.4] yield k∇(˜ y

^h₀

− y ˜

_0h

)k

_L2(Ω_J)

≤ c min

v_h∈Y_0h

k∇(˜ y

^h₀

− v

_h

)k

_L2(Ω⁰_J)

+ 1

d

_J

k y ˜

₀^h

− v

_h

k

_L2(Ω⁰_J)

+ c 1

d

_J

k˜ y

₀^h

− y ˜

0h

k

_L2(Ω⁰_J)

.

By choosing v

h

≡ 0 and by applying the Poincar´ e inequality, we conclude k∇(˜ y

₀^h

− y ˜

_0h

)k

_L2(Ω_J)

≤ c

k∇˜ y

₀^h

k

_L2(Ω⁰_J)

+ 1

d

_J

k˜ y

^h₀

− y ˜

_0h

k

_L2(Ω⁰_J)

≤ c

k∇˜ y

₀^h

k

L²(Ω⁰⁰_J)

+ d

^{(−1−µ)/2}_J

kσ

^{−(1−µ)/2}

(˜ y

^h₀

− y ˜

0h

)k

L²(Ω⁰_J)

, (34)

where we used d

_J

∼ σ for x ∈ Ω

⁰_J

. Consequently, we get from (32)–(34) kσ

^{−(1−µ)/2}

(˜ y

₀^h

− y ˜

_0h

)k

²_L2(Ω)

≤ c

I

X

J=0

hkσ

^(1−µ)/2

∇˜ y

₀^h

k

_L2(Ω⁰⁰_J)

+ hd

^−µ_J

kσ

^{−(1−µ)/2}

(˜ y

^h₀

− y ˜

_0h

)k

_L2(Ω⁰_J)

|v|

_V2,2 (1−µ)/2(Ω⁰_J)

≤ c

hkσ

^(1−µ)/2

∇ y ˜

₀^h

k

_L2(Ω)

+ c

^−µ_I

kσ

^{−(1−µ)/2}

(˜ y

₀^h

− y ˜

0h

)k

_L2(Ω)

|v|

_V2,2 (1−µ)/2(Ω)

,

where we again employed d

J

∼ σ for x ∈ Ω

⁰⁰_J

, hd

^−µ_J

≤ c

^−µ_I

, which holds due to the definition (22) of d

I

, and the discrete Cauchy–Schwarz inequality. For µ ∈ (0, 2λ − 1) we infer by the a priori estimate (31) that

kσ

^{−(1−µ)/2}

(˜ y

^h₀

− y ˜

0h

)k

_L2(Ω)

≤ c

hkσ

^(1−µ)/2

∇ y ˜

₀^h

k

L²(Ω)

+ c

^−µ_I

kσ

^{−(1−µ)/2}

(˜ y

₀^h

− y ˜

_0h

)k

L²(Ω)

.

By choosing c

_I

large enough we can kick back the second term in the above inequality such that Lemma 2.4 yields the desired result.

Theorem 2.6. For µ ∈ (0, 2λ − 1) we get ky − y

h

k

L²(Ω)

≤ ch

^1/2

kuk

L²(Ω)

+ h

^1/2

kf k

_H⁻¹_(Ω)

. (35)

Proof. Due to the boundedness of σ

^(1−µ)/2

independent of h for all µ ∈ (0, 1] we obtain from Lemma 2.5

k y ˜

₀^h

− y ˜

0h

k

L²(Ω)

≤ kσ

^(1−µ)/2

k

L^∞(Ω)

kσ

^{−(1−µ)/2}

(˜ y

₀^h

− y ˜

0h

)k

L²(Ω)

≤ ch

^1/2

kuk

L²(Γ)

. (36)

In view of (17) we get by using the triangle inequality

ky − y

h

k

_L2(Ω)

≤ ky − y

^h

k

_L2(Ω)

+ ky

f

− y

f h

k

_L2(Ω)

+ k y ˜

₀^h

− y ˜

0h

k

_L2(Ω)

.

Using Lemmas 2.1 and 2.2 as well as (36) we get (35).

