Analysis and Finite Element Approximation of a Nonlinear Stationary Stokes Problem Arising in Glaciology

The aim of this paper is to study a nonlinear stationary Stokes problem with mixed boundary conditions that describes the ice velocity and pressure fields of grounded glaciers under Glen’s flow law. Using convex analysis arguments, we prove the existence and the uniqueness of a weak solution. A finite element method is applied with approximation spaces that satisfy the inf-sup condition, and a priori error estimates are established by using a quasinorm technique. Several algorithms including Newton’s method are proposed to solve the nonlinearity of the Stokes problem and are proved to be convergent. Our results are supported by numerical convergence studies.


Introduction
In this paper we consider a model problem that is commonly used by glaciologists to compute the motion of glaciers.Ice is assumed to be an incompressible non-Newtonian fluid governed by Glen's law 1 .Glen's law and the mass momentum equation lead to a nonlinear stationary Stokes problem with a strain-dependent viscosity.
Glacier models based on Glen's law have already been studied by several authors.However, all of them have considered a simplified model, called first-order approximation 2 .This model is obtained by rewriting the Stokes equations into a dimensionless form and by dropping all terms of order O 2 , where is the typical aspect ratio of glaciers.This simplification results into a nonlinear elliptic problem for the horizontal velocity field, the vertical component, and the pressure field being determined a posteriori.Colinge and Rappaz first demonstrated the well-posedness of this problem and proved the convergence of

The Model
Let us suppose that ice occupies the domain Ω ⊂ R d , with d 2 or 3. Ice can be considered as an incompressible non-Newtonian fluid with negligible inertial effects 11 .It follows that the velocity u and the pressure p of ice solve the stationary nonlinear Stokes problem in Ω: where ε u 1/2 ∇u ∇u T denotes the rate of strain tensor, μ the viscosity of ice, and f the gravity force.Here above, the viscosity μ depends on |ε u | : ε u : ε u and is defined by the regularised Glen's flow law 11 .More precisely, for a given velocity field u, the viscosity μ satisfies the following nonlinear equation: where s |ε u |, A is a positive parameter, n ≥ 1 is Glen's exponent, and τ 0 > 0 is a small regularization parameter which prevents infinite viscosity for zero strain τ 0 0 in the original Glen's law 1 .When n 1, then the viscosity μ is constant and 2.1 correspond to the classical linear Stokes problem related to a Newtonian fluid.In the framework of glaciology, n is often taken equal to 3; see 12 .Let us set the boundary conditions for the system of 2.1 .Three mechanical circumstances may occur at the boundary of a glacier: i no force applies on the ice-air interface; ii ice slides on the bedrock-ice interface; iii ice is stuck to the bedrock-ice interface.The boundary of Ω is thus split into three parts: Γ N , Γ R , and Γ D , referring to circumstances i , ii , and iii , respectively.We assume throughout that Ω is bounded, its boundaries Γ N and Γ R , are C 1 and Γ D / ∅.We consider the free surface condition: where n is the unit outward normal vector along the boundary of the domain Ω.We apply the nonlinear sliding condition 10, 13, 14 : where {t i } i 1,d−1 are the orthogonal vectors tangent to the boundary Γ R , that is, t 1 when d 2 and t 1 , t 2 when d 3.Here above, α α |u| is the sliding coefficient that is given by where t |u| is the Euclidean norm of u, n is Glen's exponent, c is a positive parameter, and t 0 > 0 is a small parameter which prevents infinite α for zero velocity.The no-sliding condition writes u 0, on Γ D .

2.6
Note that the conditions applied on boundaries Γ N , Γ R , and Γ D are Neumann, Robin-Dirichlet, and Dirichlet conditions, respectively.When n 1 Newtonian flow and Γ R ∅, the problem 2.1 with boundary conditions 2.3 , 2.6 has already been widely studied; see, for instance, 15-17 .

