on adaptive anisotropic mesh optimisation for convection...

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Technische Universität DresdenHerausgeber: Der Rektor

On adaptive anisotropic mesh optimisation for convection-diffusion problems

Petr Knobloch and René Schneider

MATH-NM-06-2012

27/09/2012

On adaptive anisotropic mesh optimisation forconvection-diffusion problems

Petr Knobloch and René Schneider

28th September 2012

Abstract

Numerical solution of convection-dominated problems requires the use of layer-adapted aniso-tropic meshes. Since a priori construction of such meshes is difficult for complex problems, it isproposed to generate them in an adaptive way by moving the node positions in the mesh such that ana posteriori error estimator of the overall error of the approximate solution is reduced. This approachis formulated for a SUPG finite element discretisation of a stationary convection-diffusion problemdefined in a two-dimensional polygonal domain. The optimisation procedure is based on the discreteadjoint technique and a SQP method using the BFGS update. The optimisation of node positions isapplied to a coarse grid only and the resulting anisotropic mesh is then refined by standard adaptivered-greed refinement. Four error estimators based on the solution of local Dirichlet problems aretested and it is demonstrated that an L2 norm based error estimator is the most robust one. Theefficiency of the proposed approach is demonstrated on several model problems whose solutionscontain typical boundary and interior layers.

1 IntroductionWe consider the stationary convection-diffusion problem

−ε∆u+ b · ∇u = f in Ω ,

u = ub on ΓD ,

ε∂u

∂n= g on ΓN

(1)

in a polygonal domain Ω ⊂ R2 with the boundary Γ = ΓD ∪ΓN , where ΓD and ΓN are disjoint andthe one-dimensional measure of ΓD is positive. We denote by n the outward unit normal vector to Γ.We assume that ε is a positive constant, b ∈ W 1,∞(Ω)2, divb ≤ 0, f ∈ L2(Ω), ub ∈ H1/2(Γ) andg ∈ L2(ΓN ). Moreover, we assume that the inflow boundary x ∈ Γ : b(x) · n(x) < 0 is a subsetof ΓD. Then the problem (1) has a unique solution in H1(Ω).

Of special interest is the singularly perturbed case 0 < ε ‖b‖L∞(Ω)2 , i.e., the convection-dominated case. In this case boundary and interior layers may appear, meaning narrow regionsof the domain across which the solution changes dramatically. Boundary layers occur where the

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solution of the transport equation b · ∇u = f , to which the PDE from (1) reduces in most of Ωdue to the dominating convection part, conflicts with the boundary conditions, thus the diffusionpart of the equation becomes active to conform with the boundary conditions. One distinguishesexponential boundary layers at the outflow Dirichlet boundary and characteristic layers along partsof the boundary that are aligned to the flow direction b. Interior layers appear where discontinuitiesare transported in the domain, e.g. discontinuous Dirichlet data on an inflow boundary or at pointswhere flow from different directions arrives. They may also be generated by the right-hand side f .

The convection-diffusion equation is of importance of its own right, for example as a model ofthe transport of chemicals in flow fields, but also as a simplified model for effects that occur in moregeneral PDEs, e.g., in the Navier-Stokes equations.

To capture the layer phenomena accurately by numerical methods is extremely challenging,as they typically occur on length scales which are prohibitive to be resolved by standard uniformmeshes. As a remedy layer-adapted anisotropic meshes of different types have been developed, seee.g. [12] for an overview. While these meshes are fairly simple to create and very successful inovercoming the problems associated with the layer phenomena, they rely on a priori analysis of theproblem, which may be difficult for more complex PDEs. On the other hand standard adaptive meshrefinement techniques based on a posteriori error estimators and subdivision of mesh cells generallylead to locally isotropic meshes and unnecessary overrefinement along the layer, thus rendering thisapproach inefficient compared to the anisotropic meshes based on a priori analysis.

