A priori and a posteriori error analysis for a

large-scale ocean circulation finite element

model

J. M. Cascon a,1,2, G. C. Garcıa b,1,3, and R. Rodrıguez b,1

aDepartamento de Matematica Aplicada, Universidad de Salamanca, 37008

Salamanca, Espana

bGI2MA, Departamento de Ingenierıa Matematica, Universidad de Concepcion,

Casilla 160–C, Concepcion, Chile.

Abstract

We consider the finite element solution of the stream function-vorticity formulationfor a large-scale ocean circulation model. First, we study existence and uniquenessof solution for the continuous and discrete problems. Under appropriate regularityassumptions we prove that the stream function can be computed with an error oforder h in H1–seminorm. Second, we introduce and analyze an h–adaptive meshrefinement strategy to reduce the spurious oscillations and poor resolution whicharise when convective terms are dominant. We propose an a posteriori anisotropicerror indicator based on the recovery of the Hessian from the finite element solution,which allows us to obtain well adapted meshes. The numerical experiments show anoptimal order of convergence of the adaptive scheme. Furthermore, this strategy isefficient to eliminate the oscillations around the boundary layer.

Key words: quasi-geostrophic model, stream function-vorticity formulation,anisotropic mesh-refinement

1 Introduction

Different models have been proposed for large-scale horizontal ocean dynamics.Among them, the quasi-geostrophic model is one of the most widely used by

1 This work was partially supported by FONDAP in Applied Mathematics, Chile.2 Partially supported by AECI, Spain.3 Partially supported by CONICYT, Chile.

Preprint submitted to Elsevier Science 5 August 2003

oceanographers to predict wind-stress driven circulation at mid-latitudes; see[26–29] and references therein. To study this model, it is usual to work withthe stream function-vorticity mixed equations [12,26–28].

In the context of the Stokes equation, this formulation and an associated fi-nite element scheme have been introduced by Ciarlet and Raviart in [17]. Sincethen, several authors have studied extensively this subject. A detailed math-ematical analysis can be found in Girault and Raviart’s book [21], which alsoincludes further references. More recently, Amara et. al. [5,6] have introducedand analyzed improved finite element methods to deal with this formulation.

For a large-scale ocean model, a typical phenomenon is the formation of theWestern boundary currents, as in the North Pacific and the North Atlantic.These currents have a typical horizontal scale of about one thousand kilome-ters and are persistent and dominant [12]. Because of them, the solutions ofthese models present boundary layers that, when numerically solved, lead tospurious oscillations and poor resolution. Nevertheless, the accuracy can beremarkably improved if correctly refined meshes are used where the boundarylayer arises. Because of this, mesh-refinement strategies to create well-adaptedmeshes in an automatic manner are particularly useful for the numerical so-lution of the quasi-geostrophic model.

Mesh-refinement strategies are typically based on a posteriori error indicators.This subject has been introduced for finite element methods by Babuska andRheinboldt [10] long time ago. Since then, many different approaches havebeen devised for many different problems. See for instance the monographs byVerfurth [35], Ainsworth and J.T. Oden [1], and Babuska and Strouboulis [11].In particular, for the stream function-vorticity formulation of Stokes problem,an error indicator have been analyzed in [4].

However, most of the existing adaptive techniques do not take care of thepresence of boundary layers. In a boundary layer zone, the solution typicallyhave strong gradients in one direction and almost no variation in the orthog-onal one. Then, in this case, it turns out convenient to use non-shape regularstretched elements aligned with these layers.

Several alternatives have been proposed to create such “anisotropic” meshes.Some of them are based on appropriate anisotropic error indicators (see [8,9,18–20,22,23,31]). In practice, error indicators based on a post-processed Hessianof the computed solution are extensively used (see, for instance, [14,16,2,30]).In particular, Almeida et. al. have introduced in [3] an anisotropic mesh adap-tation process guided by a directional error estimator which is based on therecovery of the second derivatives of the finite element solution.

The goal of this paper is two-fold. On one hand, we analyze a finite element dis-cretization of the stream function-vorticity formulation of a large-scale ocean

2

circulation model. We show that under appropriate regularity assumptionsthis numerical method leads to an optimal order approximation of the veloc-ity, which is the variable of interest.

However, in real problems, when the convective terms in the model are dom-inant, these estimates become meaningless. There is a vast literature dealingwith methods for convection-dominated problems; see [32] for a survey. Thesemethods invoke some form to boost the stability. The streamline diffusion fi-nite element methods and the upwind finite element methods are two such.Frequently, these methods are used on Shishkin-type meshes; i.e., meshes veryfine inside on the layer and coarse otherwise. These meshes yield much betterresults that uniform and adaptively-isotropic refined meshes containing a sim-ilar number of nodes [25], but their construction requires a priori knowledgeabout the behavior of the derivatives of the exact solution. Optimal (or quasi-optimal) convergence results are known for such methods in several norms,assuming that the solution can be decomposed in a regular part and layerterms [24,33].

Our approach, instead, consists in using an a posteriori indicator to locate theboundary layer without any a priori information and to create meshes welladapted to the solution. We introduce a mesh refinement technique relyingon an a posteriori anisotropic error indicator based on the recovery of theHessian from the finite element solution, together with the anisotropic meshgenerator BL2D [13]. We assess the efficiency of our strategy by means of severalnumerical experiments.