Existence and Uniqueness
In this section, we prove that there exists a unique weak solution to problem 2.1 with mixed boundary conditions 2.3 , 2.4 , and 2.6 .Pressure is first eliminated from the system by restricting the velocity space to divergence-free fields.Afterwards, the reduced problem is transformed into a minimisation problem.Following 3, 8 , its well-posedness is proved by using convex analysis arguments.The existence and the uniqueness of the pressure field are ensured by an inf-sup condition.We now state in the next lemma several properties of the function μ that will often be used in Sections 3, 4 and 5. Lemma 3.1.For all s ∈ R , there exists a unique μ μ s ∈ R satisfying 2.2 .The function s → μ s is C ∞ 0, ∞ and decreasing.There exist D 1 , D 2 , D 3 , D 4 > 0 such that: Proof.The properties of μ and inequalities 3. Let us notice that property 3.1 was introduced by Barrett and Liu see 5 in order to obtain a priori error estimates of a similar problem to the one treated in this paper.Define the Banach spaces: where are conjugate exponents and n is Glen's exponent.By using 3.2 , we have μ s s ≤ Cs r−1 for all s > 0.Then, if u ∈ V , we have Owing to H ölder's inequality, the mixed formulation of problem 2.1 with boundary conditions 2.3 , 2.4 , and 2.6 that consists of finding u, Moreover, if n 1, then r 2 and μ is constant and if Γ R ∅, then the linear problem 3.8 is well posed; see, for instance, 15-17 .
The next lemma states the equivalence of norms |ε • | L r and • W 1,r on space V . 3.9 Proof.We apply Corollary 4.1 in 18 F being the identity matrix and Lemma 3.1 page 40 in 16 .
We consider the divergence-free velocity space:

3.10
In V div , the pressure field p vanishes of the variational formulation 3.8 .The reduced formulation consists then of finding u ∈ V div such that

3.11
To transform problem 3.11 into a minimisation problem, we introduce the functional where x 0 tα t dt.

3.13
The functional J is Gâteaux differentiable, and its first derivative DJ, at point u ∈ V div , in direction v ∈ V div , is given by

3.14
Clearly, any minimiser of J in V div satisfies 3.11 .We now establish several lemmas that allow us to prove the existence and the uniqueness of this minimiser in Theorem 3.8.We show Advances in Numerical Analysis the continuity of J in Lemma 3.5, the strict convexity of J in Lemma 3.6, and the coercivity in the sense of 3.18 of J in Lemma 3.7.The continuity of J requires the following result Lemma 4 in 3 Lemma 3.4.Let O be a measurable set of R d and f, g ∈ L r O , then one has the following inequality: Proof.By using 3.2 , 2.5 , and 1 − 1/n 2 − r, we have, for all u, v ∈ V div

3.16
These two inequalities together with Lemma 3.4 imply the • W 1,r -continuity of J.
Lemma 3.6.The functional J is strictly convex on V .
Proof.Clearly, M s sμ s and M s sμ s μ s .From 3.3 , we have M s sμ s μ s ≥ 1/n μ s > 0 if s > 0, and then M is strictly convex.Since M is an increasing function, M | • | is strictly convex.In the same way, we can show that N | • | is strictly convex by using 2.5 .Let u, v ∈ V div satisfying u / v and θ ∈ 0, 1 .From Korn's inequality Lemma 3.3 , we have ε u / ε v in L r .As a consequence, 3.17 The strict convexity of J follows from the previous inequality and the convexity of N |•| .
Since J is convex, u ∈ V div satisfies 3.11 if and only if J u ≤ J v , ∀v ∈ V div .

Lemma 3.7.
There exist two constants D 1 , D 2 > 0 such that, for all u ∈ V ,

3.19
Advances in Numerical Analysis 7 As a consequence, there exist two constants By using Korn's inequality Lemma 3.3 , there exists

3.20
From Young's inequality, we have, for all δ > 0, where C 4 , C 5 > 0. We set δ small enough such that C 3 − δ r C 5 > 0. From inequalities 3.20 , 3.21 , and N ≥ 0, we obtain which is exactly 3.18 with D 1 : Theorem 3.8.There exists a unique u ∈ V div such that J u inf{J v ; v ∈ V div }.Moreover, u is the unique solution of 3.11 .