In [13, Chapter 3] it was proposed to generate finite element meshes with suitable anisotropyby moving the node positions in the mesh such that an a posteriori error estimator for a quantity ofinterest is reduced. This technique was applied to the solution of reaction–diffusion problems andthe a posteriori error estimation was based on the dual weighted residual method. The aim of thepresent paper is to apply the general idea of [13] to problem (1) and to adapt the mesh in such away that an a posteriori estimator of the overall error of the approximate solution is reduced. Forthis approach to be successful, the a posteriori error estimator has to meet certain requirements.Error estimators which are based on reducing the properties of an element of the triangulation to itsdiameter (e.g., the standard residual type error estimator [1, Section 2.2]) are generally not suitableto exploit the anisotropy of the solution. Extended versions of this approach define a longest and ashortest directional size of an element, e.g. [9], to allow utilisation of the anisotropy. Unfortunately,the definition of the length scales of the elements in [9] and related works is based on choosingthe edge of maximum length, and is thus not continuously differentiable with respect to the nodepositions. This renders this whole class of error estimators inappropriate for the optimisation due tothe requirements of the optimisation techniques in [13]. On the other hand error estimators basedon the solution of local problems (see, e.g., [1, Section 3]) appear to take account of the elementgeometry in a very natural way, and thus seem to be preferable for this type of mesh optimisation.

Therefore, in the present paper, estimators based on the solution of local problems will be consid-ered. We employ local problems proposed in [18], however, since the resulting error estimator of [18]was analysed for shape-regular meshes only, we consider also variants of this estimator that differ inthe norm used to measure the solutions of the local problems. In the context of the present paper, acrucial property of error estimators is the robustness with respect to mesh deformations, as the esti-mator alone is used to guide the movement of mesh nodes to reduce the error and this is performedpreferably on relatively coarse meshes to achieve optimal efficiency. Numerical tests presented inthis paper demonstrate that many of the estimators guide the node movement in the opposite of thedesired direction. It was found that if only the L2 norm (a weaker norm than in the other estimators)

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of the solution of the local problems is used in the error estimator, then the directional informationbecomes more reliable, and indeed appears to be sufficient for the optimisation approach.

For clarity of the presentation, the problem (1), its discretisation and the error estimators areformulated in a simple form corresponding to the setting of numerical experiments presented in thispaper. Nevertheless, the paper can be easily generalised in various ways: to the three-dimensionalcase, to convection–diffusion–reaction problems, or to higher order finite elements. Since such gen-eralisations are really straightforward, we decided to omit them.

The underlying basic techniques are discussed in the first three sections of this paper, the finiteelement discretisation of (1) in Section 2, a class of corresponding a-posteriori error estimators inSection 3, mesh adaptation by optimisation of node positions in Section 4. Thereafter, in Section 5,suitability of the estimators for the optimisation of node positions is investigated. Finally the refine-ment is tested in numerical experiments in Section 6, before the paper is closed with conclusions inSection 7.

2 Finite element discretisationDenoting

V := v ∈ H1(Ω) : v = 0 on ΓD ,

the standard variational formulation of the convection-diffusion problem (1) reads: Find u ∈ H1(Ω)such that u = ub on ΓD and

a(u, v) = (f, v) + (g, v)ΓN∀ v ∈ V , (2)

where (., .) is the inner product in L2(Ω) or L2(Ω)2, (., .)ΓNis the inner product in L2(ΓN ) and

a(u, v) := ε (∇u,∇v) + (b · ∇u, v) .

Since a(v, v) ≥ ε |v|21,Ω for any v ∈ V , it follows from the Lax–Milgram lemma that the variationalformulation (2) has a unique solution.

We denote by Th a triangulation of Ω consisting of open triangles possessing the usual compati-bility properties and approximate the space H1(Ω) by the finite element space

Wh := vh ∈ C(Ω) : vh|K ∈ P1(K) ∀ K ∈ Th ,

where P1(K) is the space of linear functions on K. We denote by ubh ∈ Wh a function whosetrace approximates ub. Then a Galerkin finite element discretisation of the problem (1) reads: Finduh ∈Wh such that uh = ubh on ΓD and

a(uh, vh) = (f, vh) + (g, vh)ΓN∀ vh ∈ Vh := Wh ∩ V . (3)

Again, this problem is uniquely solvable.It is well known (cf., e.g., [11]) that the Galerkin discretisation of (1) provides solutions polluted

by spurious oscillations in the singularly perturbed case. A possible remedy is to add a stabilisingterm to (3). One of the most popular stabilised methods for convection-dominated problems is the

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SUPG method [2] which combines good stability properties with a high accuracy outside layers. Inour case, it can be formulated in the form: Find uh ∈Wh such that uh = ubh on ΓD and

a(uh, vh) + (b · ∇uh, δ b · ∇vh)

= (f, vh + δ b · ∇vh) + (g, vh)ΓN∀ vh ∈ Vh , (4)

where δ ∈ L∞(Ω) is a nonnegative stabilisation parameter. It is obvious that the problem (4) alsohas a unique solution.