The article is organized as follows: In Section 2, we recall the formulation ofthe steady-state linear quasi-geostrophic ocean model and its standard varia-tional formulation. Then, in Section 3, we present its stream function-vorticityformulation and a finite element discretization. A priori error estimates areestablished in Section 4. We also present in this section some numerical ex-periments confirming the theoretical results for sufficiently smooth solutionsand showing the need of correctly refined meshes on the wind-driven modelsubject to realistic values of the physical parameters. In Section 5, we intro-duce error indicators for the L2 norms of the stream function and the velocityfield. Then, we present a mesh adaptation technique. Finally, in Section 6,we report several numerical experiments exhibiting the performance of theproposed strategy.

2 Simplified ocean model

Let Ω be an open bounded and connected, although not necessarily simplyconnected, subset of R

2 with a Lipschitz continuous boundary Γ. We consider

3

the steady-state linear quasi-geostrophic ocean model [12,28] described by thefollowing equations

−AH∆u+ γu+ f(x2)k × u+1

ρ0

∇p =1

ρ0D0

τ in Ω,

div u = 0 in Ω,

u = 0 on Γ,

(2.1)

where u and p are the velocity and the pressure of the fluid for each pointx = (x1, x2) ∈ Ω, τ = (τ1, τ2) ∈ L2(Ω) is the surface wind stress, AH theconstant horizontal coefficient of eddy viscosity, γ the bottom friction, D0 andρ0 the depth and the density of the ocean, respectively. We denote by Γ0 theexterior boundary and Γi, 1 ≤ i ≤ p, the other connected components of Γif any. On the other hand, k × u denotes the 90 rotation of the vector u;namely, k× u = (−u2, u1). The Coriolis parameter f represents the planetaryvorticity of the motion due to the rotation of the Earth.

The β-plane approximation is assumed (see, for instance, [29]). It consists ofsubstituting the Coriolis parameter f by a linear approximation

f = f0 + βx2, f0 = 2ω0 sin θ0, β =2ω0

Rcos θ0,

where ω0 is the angular rotation rate of the Earth (7.24 × 10−5 s−1), and R isthe radius of the Earth (6.371× 106 m). This is a first order approximation tostudy the large-scale ocean dynamics valid at mid-latitudes (20 ≤ θ0 ≤ 50).

We use standard notation for Sobolev spaces, norms and seminorms. Moreover,we introduce the space

V :=v ∈ H1

0 (Ω)2 : div v = 0.

A standard variational formulation of (2.1) is as follows:

Problem 1 Find u ∈ V such that

AH(curlu, curl v) + γ(u, v) + ((f0 + βx2)k × u, v) =1

ρ0D0

(τ, v) ∀v ∈ V.

Here and thereafter (·, ·) denotes the inner product of L2(Ω) or L2(Ω)2, ascorresponds.

4

The arguments of Theorem I.5.1 in [21] for the Stokes problem, can be ex-tended to our case to prove that Problem 1 is equivalent to (2.1), in thesense that u is a solution of Problem 1 if and only if there exists a functionp ∈ L2

0(Ω) such that (u, p) is a weak solution of (2.1). Moreover, both attainunique solutions.

Remark 1 In the case that either Ω is a convex polygon or its boundary Γ is ofclass C2, the arguments of Theorem I.5.4 and Remark I.5.6 in [21] allow us toprove that the unique solution of (2.1) satisfies (u, p) ∈ H2(Ω)2×H1(Ω)∩L2

0(Ω)and

‖u‖2,Ω + ‖p‖1,Ω ≤ C‖τ‖0,Ω,

where C is a constant independent of τ .

3 Stream function-vorticity formulation

3.1 The continuous problem

It is well-know that the divergence-free condition can be expressed by intro-ducing a stream function ψ of u:

u = ~curlψ :=

(∂ψ

∂x2

,− ∂ψ

∂x1

).

Since u vanishes on Γ, ψ must be constant on each of its connected componentsΓi. Moreover, ψ is uniquely determined if we set ψ = 0 on Γ0. Thus the streamfunction belongs to the space

Φ :=φ ∈ H1(Ω) : φ|Γ0

= 0, φ|Γi= ci, 1 ≤ i ≤ p

,

where ci denote arbitrary constants.

Let us introduce the vorticity ω of u:

ω := curlu :=

(∂u2

∂x1

− ∂u1

∂x2

)= −∆ψ. (3.1)

If ω ∈ H1(Ω), by choosing v = ~curlφ in Problem 1 and integrating by parts,we obtain

5

AH( ~curl ω, ~curl φ) + γ(ω, φ) − β

(∂ψ

∂x1

, φ

)=

1

ρ0D0

(τ, ~curlφ) ∀φ ∈ Φ,

where we have also used that f0 is constant.

After scaling the equations by introducing non-dimensional variables, we areled to the following problem:

Problem 2 Find (ψ, ω) ∈ Φ ×H1(Ω) such that

εm( ~curlω, ~curlφ) + εs(ω, φ) −(∂ψ

∂x1

, φ

)= (τ, ~curlφ) ∀φ ∈ Φ,

−(ω, µ) + ( ~curlψ, ~curlµ) = 0 ∀µ ∈ H1(Ω).

The last equation is a weak formulation of (3.1). The parameters εs and εmare the non-dimensional Stommel and Munk numbers, respectively, which aredefined by

εs :=γ

βLand εm :=

AHβL3

,

with L being a typical horizontal length scale of the domain (see [28]).