Proof. Clearly, there exists
There exists an integer K such that, for all ν > K, we have m 1 > J u ν .Owing to Lemma 3.7, the sequence {u ν } is bounded in V div .Since V div is a closed subspace of V , V div is reflexive.Consequently, there exist u ∈ V div and a subsequence of {u ν } still denoted {u ν } that converges weakly to u in V div .By Lemmas 3.5 and 3.6, J is weakly lower semicontinuous; see, for instance, Corollary III.8 in 19 page 38.Then, we have and J possesses at least one minimum u ∈ V div .Since J is strictly convex Lemma 3.6 , this minimum is unique.Moreover, u is the unique solution of 3.11 .
Spaces V and Q are required to satisfy the inf-sup condition, see 5, 20 , to ensure the existence and the uniqueness of p ∈ Q such that u, p satisfies the mixed formulation 3.8 .The inf-sup condition is proved in 15, 21 when Γ D ∂Ω or, equivalently, By following the proof of Proposition 5.3.2 in 22 , we can easily generalise this result when Γ R ∪ Γ N / ∅; see details in 12 .
Lemma 3.9.Spaces V and Q satisfy the inf-sup condition; that is, there exists C > 0 such that Theorem 3.10.There exists a unique couple u, p ∈ V, Q satisfying 3.8 .
Proof.Although the result is a straightforward application of Theorem 2.1 in 20 together with Theorem 3.8 and Lemma 3.9, we give all the arguments of the proof.Let A : V → V and B : V → Q be the operators defined by

Finite Element Approximation and A Priori Estimates
We assume that Ω is a convex polygonal or polyhedral domain and T h is a regular mesh of Ω parametrized by h, the highest diameter of the elements of T h .We say that V h ⊂ V and Q h ⊂ Q, some finite-dimensional approximation spaces on T h of V and Q, satisfy the inf-sup condition if, for all κ ∈ 1, ∞ , there exists a constant C h > 0 such that Advances in Numerical Analysis 9 The discrete problem is obtained by replacing the spaces V and Q by V h and Q h , respectively.

4.2
The discrete similar space to V div is

4.4
Since V div,h is a closed subspace of V , Theorem 3.8 and the proof can be rewritten by replacing V div by V div,h .

Theorem 4.1. There exists a unique
where C > 0 does not depend on u h .
From Theorem 4.1 and the inf-sup condition 4.1 , we can rewrite Theorem 3.10 and its proof for the discrete mixed problem.The error analysis that follows is partly inspired from 5, 7 .We give a priori estimates for the numerical approximation of the stationary Stokes problem in Theorem 4.9.
For the sake of simplicity, we suppose Γ R ∅; that is, the boundary Robin-Dirichlet condition is not considered; see also Remark 4.11.The nonlinearity of problem 3.8 is treated by introducing in Lemma 4.5 a quasi-norm that depends on the solution; see 5 .The orthogonality of the error Lemma 4.6 together with properties 3.4 and 3.5 of the function μ allow quasi-norm estimates to be established in Theorem 4.7.The properties of the quasi-norm given in Lemma 4.5 allow estimates with standard norms to be proved in Theorem 4.8.Eventually, these estimates together with interpolation inequalities yield to the main Theorem 4.9.
Lemma 4.5.Let u, p be the solution of 3.8 ; the application is a quasi-norm of V ; that is, it satisfies all properties of norms, except homogeneity.Moreover, there exists D 1 > 0 such that, for all v ∈ W 1,r Ω , one has and there exists D 2 > 0 such that, for all κ ∈ r, 2 and for all v ∈ W 1,κ Ω , one has Proof.The quasi-norm properties are shown in Lemma 3.1 in 5 .Inequalities 4.7 and 4.8 result from Korn and H ölder's inequalities; see details in 12 .
By setting v v h ∈ V div,h in 3.8 it is easy to prove the next lemma.
Lemma 4.6.Let u, p ∈ V, Q be the solution of problem 3.8 and u h ∈ V div,h the solution of problem 4.4 , then where D 1 > 0.Moreover, if the spaces V h and Q h satisfy the inf-sup condition 4.1 , then the solution u h , p h of 4.2 satisfies, for all κ ∈ r, 2 and for all q h ∈ Q h , where D 2 > 0. The constants D 1 , D 2 do not depend on u h and v h ; however, D 2 increasingly depends on C h −1 .
Proof.By using, respectively, the definition 4.6 of the quasi-norm ||| • |||, inequality 3.4 with 1 − 1/n 2 − r, and 4.9 , there exists where v h , q h ∈ V div,h × Q h .For the sake of simplicity, A 1 and A 2 are handled separately.By using inequality 3.5 with 1 − 1/n 2 − r, there exists C 2 > 0 such that

