Analysis and Finite Element Approximation of a Nonlinear Stationary Stokes Problem Arising in Glaciology
Guillaume Jouvet1,2and Jacques Rappaz2
Academic Editor: Michele Benzi
Received13 Sept 2011
Accepted25 Oct 2011
Published12 Jan 2012
Abstract
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.
1. 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 , 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 finite element approximation with piecewise linear continuous functions in [3]. Inspired by the work of Baranger/Najib [4] and Barrett/Liu [5] on non-Newtonian problems, a priori and a posteriori error estimates were obtained later in [6โ8].
Unlike the first-order approximation, the original Stokes model which is considered in this paper is a saddle point problem for the velocity and the pressure fields. We prove the existence and the uniqueness of a weak solution using an equivalent minimisation problem and an inf-sup stability condition. Next, we establish a priori estimates for a finite element approximation using a quasinorm technique [5]. Eventually, we investigate several successive approximation algorithms to solve the system nonlinearity. In particular, we upgrade by using Newton's method the fixed point algorithm, given in [3] and proved to be convergent in [6, 9].
Boundary conditions describe the basal sliding phenomena that can significantly influence the glacier ice flows. In [3, 6โ8], the first order approximation model was coupled to a Dirichlet condition. However, this approach requires the basal velocity distribution which is unknown. To overcome this difficulty, several sliding lawsโincluding a Coulomb-type lawโwere considered in [10]. In our model, we use a sliding law that results in a nonlinear Dirichlet-Robin boundary condition.
This paper is organised as follows: the physical model is presented in Section 2. We prove the well-posedness of the weak problem in Section 3. In Section 4, we apply a finite element method and establish a priori error estimates. Successive approximation algorithms to solve the system nonlinearity are proposed and proved to be convergent in Section 5. In Section 6, convergence studies are performed to support the results of Sections 4 and 5.
2. The Model
Let us suppose that ice occupies the domain , with or 3. Ice can be considered as an incompressible non-Newtonian fluid with negligible inertial effects [11]. It follows that the velocity and the pressure of ice solve the stationary nonlinear Stokes problem in :
where denotes the rate of strain tensor, the viscosity of ice, and the gravity force. Here above, the viscosity depends on and is defined by the regularised Glen's flow law [11]. More precisely, for a given velocity field , the viscosity satisfies the following nonlinear equation:
where , is a positive parameter, is Glen's exponent, and is a small regularization parameter which prevents infinite viscosity for zero strain ( in the original Glen's law [1]). When , then the viscosity is constant and (2.1) correspond to the classical linear Stokes problem related to a Newtonian fluid. In the framework of glaciology, 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 is thus split into three parts: , , and , referring to circumstances (i), (ii), and (iii), respectively. We assume throughout that is bounded, its boundaries and , are and . We consider the free surface condition:
where is the unit outward normal vector along the boundary of the domain . We apply the nonlinear sliding condition [10, 13, 14]:
where are the orthogonal vectors tangent to the boundary , that is, when and when . Here above, is the sliding coefficient that is given by
where is the Euclidean norm of , is Glen's exponent, is a positive parameter, and is a small parameter which prevents infinite for zero velocity. The no-sliding condition writes
Note that the conditions applied on boundaries , , and are Neumann, Robin-Dirichlet, and Dirichlet conditions, respectively. When (Newtonian flow) and , the problem (2.1) with boundary conditions (2.3), (2.6) has already been widely studied; see, for instance, [15โ17].
3. 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 that will often be used in Sections 3, 4 and 5.
Lemma 3.1. For all , there exists a unique satisfying (2.2). The function is and decreasing. There exist such that:
Proof. The properties of and inequalities (3.1) and (3.2) can be easily deduced from Lemmasโโ1 and 2 of [7]. Inequality (3.3) is obtained by differentiating (2.2) with respect to . Inequalities (3.4) and (3.5) result from inequality (3.1), Lemmaโโ2.1 in [5] and inequality . Details are given in [12].
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 is Glen's exponent. By using (3.2), we have for all . Then, if , we have . By using the trace inequality for all , see [19 page 197], we obtain . Similarly, we can show . Owing to Hรถlder's inequality, the mixed formulation of problem (2.1) with boundary conditions (2.3), (2.4), and (2.6) that consists of finding such that
is meaningful.
Remark 3.2. If , pressure in (3.8) is defined up to a constant. In that case, is replaced by . Moreover, if , then and is constant and if , then the (linear) problem (3.8) is well posed; see, for instance, [15โ17].
The next lemma states the equivalence of norms and on space .
Lemma 3.3 (Korn's inequality). If and if , then there exists a constant such that
for all such that on .