Notice that a solution of Problem 1 provides a solution of Problem 2 only ifω = curlu is smooth enough, for instance, if u ∈ H2(Ω). The following theoremshows existence and uniqueness of solution of Problem 2 under regularitygeometric constraints.

Theorem 1 Let Ω be either a convex polygon or such that its boundary Γ isof class C2. Then, Problem 2 attains a unique solution (ψ, ω) ∈ Φ ×H1(Ω).

PROOF. Let us consider the following Stokes-like problem: Find (u, p) ∈H1

0 (Ω)2 × L20(Ω) such that

a(u, v) + (p, div v) = (τ, v) ∀v ∈ H1

0 (Ω)2,

(div u, q) = 0 ∀q ∈ L20(Ω),

(3.2)

with

a(u, v) = εm(curlu, curl v) + εs(u, v) + (x2k × u, v).

6

This bilinear form is V -elliptic. In fact, ∀v ∈ V

a(v, v) = εm|v|21,Ω + εs‖v‖20,Ω + (x2k × v, v) ≥ minεm, εs‖v‖2

1,Ω,

since (x2k × v, v) vanishes.

Then, the standard abstract theory (see for instance Theorem I.5.1 in [21])applies to show that (3.2) attains a unique solution. Moreover, under thegeometric assumptions on Ω, (u, p) ∈ H2(Ω)2 ×H1(Ω) and

‖u‖2,Ω + ‖p‖1,Ω ≤ C‖τ‖0,Ω,

with C independent of τ .

In fact, this is a consequence of Theorem I.5.4 and Remark I.5.6 in [21] appliedto the equivalent formulation

εm(curlu, curl v) + (p, div v) = (τ − εs(u, v) − x2k × u, v)

∀v ∈ H10 (Ω)2,

(div u, q) = 0 ∀q ∈ L20(Ω).

Now, the abstract framework in Section III.1.1 and III.2.1 of [21] (in particularTheorem 2.1, 2.2, 2.3 and 2.4) applies to our problem allowing us to prove theequivalence between Problem 2 and (3.2) and, consequently, the theorem. 2

Remark 2 The geometric assumptions on Ω have been used only to ensurethat (u, p) ∈ H2(Ω)2 × H1(Ω). Therefore, if (3.2) has such a smooth solu-tion, we attain the same conclusion regarding existence and uniqueness forProblem 2.

3.2 The discrete problem

Let Th be a regular family of triangulations of Ω, where h = maxT∈ThhT ,

with hT = diam(T ) ∀T ∈ Th. Let Pl(T ) be the space of polynomial functionsof degree at most l defined on the triangle T ; we set

Lh := φh ∈ C(Ω) : φh|T ∈ P1(T ) ∀T ∈ Th

and define

7

Φh := Lh ∩ Φ and Θh := Lh.

The corresponding Galerkin scheme is:

Problem 3 Find (ψh, ωh) ∈ Φh × Θh such that

εm( ~curlωh, ~curlφh) + εs(ωh, φh) −(∂ψh∂x1

, φh

)= (τ, ~curlφh) ∀φh ∈ Φh,

−(ωh, µh) + ( ~curlψh, ~curlµh) = 0 ∀µh ∈ Θh.

The arguments of Section III.2.2 of [21] leading to Lemma III.2.4 and Theo-rem III.2.5 of this reference apply to our case allowing us to prove the followingresult:

Theorem 2 Problem 3 attains a unique solution (ψh, ωh) ∈ Φh × Θh.

4 A priori error estimates

We introduce the following elliptic projection operators:

• Ph : H1(Ω) −→ Θh defined by

εm( ~curl(Phµ− µ), ~curl θh) + εs(Phµ− µ, θh) = 0 ∀θh ∈ Θh;

• P oh : Φ −→ Φh defined by

εm( ~curl(P ohφ− φ), ~curlφh) = 0 ∀φh ∈ Φh.

It is proved in [34] that, when Ω is a convex polygonal domain in R2 and

Th a family of quasi-uniform triangulations, given s and p in R such that0 ≤ s ≤ 1 and 2 ≤ p < ∞, there exists a constant C > 0 such that theprojections satisfy the following error estimates:

‖v − Phv‖0,p,Ω + h|v − Phv|1,p,Ω ≤ Chs+1‖v‖s+1,p,Ω ∀v ∈ W s+1,p(Ω),

‖v − P ohv‖0,p,Ω + h|v − P o

hv|1,p,Ω ≤ Chs+1‖v‖s+1,p,Ω

∀v ∈ W s+1,p(Ω) ∩W 1,p0 (Ω).

Here and thereafter C denotes a generic constant not necessarily the same ateach occurrence but always independent of the mesh size h.

Let

8

Wh : =(φh, θh) ∈ Φh × Θh : ( ~curlφh, ~curlµh) = (θh, µh) ∀µh ∈ Θh

.

We prove the following theorem by adapting the arguments in Theorem III.2.6of [21] to our case.

Theorem 3 Let (ψ, ω) and (ψh, ωh) be the respective solutions of Problems 2and 3. If Ω is a convex polygon, then there exists a constant C > 0 dependingon εm and εs such that the following error estimate holds:

|ψ − ψh|1,Ω + ‖ω − ωh‖0,Ω ≤C

[inf

(φh,θh)∈Wh

(|ψ − φh|1,Ω + ‖ω − θh‖0,Ω)

+ |ψ − P ohψ|1,Ω + ‖ω − Phω‖0,Ω

].