4.18
By moving 1/2 |||u − u h ||| 2 to the left-hand side, we obtain 4.11 .From the inf-sup condition 4.1 , we have, for all q h ∈ Q h , From 4.10 , we have, for all

4.20
Advances in Numerical Analysis 13 From 4.19 , 4.20 , and 3.5 with 1 − 1/n 2 − r, there exist C 6 , C 7 > 0 such that where C 8 : Eventually, the previous inequality together with p − p h L κ ≤ q h − p h L κ p − q h L κ leads to 4.12 .
Theorem 4.8.Let u, p ∈ V, Q be the solution of 3.8 , and u h ∈ V div,h the solution of 4.4 .For all v h , q h ∈ V div,h × Q h and for all κ ∈ r, 2 , assuming u ∈ W 1,κ Ω , one has where D 1 > 0.Moreover, if the spaces V h and Q h satisfy the inf-sup condition 4.1 , then the solution u h , p h of 4.2 satisfies for all v h , q h ∈ V h × Q h and for all κ ∈ r, 2 , assuming u ∈ W 1,κ Ω , where C depends on the inf-sup constant C h .Eventually, 4.24 follows from 4.12 , 4.11 , and 4.8 .

Advances in Numerical Analysis
Theorem 4.9.Assume that, for all κ ∈ r, 2 , there exists a continuous operator and a continuous operator where h is the size of the higher diameter of the elements of T h .Assume that V h and Q h satisfy the inf-sup condition 4.1 .Let u, p be the solution of problem 3.8 and let u h , p h be the solution of problem 4.2 .Assume that u, p ∈ W 2,κ d , W 1,κ , where κ ∈ r, 2 , then one has where Proof.Apply 4.23 and 4.24 with v h π h u and q h ρ h p .By using the continuity of π h , 4.5 , 4.26 , and 4.27 , there exist 1 |u| |v| 2−r dS.4.30

Successive Approximations
In this section, several successive approximation algorithms are proposed for solving the nonlinearity of the discrete problem 4.4 when n > 1.For the sake of simplicity, we suppose Γ R ∅ in this section; see Remark 5.8.We present a unified scheme that contains the classical fixed point method together with Newton's method.The mesh T h is fixed, and we assume that the approximation spaces satisfy • denotes an arbitrary norm of V div,h .Since V div,h is a finite-dimensional space, all norms are equivalent.Let γ ∈ 0, 1 .We define

5.2
The application E is well defined.Indeed, by using, respectively, μ < 0, inequalities 3.3 and 3.2 , there exist

5.3
As a consequence, the problem 5.2 is coercive.From the Lax-Milgram Theorem, see 15 page 83, there exists a unique solution In what follows, u h denotes the solution 4.4 , which is also the unique fixed point of E. Assume that u h,0 is given; we define iteratively a sequence u h,k , for all k ≥ 1, by Our goal is to prove that u h,k converges to u h when k goes to the infinity.When γ 0, we obtain the classical fixed point method, widely used to solve the nonlinearity of Glen's law; see 6, 9, 14 .When γ 1, we have an additional term in 5.2 which corresponds to Newton's method; see Remark 5.5.The case γ ∈ 0, 1 corresponds to a hybrid fixed point-Newton's method.The convergence of sequence u h,k requires several preliminary results.We compute the first derivative of E in Lemma 5.1.Lemma 5.2 provides an upper bound of the first derivative.Eventually, Theorem 5.3 states the linear convergence of u h,k by using the Banach fixed point theorem.Theorem 5.7 states the second-order convergence when γ 1.By differentiating formally 5.2 at point u h in direction u h , with w h E u h , we obtain the following lemma.

5.5
where w h solves

5.6
The problems 5.2 and 5.6 have the same coercivity properties to compute w h resp., w h from u h resp., u h .As a consequence, the problem 5.6 is well-posed by the Lax-Milgram theorem.To prove the convergence of the sequence u h,k , we look for a norm that makes E a contraction at point u h .Lemma 5.2.Let γ ∈ 0, 1 , and let u h be the fixed point of E. The application DE u h satisfies Proof.Since E u h u h , then 5.6 , with u h u h and w h u h , is rewriten as for all v h ∈ V div,h .From 5.8 , μ < 0, and 3.3 , we have

5.9
By setting v h w h in 5.9 , we obtain
Theorem 5.3.Let γ ∈ 0, 1 , let T h be a given mesh of Ω, and let • be a norm of V div,h .There exist δ > 0 and C > 0 such that if u h,0 − u h < δ, then one has 12 for all k ≥ 0, and u h,k is linearly convergent to u h .
Proof.From Lemma 5.2, the spectral radius of DE u h is lower than constant: which is lower than 1.The theorem is then a direct application of the Banach fixed point theorem in V div,h .
Remark 5.4.In Theorem 5.3, it should be stressed that δ depends on h.When h → 0 i.e., if we replace u h by u , we cannot ensure Theorem 5.3 to remain true.Nevertheless, in practise, δ seems to be independent of h, see Section 6.