Proof. We apply Corollaryโโ4.1 in [18] ( being the identity matrix) and Lemmaโโ3.1 page 40 in [16].
We consider the divergence-free velocity space:
In , the pressure field vanishes of the variational formulation (3.8). The reduced formulation consists then of finding such that
To transform problem (3.11) into a minimisation problem, we introduce the functional
where
The functional is Gรขteaux differentiable, and its first derivative , at point , in direction , is given by
Clearly, any minimiser of in 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 the continuity of in Lemma 3.5, the strict convexity of in Lemma 3.6, and the coercivity (in the sense of (3.18)) of in Lemma 3.7. The continuity of requires the following result (Lemmaโโ4 in [3])
Lemma 3.4. Let be a measurable set of and , then one has the following inequality:
Lemma 3.5. The functional is -continuous.
Proof. By using (3.2), (2.5), and , we have, for all
These two inequalities together with Lemma 3.4 imply the -continuity of .
Lemma 3.6. The functional is strictly convex on .
Proof. Clearly, and . From (3.3), we have if , and then is strictly convex. Since is an increasing function, is strictly convex. In the same way, we can show that is strictly convex by using (2.5). Let satisfying and . From Korn's inequality (Lemma 3.3), we have in . As a consequence,
The strict convexity of follows from the previous inequality and the convexity of .
Since is convex, satisfies (3.11) if and only if .
Lemma 3.7. There exist two constants such that, for all ,
Proof. Let . From (3.2) and , there exists such that
As a consequence, there exist two constants such that . By using Korn's inequality (Lemma 3.3), there exists such that
From Young's inequality, we have, for all ,
where . We set small enough such that . From inequalities (3.20), (3.21), and , we obtain
which is exactly (3.18) with and .
Theorem 3.8. There exists a unique such that . Moreover, is the unique solution of (3.11).
Proof. Clearly, there exists such that . Lemma 3.7 ensures the existence of . Let be a sequence of such that . There exists an integer such that, for all , we have . Owing to Lemma 3.7, the sequence is bounded in . Since is a closed subspace of , is reflexive. Consequently, there exist and a subsequence of (still denoted ) that converges weakly to in . By Lemmas 3.5 and 3.6, is weakly lower semicontinuous; see, for instance, Corollary III.8 in [19] page 38. Then, we have
and possesses at least one minimum . Since is strictly convex (Lemma 3.6), this minimum is unique. Moreover, is the unique solution of (3.11).
Spaces and are required to satisfy the inf-sup condition, see [5, 20], to ensure the existence and the uniqueness of such that satisfies the mixed formulation (3.8). The inf-sup condition is proved in [15, 21] when (or, equivalently, ). By following the proof of Propositionโโ5.3.2 in [22], we can easily generalise this result when ; see details in [12].
Lemma 3.9. Spaces and satisfy the inf-sup condition; that is, there exists such that
Theorem 3.10. There exists a unique couple satisfying (3.8).
Proof. Although the result is a straightforward application of Theoremโโ2.1 in [20] together with Theorem 3.8 and Lemma 3.9, we give all the arguments of the proof. Let and be the operators defined by
where and are dual to and , respectively. From Theorem 3.8, there exists a unique such that for all , which means that . Owing to the inf-sup condition (4.1), the operator is surjective, , and is closed; see LemmaโโA.40 in [15]. As a consequence, and there exists such that . Since , the pressure is necessarily unique. Eventually, there exists a unique couple satisfying
or equivalently (3.8).
4. Finite Element Approximation and A Priori Estimates
We assume that is a convex polygonal or polyhedral domain and is a regular mesh of parametrized by , the highest diameter of the elements of . We say that and , some finite-dimensional approximation spaces on of and , satisfy the inf-sup condition if, for all , there exists a constant such that
The discrete problem is obtained by replacing the spaces and by and , respectively. It consists of finding such that
The discrete similar space to is
Note that is not necessarily included in . The discrete reduced problem consists of finding such that
Since is a closed subspace of , Theorem 3.8 and the proof can be rewritten by replacing by .
Theorem 4.1. There exists a unique such that . Moreover, is the unique solution of (4.4).
Remark 4.2. By setting in (4.4) and by using inequality (3.2), (2.5), and Korn's inequality (Lemma 3.3), we can show that the solution of problem (4.4) satisfies
where does not depend on .
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.
Theorem 4.3. If and satisfy the inf-sup condition (4.1), then there exists a unique couple satisfying (4.2).
Remark 4.4. The spaces and are two examples that satisfy the inf-sup condition (4.1) while does not satisfy (4.1); see [15].