PROOF. Let (φh, θh) ∈Wh. We have

εm( ~curl(ω − ωh), ~curlφh) + εs(ω − ωh, φh) −(∂

∂x1

(ψ − ψh), φh

)= 0.

Then, by using the definition of Ph and the fact that φh ∈ Θh,

εm( ~curl(Phω−ωh), ~curlφh) + εs(Phω − ωh, φh)

−(∂

∂x1

(P ohψ − ψh), φh

)−(∂

∂x1

(ψ − P ohψ), φh

)= 0.

Now, the definition of Wh implies that

εm(Phω − ωh, θh) + εs(Phω − ωh, φh)

−(∂

∂x1

(P ohψ − ψh), φh

)−(∂

∂x1

(ψ − P ohψ), φh

)= 0.

If we add and subtract φh and θh, we obtain

εm(Phω − θh, θh) + εs(Phω − θh, φh) −(∂

∂x1

(P ohψ − φh), φh

)

−(∂

∂x1

(ψ − P ohψ), φh

)

= εm(ωh − θh, θh) + εs(ωh − θh, φh) −(∂

∂x1

(ψh − φh), φh

).

9

This equation is valid in particular for (φh, θh) = (ψh, ωh), since (ψh, ωh) ∈ Wh

because of the second equation of Problem 3. Thus, we obtain

εm(Phω − θh, ωh − θh) + εs(Phω − θh, ψh − φh)

−(∂

∂x1

(P ohψ − φh), ψh − φh

)−(∂

∂x1

(ψ − P ohψ), ψh − φh

)

= εm‖ωh − θh‖2 + εs(ωh − θh, ψh − φh) −(∂

∂x1

(ψh − φh), ψh − φh

).

It is easy to see that the last term vanishes. Then the definition of Wh implies

εs‖ curl v(ψh − φh)‖20,Ω + εm‖ωh − θh‖2

0,Ω

≤ εm‖Phω − θh‖0,Ω‖ωh − θh‖0,Ω

+ (εs‖Phω − θh‖0,Ω + |P ohψ − φh|1,Ω + |ψ − P o

hψ|1,Ω) ‖ψh − φh‖0,Ω.

On the other hand, since ψh − φh = 0 on Γ0, Poincare’s Lemma yields

‖ψh − φh‖0,Ω ≤ C‖ ~curl(ψh − φh)‖0,Ω.

Then, the following inequality is a consequence of the two previous estimatesand straightforward computations:

|ψh − φh|1,Ω + ‖ωh − θh‖0,Ω ≤C (‖Phω − θh‖0,Ω

+|P ohψ − φh|1,Ω + |ψ − P o

hψ|1,Ω) .

Since this is valid ∀(φh, θh) ∈Wh, the triangle inequality allows us to concludethe proof. 2

Finally, the following theorem provides error estimates for our problem.

Theorem 4 Assume that Ω is a bounded, convex polygon and Th a quasi-uniform family of triangulations of Ω. If ψ ∈ H3(Ω), then, for all ε > 0,

|ψ − ψh|1,Ω ≤C(ε)h1−ε‖ψ‖3,Ω,

‖ω − ωh‖0,Ω ≤C(ε)h1/2−ε‖ψ‖3,Ω.

PROOF. The arguments of Section III.3.1 of [21] apply to our case with obvi-ous modifications yielding the second estimate (see in particular Remark III.3.3

10

of this reference). On the other hand, the first estimate follows from a dualityargument similar to that in Theorem III.3.3 of [21] but based on the followingauxiliary problem: Given g ∈ L2(Ω)2, find φg ∈ H1

0 (Ω) and λg ∈ H1(Ω) suchthat

( ~curlλg, ~curlχ) −(∂χ

∂x1

, φg

)= (g, ~curlχ) ∀χ ∈ H1

0 (Ω),

εm( ~curlφg, ~curlµ) + εs(φg, µ) − (λg, µ) = 0 ∀µ ∈ H1(Ω). 2

4.1 Numerical experiments

The goal of our first experiment is to verify the error estimate above for thestream function. Let Ω = [0, 3]× [0, 1], εm = εs = 1 and a right hand side suchthat the exact solution is the smooth function

ψ(x) = sin2 πx1

3sin2 πx2.

We have used several quasi-uniform meshes with different degrees of refine-ment. Fig. 1 shows the error curve of the method which exhibits an O(h) ofconvergence, confirming the theoretical result of Theorem 4.

102

103

104

105

10−2

10−1

100

Numer of nodes (N)

Str

ea

m f

un

ctio

n e

rro

r

|ψ−ψh|1,Ω

C N−1/2

Fig. 1. Error curve for |ψ − ψh|1,Ω on a quasi-uniform mesh. Smooth solution.

In the next experiment we consider the same domain but more realistic valuesfor the Munk and Stommel numbers, taken from Myers and Weaver [28]:

11

εm = 6 × 10−5, εs = 0.05,

which correspond to

γ = 7 × 10−6 s−1, AH = 1.2 × 103m2s−1, L = 106 m, D0 = 800 m.

For the wind stress we use, also as in reference [28],

(τ1, τ2) =(− 1

πcos πx2, 0

).

We have used the uniform mesh shown in Fig. 2. Since εm and εs are small, theconvective term in Problem 2 is dominant and introduces a strong boundarylayer on the left side of the domain. This can be seen in the computed streamlines shown in Fig. 3. This figure also shows a poor resolution of this layer andthat spurious oscillations arise.

Fig. 2. Uniform mesh.

Fig. 3. Stream lines obtained with a uniform mesh.