Advances in Numerical Analysis
When γ 1, we have DE u h 0 from Lemma 5.2.It suggests that the convergence of sequence u h,k is quadratic.To establish the second-order convergence, we define, for all

5.14
Let u h , u h , u h ∈ V div,h .We compute formally the first-order derivative of F at point u h in direction u h :

5.15
and the second-order derivative of F at point u h in direction u h , u h :

5.16
Remark 5.5.If γ 1, we have, from the definition of E 5.2 , of F 5.14 , and of DF 5.15 , which highlights Newton's method.
Lemma 5.6.The following inequalities hold, for all u h , u h , u h ∈ V div,h :

5.19
Advances in Numerical Analysis 19 Proof.Inequality 5.18 follows from 5.15 and 3.3 , while inequality 5.19 directly follows from 5.16 .Computational details are given in 12 .
Theorem 5.7.Suppose γ 1, let T h be a given mesh of Ω, and let • be a norm of V div,h .There exist δ > 0 and C > 0 such that if u h,0 − u h < δ, then one has for all k ≥ 0, and u h,k is quadratically convergent to u h .
Proof.Owing to Theorem 5.3, there exists δ > 0 such that if By writing the Taylor expansion of F at point u h,k , there exists u h,k ∈ V div,h such that

5.22
Since u h solves 4.4 , then F u h ; v h 0 in 5.22 .By setting v h u h − u h,k 1 , we obtain, from 5.22 and 5.17 ,

5.23
Thanks to 3.2 , there exists

5.25
By applying 5.18 with u h u h,k and u h u h − u h,k 1 and 5.24 , we obtain

Advances in Numerical Analysis
By applying 5.19 with u h u h,k , u h u h − u h,k 1 , and u h u h − u h,k and 5.25 , we obtain:

5.28
By using Cauchy-Schwarz's inequality and the equivalence of norms, there exists C 3 > 0 such that
Remark 5.8.If Γ R / ∅, the nonlinear Robin-Dirichlet condition can be handled in the same way as for the viscosity function.In that case, we modify the application E by adding to the left-hand side of 5.2 .Theorems 5.3 and 5.7 can be easily extended to this case.

Numerical Results
In this section, numerical experiences are performed in two dimensions d 2 to validate the results of Theorems 4.9, 5.3, and 5.7.An exact solution of the Stokes problem 2.1 in the square Ω 0, 1 2 is considered in the pure Dirichlet case, that is, ∂Ω Γ D .Let Γ D , a penalisation term is added in the variational formulation to constrain the pressure average to be close to zero.For all norms • , the error between u and u h,k has two components: where u h is the exact solution of the nonlinear discrete problem.The convergence of the first component u − u h with respect to h is the concern of Theorem 4.9, while the convergence of the second component u h − u h,k with respect to k is the concern of Theorems 5.3 and 5.7.Let k be an integer large enough such that u h and u h,k can be confused, that is, such that To check the convergence of the second component, we compute the following error: where the norm • L r is evaluated by using the trapezoidal rule.For a fixed h, Theorem 5.3 states the linear convergence of E u k that depends on constant 5.13 when γ ∈ 0, 1 and the quadratic convergence when γ 1.Three values of γ are considered: γ 0 to test the fixed point algorithm, γ 0.5 to test the hybrid method, and γ 1 to test Newton's method.Figure 1 displays E u k according to k for each method: γ ∈ {0, 0.5, 1}, and for two different meshes.The recorded orders of convergence are consistent with Theorem 5.3: Newton's method γ 1 converges quadratically, the fixed point method and the hybrid method γ < 1 converge linearly, and convergence is faster for bigger γ and then smaller constant 5.13 .Newton's method is especially very efficient: in our example, only 3 iterations are needed against 8 for the fixed point algorithm to obtain the same accuracy of the numerical solution.Figure 1 also shows that the convergence of E u k with respect to k is not affected by any mesh refinement, as noticed in 3 .Moreover, the addition of supplementary terms in the Stokes system does not increase significantly the computational time for solving the linear system with a direct method.
The estimate of Theorem 4.9 is now tested by computing the following errors: We can change the regularity of u by changing the parameter θ in 6.1 from 2 to 1.34.Indeed, if θ 2, then u ∈ C ∞ Ω , while if θ 1.34, then u / ∈ W 2,2 Ω 2 , but u ∈ W 2,r Ω 2 , where r 3/2.In any case p ∈ C ∞ Ω .Figure 2 displays E u h and E The error E u k with respect to k for each method fixed point: γ 0, hybrid: γ 0.5, Newton: γ 1 .The error E u k obtained with the coarsest mesh is displayed on a and the error E u k obtained with the finest mesh is displayed on b .