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 ; 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 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 be the solution of (3.8); the application
is a quasi-norm of ; that is, it satisfies all properties of norms, except homogeneity. Moreover, there exists such that, for all , one has
and there exists such that, for all and for all , one has
Proof. The quasi-norm properties are shown in Lemmaโโ3.1 in [5]. Inequalities (4.7) and (4.8) result from Korn and Hรถlder's inequalities; see details in [12].
By setting in (3.8) it is easy to prove the next lemma.
Lemma 4.6. Let be the solution of problem (3.8) and the solution of problem (4.4), then
holds for all . Moreover, if the spaces and satisfy the condition (4.1), then the solution of (4.2) satisfies
for all .
Theorem 4.7. Let be the solution of (3.8) and the solution of (4.4). For all , one has
where . Moreover, if the spaces and satisfy the condition (4.1), then the solution of (4.2) satisfies, for all and for all ,
where . The constants do not depend on and ; however, increasingly depends on .
Proof. By using, respectively, the definition (4.6) of the quasi-norm , inequality (3.4) with , and (4.9), there exists such that
where . For the sake of simplicity, and are handled separately. By using inequality (3.5) with , there exists such that
By using the inequality (see Lemmaโโ2.2 in [5] orโโ(3.10) in [8]),
with , , , and , we obtain, for all ,
We now use respectively Hรถlder's inequality, Young's inequality, and (4.7); there exist such that
By setting and , we obtain
By moving to the left-hand side, we obtain (4.11). From the inf-sup condition (4.1), we have, for all ,
From (4.10), we have, for all ,
From (4.19), (4.20), and (3.5) with , there exist such that
where . Eventually, the previous inequality together with leads to (4.12).
Theorem 4.8. Let be the solution of (3.8), and the solution of (4.4). For all and for all , assuming , one has
where . Moreover, if the spaces and satisfy the inf-sup condition (4.1), then the solution of (4.2) satisfies for all and for all , assuming ,
where . The constants do not depend on and ; however, and increasingly depends on .
Proof. On one hand, inequality (4.22) follows from inequalities (4.7), (4.11), and (4.8). On the other hand, (4.23) follows from (4.22) and from the following property (see (1.16), page 115 in [16]): for all and for all , there exists such that
where depends on the inf-sup constant . Eventually, (4.24) follows from (4.12), (4.11), and (4.8).
Theorem 4.9. Assume that, for all , there exists a continuous operator that satisfies
and a continuous operator that satisfies
where is the size of the higher diameter of the elements of . Assume that and satisfy the inf-sup condition (4.1). Let be the solution of problem (3.8) and let be the solution of problem (4.2). Assume that , where , then one has
where .
Proof. Apply (4.23) and (4.24) with and . By using the continuity of , (4.5), (4.26), and (4.27), there exist such that
The estimate (4.28) directly follows from (4.29).
Remark 4.10. The combination for spaces and , introduced in [23], satisfies the assumptions of Theorem 4.9; see Lemmaโโ4.20 page 190 of [15] for the inf-sup condition (4.1) and [15, 16] for the interpolation inequalities (4.26) and (4.27).
Remark 4.11. If , a similar analysis can be led by replacing the norm defined by (4.6) by
5. Successive Approximations
In this section, several successive approximation algorithms are proposed for solving the nonlinearity of the discrete problem (4.4) when . For the sake of simplicity, we suppose 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 is fixed, and we assume that the approximation spaces satisfy and . In what follows, denotes an arbitrary norm of . Since is a finite-dimensional space, all norms are equivalent. Let . We define
where solves
The application is well defined. Indeed, by using, respectively, , inequalities (3.3) and (3.2), there exist such that
As a consequence, the problem (5.2) is coercive. From the Lax-Milgram Theorem, see [15] page 83, there exists a unique solution of (5.2).
In what follows, denotes the solution (4.4), which is also the unique fixed point of . Assume that is given; we define iteratively a sequence , for all , by
Our goal is to prove that converges to when goes to the infinity. When , we obtain the classical fixed point method, widely used to solve the nonlinearity of Glen's law; see [6, 9, 14]. When , we have an additional term in (5.2) which corresponds to Newton's method; see Remark 5.5. The case corresponds to a hybrid fixed pointโNewton's method. The convergence of sequence requires several preliminary results. We compute the first derivative of in Lemma 5.1. Lemma 5.2 provides an upper bound of the first derivative. Eventually, Theorem 5.3 states the linear convergence of by using the Banach fixed point theorem. Theorem 5.7 states the second-order convergence when . By differentiating formally (5.2) at point in direction , with , we obtain the following lemma.
Lemma 5.1. Let satisfy . The application is Gรขteaux differentiable at point and its derivative is given by
where solves
The problems (5.2) and (5.6) have the same coercivity properties to compute (resp., ) from (resp., ). As a consequence, the problem (5.6) is well-posed by the Lax-Milgram theorem. To prove the convergence of the sequence , we look for a norm that makes a contraction at point .