A typical strategy to avoid these defects is to use non shape-regular stretchedelements aligned with the stream lines in the zone of the boundary layer(see [3]). We have done this with the mesh shown in Fig. 4. Thus, we haveobtained a much better resolution of the boundary layer and avoided spuriousoscillations, as shown in Fig. 5.

The remainder of this work is devoted to design an algorithm for creatingwell-adapted meshes in an automatic manner by using the computed solution

12

Fig. 4. Graded mesh over-refined around the boundary layer.

Fig. 5. Stream lines obtained with the graded mesh.

on coarser triangulations.

5 A mesh-refinement strategy

We propose an h-adaptive mesh-refinement strategy, based on a posteriorierror indicators providing information on the stretching direction and ratiowhere refinement and/or coarsening are needed. Our error indicator is basedon a recovery of second derivatives (Hessian) from the finite element solution(see [3]). We use this information to devise an adaptive process leading tomeshes well adapted to the solution, correctly refined, with stretched andoriented elements aligned with its singularities.

5.1 An anisotropic error indicator

The heuristics behind our indicator is similar to that in [3]. It consists of as-suming that the finite element solution error can be reasonably approximatedby the linear Taylor expansion of the exact solution around the barycenter ofeach element. Then, given T ∈ Th, ∀x ∈ T we have

ψ(x) − ψh(x) ≈ ψ(x) − ψL(x) ≈ 1

2(x− xT )tHψ(xT )(x− xT ), (5.1)

13

where ψ is the exact solution, ψh the finite element solution, ψL the linearTaylor expansion of ψ around the barycenter xT of T , and Hψ denotes theHessian matrix of ψ. Therefore, straightforward computations lead to

‖ψ − ψh‖0,T ≈ 1

2

∫

T

[(x− xT )tHψ(xT )(x− xT )

]2dx

1/2

. (5.2)

Our error indicator consists of computing the last expression by using anapproximate Hessian. This is obtained by means of a post-process of the com-puted solution ψh, based on applying twice a technique to recover the gradient.

First, we smooth ∇ψh. To do this we use the Clement interpolation operatorΠ : L2(Ω) → Lh, which is defined by

Πv :=N∑

i=1

Piv(Qi)ϕi,

where, for i = 1, . . . , N , Qi are the nodes of the triangulation Th, ϕi arethe standard nodal basis functions of Lh, and Pi : L2(Si) → P0(Si) denotesthe orthogonal projection onto the subspace of constant functions defined inSi = suppϕi =

⋃T ∈ Th : T 3 Qi.

Now, we introduce the gradient-recovery operator ∇R : Φh −→ L2h given by

∇Rψh :=

Π∂ψh∂x1

Π∂ψh∂x2

.

Notice that, since ψh is piecewise linear, we have

∇Rψh(Qi) =∑

T⊂Si

|T ||Si|

∇(ψh|T ),

where |T | and |Si| denote the corresponding areas.

Next, we define the symmetric recovered Hessian matrix HS by

HSψh :=HRψh +Ht

Rψh2

∈ L2×2h ,

where

14

HRψh :=

[∇R

(Π∂ψh∂x1

) ∣∣∣∣∣ ∇R

(Π∂ψh∂x2

) ],

namely, the columns of HRψh are the recovered gradients of Π ∂ψh

∂xi.

The numerical evidence reported in [3] shows that HS can be safely used as anapproximation of H in adaptive processes for convection dominated problems.Then, from (5.2), we define the anisotropic error indicator

ηT :=

∫

T

[(x− xT )tHSψh(xT )(x− xT )

]2dx

1/2

(5.3)

and the corresponding global error estimator

η :=

∑

T∈Th

η2T

1/2

.

The numerical experiments in Section 6 below give evidence that this estimatoris equivalent to the error for both, regular and singular solutions.

5.2 The adaptive procedure

Our next goal is to design an adaptive procedure to solve Problem 3 on asequence of meshes up to finally attain a solution with an error within a pre-scribed tolerance. To attain this purpose, the created meshes must be correctlyrefined and must contain stretched elements to take care of the anisotropy ofthe solution.

At each step, a new mesh Th′ better adapted to the solution of Problem 3 mustbe created. This will be done by using information of the solution computedon the “old” mesh Th. Two features of Th′ must be determined: the diameterhT ′ of the elements on different parts of the domain and the stretching andorientation of these elements to correctly solve the boundary layers of thesolution.

To do this we use the recovered Hessian matrix on each element T ∈ Th. Let λ1

and λ2 be the eigenvalues of the symmetric matrix HSψh(xT ), with |λ1| ≥ |λ2|.Let q1 and q2 be the corresponding orthonormal eigenvectors. Then

HSψh(xT ) = QΛQt, (5.4)

15

with Λ = diagλ1, λ2 and Q the orthogonal matrix of columns q1 and q2.Thus, by substituting (5.4) in (5.3), straightforward computations lead to

η2T =

∫

T

2∑

i=1

|λi|[(x− xT )tqi

]22

dx. (5.5)

We define hi := maxx∈T |(x− xT )tqi|, i = 1, 2, which can be seen as a typicallength of the element T in the direction qi (actually, it is proportional to thelength of the projection of T onto the span of qi). We also call x1, x2 andx3 the coordinates of the vertices of T . The following lemmas will be used toobtain a useful equivalence for the local indicator ηT .

Lemma 1 For all w ∈ R2

∫

T

[(x− xT )tw

]4dx =

|T |30

3∑

k=1

[(xk − xT )tw

]4.