Conclusions and Perspectives
We have proved the existence and the uniqueness of a weak solution of a nonlinear Stokes problem that describes the motion of glaciers.We have also proved the convergence of the finite element approximation and given a priori error estimates.New successive approximation algorithms have been proposed to solve the system nonlinearity and have been proved to be convergent.When implementing Newton's method, both theoretical and numerical studies have shown the efficiency of this method in comparison with the classical fixed point method.
Two extensions of our work should be investigated in future research.First, a posteriori estimates could be an aspect to be developed in order to implement an adaptive mesh procedure.Second, the presented Stokes model could benefit from recent improvements of the basal sliding description with Coulomb-type laws 10 .
V and Q are dual to V and Q, respectively.From Theorem 3.8, there exists a unique u ∈ ker B such that Au−f, v 0 for all v ∈ ker B, which means that Au−f ∈ ker B ⊥ .Owing to the inf-sup condition 4.1 , the operator B : V → Q is surjective, ker B T ∅, and R B T is closed; see Lemma A.40 in 15 .As a consequence, Au − f ∈ ker B ⊥ R B T R B T and there exists p ∈ Q such that Au − f B T p. Since ker B T ∅, the pressure p is necessarily unique.Eventually, there exists a unique couple u, p ∈ V × Q satisfying Au − B T p f,

Theorem 4 . 3 .
If V h and Q h satisfy the inf-sup condition 4.1 , then there exists a unique couple u h , p h ∈ V h , Q h satisfying 4.2 .Remark 4.4.The spaces P 1 /Bulle d −P 1 and P 2 d −P 1 are two examples that satisfy the inf-sup condition 4.1 while P 1 − P 1 does not satisfy 4.1 ; see 15 .

p h with respect to h in both cases 22 AdvancesFigure 1 :
Figure 1:The error E u k with respect to k for each method fixed point: γ 0, hybrid: γ 0.5, Newton: γ 1 .The error E u k obtained with the coarsest mesh is displayed on a and the error E u k obtained with the finest mesh is displayed on b .

Figure 2 :
Figure 2: The errors E u h and E p h with respect to h with θ 2 a and with θ 1.34 b .Regression straight lines have been drawn from the last three points of each set of six recorded errors. θ u h is the unique solution of 4.4 .Remark 4.2.By setting v h u h in 4.4 and by using inequality 3.2 , 2.5 , and Korn's inequality Lemma 3.3 , we can show that the solution u h of problem 4.4 satisfies Numerical solutions are obtained after several successive approximations u h,k , as described in Section 5.Each u h,k corresponds to a unique p h,k .The algorithm is initialised by u h,0 , p h,0 : 0, 0 .Each linearised problem is solved by using the finite element open source code Freefem ; see 24 .As spaces V h and Q h , we opt for the combination P 1 /Bulle d − P 1 that satisfy the inf-sup condition 4.1 and the interpolation properties 4.26 and 4.27 ; see Remark 4.10.Six Delaunay unstructured regular meshes T h of the square Ω 0, 1 2 are generated with various resolutions h.Since the Dirichlet condition u 0 is applied on the whole boundary ∂Ω O h 1/2 if θ 1.34.In both cases, the observed order of convergence for E u is close to one, which is greater or equal to the estimate.It suggests the nonoptimality of estimate 4.28 in the nonregular case, as noticed in 7 for a comparable problem.