Lemma 5.2. Let , and let be the fixed point of . The application satisfies
where is the subordinated norm to .
Proof. Since , then (5.6), with and , is rewriten as
for all . From (5.8), , and (3.3), we have
By setting in (5.9), we obtain
From (5.10) and Cauchy-Schwarz's inequality, we obtain
Eventually, (5.7) follows from the definition of norm .
Theorem 5.3. Let , let be a given mesh of , and let be a norm of . There exist and such that if , then one has
for all , and is linearly convergent to .
Proof. From Lemma 5.2, the spectral radius of is lower than constant:
which is lower than 1. The theorem is then a direct application of the Banach fixed point theorem in .
Remark 5.4. In Theorem 5.3, it should be stressed that depends on . When (i.e., if we replace by ), we cannot ensure Theorem 5.3 to remain true. Nevertheless, in practise, seems to be independent of , see Section 6.
When , we have from Lemma 5.2. It suggests that the convergence of sequence is quadratic. To establish the second-order convergence, we define, for all , the application by
Let . We compute formally the first-order derivative of at point in direction :
and the second-order derivative of at point in direction :
Remark 5.5. If , we have, from the definition of (5.2), of (5.14), and of (5.15),
which highlights Newton's method.
Lemma 5.6. The following inequalities hold, for all :
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 , let be a given mesh of , and let be a norm of . There exist and such that if , then one has
for all , and is quadratically convergent to .
Proof. Owing to Theorem 5.3, there exists such that if , then when . As a consequence, there exists such that for all . By writing the Taylor expansion of at point , there exists such that
Since solves (4.4), then in (5.22). By setting , we obtain, from (5.22) and (5.17),
Thanks to (3.2), there exists such that . As a consequence, since and when and , there exist and such that, for all ,
By applying (5.18) with and and (5.24), we obtain
By applying (5.19) with , , and and (5.25), we obtain:
By combining (5.23), (5.26), and (5.27), we obtain
By using Cauchy-Schwarz's inequality and the equivalence of norms, there exists such that
Clearly, (5.20) follows from (5.29).
Remark 5.8. If , the nonlinear Robin-Dirichlet condition can be handled in the same way as for the viscosity function. In that case, we modify the application by adding
to the left-hand side of (5.2). Theorems 5.3 and 5.7 can be easily extended to this case.
6. Numerical Results
In this section, numerical experiences are performed in two dimensions () 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 is considered in the pure Dirichlet case, that is, . Let
be a divergence-free velocity field that vanishes on the boundaries of . Let be the pressure field such that . The right-hand side term is deduced by computing (2.1). The parameters involving in Glen's law are , bar and โโ. Since , the function defined by (2.2) is given explicitly, when , by
Numerical solutions are obtained after several successive approximations , as described in Section 5. Each corresponds to a unique . The algorithm is initialised by . Each linearised problem is solved by using the finite element open source code Freefem++; see [24]. As spaces and , we opt for the combination 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 of the square are generated with various resolutions . Since the Dirichlet condition is applied on the whole boundary , 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 and has two components:
where is the exact solution of the nonlinear discrete problem. The convergence of the first component with respect to is the concern of Theorem 4.9, while the convergence of the second component with respect to is the concern of Theorems 5.3 and 5.7. Let be an integer large enough such that and can be confused, that is, such that . To check the convergence of the second component, we compute the following error:
where the norm is evaluated by using the trapezoidal rule. For a fixed , Theorem 5.3 states the linear convergence of that depends on constant (5.13) when and the quadratic convergence when . Three values of are considered: to test the fixed point algorithm, to test the hybrid method, and to test Newton's method. Figure 1 displays according to for each method: , and for two different meshes. The recorded orders of convergence are consistent with Theorem 5.3: Newton's method () converges quadratically, the fixed point method and the hybrid method () converge linearly, and convergence is faster for bigger 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 with respect to 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.
(a)
(b)
The estimate of Theorem 4.9 is now tested by computing the following errors:
We can change the regularity of by changing the parameter in (6.1) from 2 to 1.34. Indeed, if , then , while if , then , but , where . In any case . Figure 2 displays and with respect to in both cases and . The estimate (4.28) anticipates , if , and , if . In both cases, the observed order of convergence for and 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.
(a)
(b)
7. 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].
Acknowledgments
This work was funded by the Swiss National Science Foundation, Projects nos. 200021-119721 and PBELP2-133349. The authors thank H. Weaver for the English corrections.
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Copyright ยฉ 2011 Guillaume Jouvet and Jacques Rappaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