PROOF. Let x 7→ Ax+b be a transformation applying the reference triangleT of vertices (0, 0), (1, 0), and (0, 1) onto T . Then we have

∫

T

[(x− xT )tw

]4dx = det(A)

∫

T

[(x− xT )tA−tw

]4dx.

On the reference element the integration rule of the lemma can be verifiedby testing it with adequate fourth order polynomials. Then we conclude theproof by a new change of coordinates. 2

Lemma 2 For all T ∈ Th, there exists αT ∈ [1/30, 2] such that

η2T = αT |T |

2∑

i=1

λ2ih

4i .

PROOF. It is easy to see from (5.5) that

∫

T

2∑

i=1

λ2i

[(x− xT )tqi

]4dx ≤ η2

T ≤ 2∫

T

2∑

i=1

λ2i

[(x− xT )tqi

]4dx.

On one hand it is immediate that

16

∫

T

λ2i

[(x− xT )tqi

]4dx ≤ |T |λ2

ih4i .

On the other hand, from Lemma 1 we have

∫

T

λ2i

[(x− xT )t qi

]4dx =

|T |30

3∑

k=1

λ2i

[(xk − xT )tqi

]4 ≥ 1

30|T |λ2

ih4i ,

since hi = max1≤k≤3 |(xk − xT )tqi|. Thus we conclude the lemma. 2

In what follows we show that if the triangle is correctly oriented, then its areais equivalent to the product of its typical lengths h1h2. In fact, assuming thatthe triangle T is stretched in the direction of q2 (which corresponds to thedirection of smallest variation of the solution) we prove that |T | = cTh1h2

with a constant cT essentially independent of T .

First we prove this result for a conveniently scaled arbitrary triangle as thatin Fig. 6.

Fig. 6. Anisotropic reference element TA

Lemma 3 Let T be a triangle as shown in Fig. 6. If |tan β| ≤ x1/(2x2), then

8

21|TA| ≤ h1h2 ≤ |TA|,

where hi := maxx∈TA

|(x− xTA

)tqi|, i = 1, 2.

17

PROOF. First notice that according to Fig. 6 we have

1

2≤ x2 ≤ 1, 0 ≤ x1 ≤

√3

2,

and the barycenter of the triangle is xTA

= (0, 0). Moreover, the vertical axis

splits TA into two triangles of area 3x1/4. Hence, |TA| = 3x1/2.

Second, the explicit computation of the typical lengths yields

h1 = max|sin β| , |x1 cos β + x2 sin β| , |x1 cos β − (1 − x2) sin β|.h2 = maxcos β, |(1 − x2) cos β + x1 sin β| , |x2 cos β − x1 sin β|,

Then, we have

h1 ≤ max|sin β| , x1 cos β + x2 |sin β| , x1 cos β + (1 − x2) |sin β|.

Now, since x2 ≥ 1/2, for |tan β| ≤ x1/(2x2) we obtain

|sin β| ≤ x1 cos β,

x2 |sin β| ≤x1

2cos β =⇒ x1 cos β + x2 |sin β| ≤

3x1

2cos β,

(1 − x2) |sin β| ≤ x2 |sin β| ≤x1

2cos β

=⇒ x1 cos β + (1 − x2) |sin β| ≤3x1

2cos β.

Hence

h1 ≤3x1

2cos β ≤ 3x1

2= |TA|.

On the other hand

h2 = maxx∈TA

|xtq2| ≤ maxx∈TA

|x| ≤ 1,

and thus

h2h1 ≤ |TA|.

18

For the other inequality notice that, independently of the sign of β, eitherx2 sin β or −(1−x2) sin β must be non negative. Then, the explicit expressionof h1 above yields

h1 ≥ x1 cos β.

On the other hand, clearly

h2 ≥ cos β.

Hence, since x1 ≤√

3/2 and x2 ≥ 1/2, for |tan β| ≤ x1/(2x2) we obtain

h2h1 ≥ x1 cos2 β =x1

1 + tan2 β≥ x1

1 + x21/(4x

22)

≥ 4

7x1 =

8

21|TA|.

Thus we conclude the proof. 2

Notice that any triangle T ∈ Th is the image by a scaling followed by arigid motion of a triangle TA as in Fig. 6. Indeed by choosing orthogonalcoordinates centered at the barycenter of T with the vertical axis coincidingwith the longest median of the triangle, a scaling map applies T onto a trianglelike TA. Then, since the properties in Lemma 3 are rigid-motion and scaleinvariant, they are also valid for T ∈ Th. On the other hand, the constrainttan β ≤ x1/(2x2) is fulfilled whenever the triangle is reasonably stretched inthe direction q2. Then, in this case, we have

|T | = cTh1h2, with cT ∈ [8/21, 1]. (5.6)

An adaptive algorithm to minimize the error indicator (5.5) should choose thetypical lengths h′i of each element T ′ in the new mesh Th′ in order to equilibratethe two terms in this sum; namely, such that

|λ1|h′21 ≈ |λ2|h

′22 . (5.7)

Actually, the values λ1 and λ2 correspond to the “old” mesh Th. However,they are approximations of the eigenvalues of the Hessian of the exact solutionHψ(xT ). Then, it is reasonable to ask for |λ1|h′2

1 ≈ |λ2|h′22 .

Thus, it only remains to decide how to choose one of the diameters (for exampleh′1) to compute a solution with estimated error within a prescribed tolerancetol.