3. The dual singular complement method.

3.1. Analytical background and regularization. Using the notation of the previous section, we recall that the splitting (7) implies that

R := {∆v : v ∈ H

²

(Ω) ∩ H

₀¹

(Ω)}

(37)

is a closed subspace of L

²

(Ω). It is shown in [26, sect. 2.3] that L

²

(Ω) = R ⊕

^⊥

Span{p

s

}

(38)

with the dual singular function

p

s

= r

^−λ

sin(λθ) + ˜ p

s

, (39)

where ˜ p

s

∈ H

¹

(Ω) is chosen such that the decomposition (38) is orthogonal for the L

²

(Ω) inner product. Therefore, the dual singular function p

s

is a solution of

w ∈ L

²

(Ω) : (∆v, w) = 0 ∀v ∈ H

²

(Ω) ∩ H

₀¹

(Ω), (40)

which proves the nonuniqueness of the solution of (40).

Due to (38) we can split any L

²

(Ω)-function into L

²

(Ω)-orthogonal parts. To this end denote by Π

R

and Π

p_s

the orthogonal projections on R and on Span{p

s

}, respectively, i.e., for g ∈ L

²

(Ω), it is g = Π

R

g + Π

p_s

g, where

Π

p_s

g = α(g) p

s

with α(g) = (g, p

s

)

Ω

kp

s

k

²_L2(Ω)

, and Π

R

g = g − Π

p_s

g.

Since p

s

∈ L

²

(Ω) there exists

φ

s

∈ H

_∆¹

(Ω) ∩ H

₀¹

(Ω) : −∆φ

s

= p

s

; (41)

see also section 3.3 for more details on φ

_s

. For the moment we assume that p

_s

and φ

_s

are explicitly known; the decomposition g = Π

_R

g + α(g) p

_s

can be computed once g is given. Computable approximations of p

_s

and φ

_s

are discussed in section 3.3.

Now we come back to problem (2) and decompose its solution y in the form y = Π

R

y + α(y) p

s

.

(42)

From the decomposition (38) we see that problem (2) is equivalent to (y, p

_s

)

_Ω

= −(u, ∂

_n

φ

_s

)

_Γ

+ (f, φ

_s

)

_Ω

,

(y, ∆v)

Ω

= (u, ∂

n

v)

Γ

− (f, v)

Ω

∀v ∈ H

²

(Ω) ∩ H

₀¹

(Ω), and with the orthogonal splitting (42) to

α(y) (p

s

, p

s

)

Ω

= −(u, ∂

n

φ

s

)

Γ

+ (f, φ

s

)

Ω

,

(Π

_R

y, ∆v)

_Ω

= (u, ∂

_n

v)

_Γ

− (f, v)

_Ω

∀v ∈ H

²

(Ω) ∩ H

₀¹

(Ω).

The first equation directly yields α(y), namely,

α(y) = −(u, ∂

n

φ

s

)

Γ

+ (f, φ

s

)

Ω

(p

_s

, p

_s

)

_Ω

, (43)

hence the projection of y on p

s

is known. It remains to find an approximation of Π

R

y.

At this point we recall the regularization approach from [4] which we summarized already in the introduction. Let u

^h

∈ H

^1/2

(Γ) be a regularized boundary datum (this can be any, for example, Π

h

u or C

h

u from section 2, but we do not assume graded meshes here) such that we can define the regularized (weak) solution in Y

_∗^h

:=

{v ∈ H

¹

(Ω) : v|

Γ

= u

^h

},

y

^h

∈ Y

_∗^h

: (∇y

^h

, ∇v)

Ω

= (f, v)

Ω

∀v ∈ H

₀¹

(Ω).

(44)

In [4, Remark 2.13] we showed that the regularization error can be estimated by ky − y

^h

k

_L2(Ω)

≤ cku − u

^h

k

_H−s(Γ)

,

where 0 < s < λ −

¹₂

(if Ω was convex we would get s =

¹₂

, that means the regulariza- tion error is in general bigger in the nonconvex case). With the next lemma we show that Π

R

(y − y

^h

) is not affected by nonconvex corners.

Lemma 3.1. There holds the estimate

kΠ

_R

(y − y

^h

)k

_L2(Ω)

≤ Cku − u

^h

k

_H−1/2(Γ)

.

Proof. Recall V = H

_∆¹

(Ω) ∩ H

₀¹

(Ω) from (3). From (44) and the Green formula, we have for any v ∈ V

(f, v)

Ω

= (∇y

^h

, ∇v)

Ω

= −(y

^h

, ∆v)

Ω

+ (y

^h

, ∂

n

v)

Γ

.