19

If we assume that the elements T of the mesh Th are correctly stretched as tosatisfy |λ1|h2

1 ≈ |λ2|h22, then from Lemma 2 we have

η2T ≈ 2αTλ

21|T |h4

1.

Moreover, under the assumption of Lemma 3, from (5.6) there holds

|T | = cTh1h2 ≈ cT

(|λ1||λ2|

)1/2

h21.

Then

η2T ≈ 2αT cT

|λ1|5/2|λ2|1/2

h61 = CTh

61. (5.8)

Let us remark that the constant CT := 2αT cT |λ1|5/2/|λ2|1/2 depends on theHessian of the solution on T . Thus it depends on the localization of the elementT in Ω, but neither on h1 nor on the triangle shape ratio.

Therefore, if the new mesh Th′ is created as to satisfy (5.7), then we also have

η2T ′ ≈ CT ′h′61 ≈ CTh

′61 , (5.9)

for any element T ′ ∈ Th′ located in the same zone of Ω as T .

The new mesh will be created such that

η′ =

∑

T ′∈Th′

η2T ′

1/2

≈ tol,

with ηT ′ being approximately the same for all elements T ′ ∈ Th′ (the latter istypically the alternative yielding the mesh with a least number of elements).Therefore

η2T ′ ≈ tol

2

N ′,

with N ′ being the number of elements T ′ ∈ Th′ . Hence, by using (5.9) and(5.8), we have that h′1 must be chosen as follows:

h′1 ≈(

1

CT

tol2

N ′

)1/6

≈(h6

1

η2T

tol2

N ′

)1/6

≈ h1

(tol

ηT√N ′

)1/3

. (5.10)

20

This value of h′1 depends on the unknown number of elements N ′ of Th′ .However this number appears as a scale factor affecting in the same wayall the estimated h′1. Therefore, different values of N ′ will produce differentdegrees of refinement but with the same refinement pattern. (i.e., the relativesizes of the elements remain equal). Because of this, it is not so important toknow the value of N ′ in advance. In the experiment reported below, we havejust considered N ′ = N .

In practice a quasi-uniform isotropic mesh is initially used. In such case itis not convenient to prescribe a very small tolerance from the beginning. Abetter strategy consists of using a set of diminishing tolerances up to attainingthe prescribed one. Then, at each step, a new more stringent tolerance is used.

The adaptive strategy consists of obtaining an approximate solution from aninitial isotropic mesh and to recover the corresponding second derivatives ateach node. Then, we introduce this information to the mesh generator.

For our experiments we have used the mesh generator BL2D (see [13]). Theinformation that must be transmitted to this software to create a new meshis a metric at each vertex. In our case the transmitted metrics are obtainedby averaging element metrics on all the triangles sharing each vertex. Theseelement metrics are given by

Q

1/h′1 0

0 1/h′2

Qt,

where, according to (5.10) and (5.7),

h′1 = h1

(tol

ηT√N ′

)1/3

and h′2 =

√√√√ |λ2||λ1|

h′1,

and λ1, λ2, and Q are given by (5.4).

Let us remark that we have chosen to generate a new mesh at each stepinstead of refining the previous one. This choice looks much simpler and avoidseventual constraints imposed by an initial coarse isotropic mesh. On the otherhand, in all the numerical tests reported in Section 6, the elapsed time togenerate each new mesh has been always negligible compared to the timeneeded to solve the corresponding linear systems.

21

5.3 An error estimator for the velocity field

Since the variable of interest is usually the velocity field u = ~curlψ, in whatfollows we introduce an estimator for the computed velocities uh = ~curlψh.

Under the same assumptions of Section 5.2, we have from (5.1)

‖u− uh‖0,T = |ψ − ψh|1,T ≈∫

T

|Hψ(xT )(x− xT )|2 dx

1/2

,

(see also [15]). Then, we obtain the following local estimator:

ηT :=

∫

T

(x− xT )t[HRψh(xT )]t[HRψh(xT )](x− xT ) dx

1/2

.

In the experiments below we show that the local effectivity indices defined byηT/|ψ − ψh|1,T are very close to one and the estimated global error

η :=

∑

T∈Th

η2T

1/2

attains an optimal order of convergence, too.

Remark 3 In spite of the fact that ηT seems to be an excellent estimator ofthe error in H1

0 (Ω) norm, we have chosen ηT to create the meshes. Indeed, wehave tried also ηT for the adaptive procedure, but the obtained results were notso reliable, very likely because of limitations of the mesh generator.

6 Numerical results

In this section we present several numerical experiments. First, to test ourstrategy, we consider a problem with a known smooth exact solution. In thesecond example we simulate the typical Western boundary currents by usingan analytically known exponential boundary layer solution. In the third one,we consider a more realistic example with the wind stress taken from [28].Finally, in the last one, we repeat this experiment in a non-convex domain.

In all the tests we have taken the following realistic physical parameters:

22

εm = 6 × 10−5, εs = 0.05.

We have initiated always the adaptivity process with a coarse isotropic meshand a rough tolerance, which has been reduced to one half at each step.

6.1 Test 1

We take a problem with the same solution as in Section 4.1,

ψ(x) = sin2 πx1

3sin2 πx2,

but with the realistic physical parameters εm and εs indicated above. Thegeometry of the domain is shown in Fig. 7. This figure shows an intermediateobtained mesh and the corresponding stream lines solution. Notice that ourstrategy generate isotropic meshes as expected.