Note that v ∈ V is sufficient for the Green formula, and v ∈ H

²

(Ω) is not required;

see [19, Lemma 3.4]. Subtracting this expression from the very weak formulation (2), we get

(y − y

^h

, ∆v)

_Ω

= (u − u

^h

, ∂

_n

v)

_Γ

∀v ∈ V.

Restricting this identity to v ∈ H

²

(Ω) ∩ H

₀¹

(Ω), we have

(Π

_R

(y − y

^h

), ∆v)

_Ω

= (u − u

^h

, ∂

_n

v)

_Γ

∀v ∈ H

²

(Ω) ∩ H

₀¹

(Ω).

(45)

Now for any z ∈ R, we let v

z

∈ H

²

(Ω) ∩ H

₀¹

(Ω) be the unique solution of

∆v

z

= z, (46)

which satisfies

k∂

n

v

z

k

_H1/2(Γ)

≤ ckv

z

k

_H2(Ω)

≤ ckzk

_L2(Ω)

. (47)

Since for any g ∈ L

²

(Ω) the equality

(Π

R

(y − y

^h

), g)

Ω

= (Π

R

(y − y

^h

), Π

R

g)

Ω

= (y − y

^h

, Π

R

g)

Ω

holds, we get with (45)–(47) kΠ

_R

(y − y

^h

)k

_L2(Ω)

= sup

z∈R,z6=0

(y − y

^h

, z)

_Ω

kzk

L²(Ω)

= sup

z∈R,z6=0

(u − u

^h

, ∂

_n

v

_z

)

_Γ

kzk

L²(Ω)

≤ ku − u

^h

k

_H−1/2(Γ)

sup

z∈R,z6=0

k∂

n

v

z

k

_H1/2(Γ)

kzk

_L2(Ω)

≤ cku − u

^h

k

_H−1/2(Γ)

which is the estimate to be proved.

3.2. Motivation for the dual singular complement method. As already discussed in the introduction, the adapted methods are motivated by the suboptimal convergence rate of the finite element solution on a family of quasi-uniform meshes. In this subsection, we recall these results and extend them by proving an estimate for the projection of the error into the space R from (37) which yields a better convergence rate. The insight into this structure of the discretization error motivates the new method which we call the dual singular complement method.

Recall from (9) the finite element spaces

Y

h

= {v

h

∈ H

¹

(Ω) : v

h

|

T

∈ P

1

∀T ∈ T

h

}, Y

0h

= Y

h

∩ H

₀¹

(Ω), Y

_h^∂

= Y

h

|

∂Ω

, defined now on a quasi-uniform family T

_h

of conforming finite element meshes. Assume that the regularized boundary datum u

^h

is contained in Y

_h^∂

such that the estimates

ku

^h

k

L²(Γ)

≤ ckuk

L²(Γ)

, (48)

ku − u

^h

k

_H−1/2(Γ)

≤ Ch

^1/2

kuk

_L2(Γ)

, (49)

hold. It can be derived from [4, Lemma 2.14] that this can be accomplished by using the L

²

(Γ)-projection or by quasi-interpolation: The stability (48) is explicitly stated there, and the error estimate (49) follows from the definition of the H

^−1/2

(Γ)-norm and the third estimate in [4, Lemma 2.14]. A consequence of Lemma 3.1 and (49) is the estimate

kΠ

R

(y − y

^h

)k

_L2(Ω)

≤ Ch

^1/2

kuk

_L2(Γ)

. (50)

(In the case of a convex domain the operator Π

_R

is the identity, and the corresponding error estimates were already proven in [4].)

As already done in the introduction, define further the finite element solution y

_h

∈ Y

_∗h

:= Y

_∗^h

∩ Y

_h

via

y

_h

∈ Y

_∗h

: (∇y

h

, ∇v

h

)

_Ω

= (f, v

_h

)

_Ω

∀v

h

∈ Y

_0h

. (51)

We proved in [4] that in the case of a quasi-uniform family of meshes T

h

ky − y

_h

k

L²(Ω)

≤ Ch

^s

h

^1/2

kf k

H⁻¹(Ω)

+ kuk

L²(Γ)

(52)

holds for s ∈ (0, λ −

¹₂

) (again s =

¹₂

for convex domains). As before, in the next lemma we show that Π

R

(y − y

h

) is not affected by the nonconvex corners.