0 1 2 3

0.5

1

0 1 2 3

0.5

1

Fig. 7. Test 1; iteration 2: 1298 nodes mesh. Computed stream lines.

Fig. 8 shows that both, the exact and estimated global errors, attain optimalorders of convergence in L2(Ω) and H1

0 (Ω) norms in terms of the number Nof nodes:

η ≈ ‖ψ − ψh‖0,Ω = O(N−1), η ≈ |ψ − ψh|1,Ω = O(N−1/2).

23

102

103

104

10−4

10−3

10−2

10−1

Number of nodes (N)

Estimated error L2

Exact error L2

C N−1

102

103

104

10−2

10−1

100

Number of nodes (N)

Estimated error H1

Exact error H1

C N−1/2

Fig. 8. Test 1. Estimated and exact errors versus number of nodes (log-log scale).

Finally, Fig. 9 shows that the effectivity indices for ηT range between 0.4 and1 for almost all the elements.

0 1 2 3

x 10−3

0

1

2

3

x 10−3

Estimator

Lo

ca

l e

rro

r

Fig. 9. Test 1. Local estimator ηT versus local error |ψ−ψh|1,T ∀T ∈ Th (6878 nodesmesh). The slope of the solid lines are 0.4 and 1.

6.2 Test 2

We consider a problem with exact solution

24

ψ(x) =[(

1 − x1

d

) (1 − e−cx1

)sinπx2

]2.

For our experiment we have taken d = 3 and c = 20. Then the function ψexhibits a boundary layer on the left.

Fig. 10 shows an intermediate refined mesh and the corresponding stream linessolution. Let us remark that our refinement strategy recognizes the boundarylayer location and allows avoiding numerical spurious oscillations.

0 1 2 3

0.5

1

0 1 2 3

0.5

1

Fig. 10. Test 2; iteration 4: 3267 nodes mesh. Computed stream lines.

Fig. 11 shows that optimal orders of convergence are attained for estimatedand exact errors in both norms, again.

Finally, Fig. 12 shows that, in spite of the boundary layer, the effectivityindices are close to one, too.

6.3 Test 3

In this example we consider the wind stress used by Myers and Weaver (see[28]):

(τ1, τ2) =(− 1

πcos πx2, 0

).

25

102

103

104

105

10−5

10−4

10−3

10−2

10−1

Number of nodes (N)

Estimated Error L2

Exact error L2

C N−1

102

103

104

105

10−2

10−1

100

Number of nodes (N)

Estimated error H1

Exact error H1

C N−1/2

Fig. 11. Test 2. Estimated and exact errors versus number of nodes (log-log scale).

0 0.005 0.01 0.015 0.020

0.002

0.004

0.006

0.008

0.01

Estimator

Lo

ca

l e

rro

r

Fig. 12. Test 2. Local estimator ηT versus local error |ψ − ψh|1,T ∀T ∈ Th (7482nodes mesh). The slope of the solid lines are 0.3 and 1.

Fig. 13 shows a zoom of the mesh at an intermediate iteration and the corre-sponding stream lines solution. The anisotropic nature of the adaptive meshcan be clearly observed from the zoom.

In this case there is no analytical solution available. However, Fig. 14 showsthat the estimated errors attain optimal orders of convergence.

26

0 0.1 0.20.4

0.5

0.6

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Fig. 13. Test 3; iteration 6: zoom of a 9103 nodes mesh. Computed stream lines.

102

103

104

105

10−4

10−3

10−2

10−1

100

Number of nodes(N)

Estimated error L2

CN−1

102

103

104

105

10−2

10−1

100

101

Number of nodes (N)

Estimated error H1

C N−1/2

Fig. 14. Test 3. Estimated error versus number of nodes (log-log scale).

6.4 Test 4

We consider the non convex domain shown in Fig. 15. This figure shows anintermediate refined mesh and the corresponding stream lines solution.

27

0 1 2 3

0.5

1

0 1 2 3

0.5

1

Fig. 15. Test 4; iteration 4: 1522 nodes mesh. Computed stream lines.

Fig. 16 shows that, for both norms, the estimated errors attain once moreoptimal orders of convergence.

102

103

104

105

10−4

10−3

10−2

10−1

Number of nodes (N)

Estimated Error L2

C N−1

102

103

104

105

10−2

10−1

100

101

Number of nodes (N)

Estimated error H1

C N−1/2

Fig. 16. Test 4. Estimated error versus number of the nodes (log-log scale).

Let us remark that although the theory in Section 4 does not cover this case,the experimental results show that the method combined with the proposedadaptive strategy behaves as well as for convex domains.

28

7 Conclusions

A finite element method to numerically solve the stream function-vorticityformulation of the quasi-geostrophic model has been analyzed. A priori errorestimates with constants depending on the physical parameters have beenproved under appropriate regularity assumptions. Thus, results already knownfor the two-dimensional Stokes problem have been extended to this model.

For large-scale ocean dynamics, these a priori estimates become meaningless.Indeed, the Coriolis convective term dominates these equations and Western-current boundary layers appear. Then, well-adapted meshes become necessaryfor the method to work, avoiding spurious oscillations. An adaptive procedureto create such meshes in an automatic fashion is introduced. This strategyrelies on an anisotropic error indicator based on a recovered Hessian. Sev-eral numerical experiments allow assessing the efficiency of this approach. Inparticular, optimal orders of convergence are attained in all the experiments.

Acknowledgements

The authors want to thank Claudio Padra by helpful discussions and sugges-tions.

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