Lemma 3.2. The following discretization error estimate holds:

kΠ

R

(y − y

h

)k

_L2(Ω)

≤ Ch

^1/2

h

^1/2

kf k

_H−1(Ω)

+ kuk

_L2(Γ)

. Proof. By the triangle inequality we have

kΠ

_R

(y − y

_h

)k

_L2(Ω)

≤ kΠ

_R

(y − y

^h

)k

_L2(Ω)

+ kΠ

_R

(y

^h

− y

_h

)k

_L2(Ω)

. (53)

The first term is estimated in (50). For the second term we first notice that y

^h

−y

_h

∈ H

₀¹

(Ω) satisfies the Galerkin orthogonality

(∇(y

^h

− y

_h

), ∇v

h

)

_Ω

= 0 ∀v

h

∈ Y

_0h

;

(54)

see (4) and (5). With that, we estimate kΠ

R

(y

^h

− y

h

)k

_L2(Ω)

by similar arguments as kΠ

R

(y − y

^h

)k

L²(Ω)

in the proof of Lemma 3.1. Recall from (46) and (47) that v

z

∈ H

²

(Ω) ∩ H

₀¹

(Ω) is the weak solution of ∆v

z

= z ∈ R. It can be approximated by the Lagrange interpolant I

_h

v

_z

satisfying

k∇(v

z

− I

_h

v

_z

)k

L²(Ω)

≤ chkv

z

k

H²(Ω)

≤ chkzk

L²(Ω)

. We get

kΠ

R

(y

^h

− y

h

)k

_L2(Ω)

= sup

z∈R,z6=0

(y

^h

− y

h

, z)

Ω

kzk

_L2(Ω)

= sup

z∈R,z6=0

(∇(y

^h

− y

h

), ∇v

z

)

Ω

kzk

_L2(Ω)

= sup

z∈R,z6=0

(∇(y

^h

− y

h

), ∇(v

z

− I

h

v

z

))

Ω

kzk

_L2(Ω)

≤ chk∇(y

^h

− y

_h

)k

_L2(Ω)

. (55)

In order to bound k∇(y

^h

− y

h

)k

_L2(Ω)

by the data we consider the lifting ˜ B

h

u

^h

∈ Y

_∗h

defined by (16). The next steps are simpler than in section 2 since we have a quasi-uniform family of meshes and obtain a sharp estimate also by using an inverse inequality below. The homogenized solution y

^h₀

= y

^h

− B ˜

h

u

^h

∈ H

₀¹

(Ω) satisfies

(∇y

^h₀

, ∇v)

Ω

= (f, v)

Ω

− (∇( ˜ B

h

u

^h

), ∇v)

Ω

∀v ∈ H

₀¹

(Ω).

By taking v = y

^h₀

we see that

k∇y

₀^h

k

²_L2(Ω)

≤ kf k

H⁻¹(Ω)

ky

₀^h

k

H¹(Ω)

+ k∇( ˜ B

_h

u

^h

)k

L²(Ω)

k∇y

^h₀

k

L²(Ω)

. Using the Poincar´ e inequality we obtain

k∇y

^h₀

k

L²(Ω)

≤ ckf k

H⁻¹(Ω)

+ k∇( ˜ B

_h

u

^h

)k

L²(Ω)

, (56)

and with the C´ ea lemma

k∇(y

^h

− y

_h

)k

L²(Ω)

≤ k∇y

₀^h

k

L²(Ω)

≤ ckf k

H⁻¹(Ω)

+ k∇( ˜ B

_h

u

^h

)k

L²(Ω)

.

The remaining term k∇( ˜ B

h

u

^h

)k

_L2(Ω)

is estimated by using the inverse inequality k∇( ˜ B

h

u

^h

)k

_L2(T)

≤ ch

^−1/2

ku

^h

k

_L2(E)

for E ⊂ T ∩ Γ, T ∈ T

h

, which can be proved by standard scaling arguments, to get k∇( ˜ B

h

u

^h

)k

_L2(Ω)

≤ ch

^−1/2

ku

^h

k

_L2(Γ)

.

(57)

Hence we proved

k∇(y

^h

− y

h

)k

_L2(Ω)

≤ ckf k

_H−1(Ω)

+ ch

^−1/2

ku

^h

k

_L2(Γ)

. With (53), (50), (55), the previous inequality, and (48) we finish the proof.

With (42) we can immediately conclude the following result.

Corollary 3.3. Let y

h

∈ Y

∗h

be the solution of (51), then the discretization error estimate

ky − (Π

R

y

h

+ α(y)p

s

)k

L²(Ω)

≤ Ch

^1/2

h

^1/2

kf k

_H⁻¹_(Ω)

+ kuk

L²(Γ)

holds, noting that p

s

and α(y) are given by (39) and (43), respectively.

Hence the positive result is that Π

R

y

h

+ α(y)p

s

is a better approximation of y

than y

h

. The problem is that p

s

and φ

s

are used explicitly, and in practice they are

not known. A remedy of this drawback is the aim of the next section.

3.3. Approximate singular functions. Following [17], we approximate p

s

from (39) by

p

^h_s

= p

^∗_h

− r

_h

+ r

^−λ

sin(λθ), r

_h

= ˜ B

_h

r

^−λ

sin(λθ) , p

^∗_h

∈ Y

0h

: (∇p

^∗_h

, ∇v

h

)

Ω

= (∇r

h

, ∇v

h

)

Ω

∀v

h

∈ Y

0h

(58)

with ˜ B

h

from (16). The function φ

s

from (41) admits the splitting φ

s

= ˜ φ + βr

^λ

sin(λθ)

(59)

with ˜ φ ∈ H

²

(Ω) and β = π

⁻¹

kp

s

k

²_L2(Ω)

; see again [17]. It is approximated by φ

^h_s

= φ

^∗_h

− β

h

s

h

+ β

h

r

^λ

sin(λθ), s

h

= ˜ B

h

r

^λ

sin(λθ)

, β

h

= 1

π kp

^h_s

k

²_L2(Ω)

, φ

^∗_h

∈ Y

0h

: (∇φ

^∗_h

, ∇v

h

)

Ω

= (p

^h_s

, v

h

)

Ω

+ β

h

(∇s

h

, ∇v

h

)

Ω

∀v

h

∈ Y

0h

,

(60)

that means ˜ φ is approximated by ˜ φ

_h

= φ

^∗_h

− β

_h

s

_h

∈ Y

_h

. The approximation errors are bounded by

kp

_s

− p

^h_s

k

_L2(Ω)

≤ ch

^2λ−

≤ ch, (61)

|β − β

h

| ≤ ch

^2λ−ε

≤ ch, (62)

kφ

s

− φ

^h_s

k

1,Ω

≤ ch;

(63)

see [17, Lemmas 3.1–3.3], where (62) and (63) imply k φ ˜ − φ ˜

_h

k

1,Ω

≤ ch.

(64)

At the end of section 3.2 we saw that Π

_R

y

_h

+ α(y)p

_s

is a better approximation of y than y

h

. Since this function is not computable we approximate it by

z

h

= Π

^h_R

y

h

+ α

h

p

^h_s

(65)

with

Π

^h_R

y

h

= y

h

− γ

h

p

^h_s

, γ

h

= (y

h

, p

^h_s

)

Ω

kp

^h_s

k

²_L2(Ω)

, (66)

and a suitable approximation α

_h

of α(y) from (43). To this end we write the prob- lematic term by using (59) as

(u, ∂

n

φ

s

)

Γ

= (u, ∂

n

φ) ˜

Γ

+ β(u, ∂

n

(r

^λ

sin(λθ)))

Γ

,

and replace the term (u, ∂

n

φ) ˜

Γ

by (u

^h

, ∂

n

φ) ˜

Γ

. Since ˜ φ belongs to H

²

(Ω) and u

^h

is the trace of ˜ B

h

u

^h

, we get by using the Green formula

(u

^h

, ∂

n

φ) ˜

Γ

= ( ˜ B

h

u

^h

, ∆ ˜ φ)

Ω

+ (∇ B ˜

h

u

^h

, ∇ φ) ˜

Ω

= −( ˜ B

h

u

^h

, p

s

)

Ω

+ (∇ B ˜

h

u

^h

, ∇ φ) ˜

Ω

(67)

as ∆ ˜ φ = ∆φ

_s

= −p

_s

. With all these notations and results, we define

α

h

= ( ˜ B

_h

u

^h

, p

^h_s

)

_Ω

− (∇ B ˜

_h

u

^h

, ∇ φ ˜

_h

)

_Ω

− β

_h

(u, ∂

_n

(r

^λ

sin(λθ)))

_Γ

+ (f, φ

^h_s

)

_Ω

(p

^h_s

, p

^h_s

)

²_Ω

.

(68)

Note that α

h

can be computed explicitly and therefore z

h

as well.

Let us estimate the approximation errors made.