The choice of δ strongly influences the quality of the approximate solution, but an optimal wayof choosing δ is not known, see the discussion in [7]. In this paper δ will be defined on each triangleK ∈ Th by the constant value

δ|K =hK

2 ‖b‖0,∞,K

(coth PeK −

1

PeK

)with PeK =

hK ‖b‖0,∞,K

2 ε, (5)

where hK is a characteristic dimension of K (a local length scale) and PeK is the local Pécletnumber that determines whether the problem is locally (i.e., within a particular element) convectiondominated or diffusion dominated. The formula (5) is a generalisation of a formula derived in theone–dimensional case [3] to obtain a nodally exact solution for constant data and a uniform divisionof Ω.

Let bK be the value of b at the barycentre of K. If bK 6= 0, the parameter hK is often definedas the diameter of K in the direction of bK , i.e., by the value

diam(K,bK) := sup|x− y| : x,y ∈ K, x− y = αbK , α ∈ R .

Is is easy to verify that

diam(K,bK) =2 |bK |

|bK · ∇ϕ1|+ |bK · ∇ϕ2|+ |bK · ∇ϕ3|,

whereϕ1, ϕ2, ϕ3 are the usual nodal basis functions of P1(K). However, since the mesh optimisationconsidered in this paper requires that hK is continuously differentiable with respect to node positions,we set

hK :=

(2 |bK |2

|bK · ∇ϕ1|2 + |bK · ∇ϕ2|2 + |bK · ∇ϕ3|2

)1/2

. (6)

Then √2

2diam(K,bK) ≤ hK ≤

√3

2diam(K,bK)

and hK = diam(K,bK) if bK is aligned with an edge of K. Thus, since also the choice hK =diam(K,bK) is based on rather heuristic arguments, the formula (6) seems to be still reasonable.Note that we set hK := diam(K) if bK = 0.

Using the techniques of [11], an a priori error estimator of the error of the approximate solutionobtained using the discretisation (4) can be derived. Such an estimator is important from the theoreti-cal point of view but is of little use in practice since it depends on the unknown solution u. Therefore,in the next section a posteriori error estimators will be described.

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3 A posteriori error estimatorsIn this section we present a posteriori error estimators for the solution of (4) based on the solution oflocal problems proposed in [18]. We denote by Eh the set of the edges of Th and, for any E ∈ Ehand K ∈ Th, we set

ωE := int⋃

K∈Th, E⊂∂K

K , ωK := int⋃

K′∈Th, ∂K∩∂K′∈Eh

K ′ ,

where “int” denotes the interior of the respective set and ∂K is the boundary of K. For any edgeE ∈ Eh, we define an operator PE : C(E) → C(ωE) in the following way. Let K ⊂ ωE andlet E′ ∈ Eh \ E be an edge of K such that the order E′, E corresponds to the counterclockwisedirection with respect to the boundary of K. Then, for v ∈ C(E), we require that PE v = v on Eand PE v is constant in K along lines parallel to E′. For any E ∈ Eh, we denote by ψE ∈ H1(Ω)a piecewise quadratic function with suppψE = ωE that vanishes on (∂ωE) \ E and equals 1 at themidpoint of E. For any K ∈ Th, we denote by ψK ∈ H1(Ω) a function with suppψK = K that iscubic in K, vanishes on ∂K and equals 1 at the barycentre of K. Finally, for any K ∈ Th, we set

VK := spanψK′ v, ψE PE σ : K ′ ⊂ ωK , E ⊂ ∂K \ ΓD, v ∈ P1(K ′), σ ∈ P1(E)

and denote by vK ∈ VK the uniquely determined function satisfying the local problem

ε (∇vK ,∇w)ωK+ (b · ∇vK , w)ωK

= (fh, w)ωK+ (gh, w)∂K∩ΓN

− ε (∇uh,∇w)ωK− (bh · ∇uh, w)ωK

∀ w ∈ VK ,

where (·, ·)ωKdenotes the inner product in L2(ωK) or L2(ωK)2, (·, ·)∂K∩ΓN

denotes the inner prod-uct in L2(∂K ∩ ΓN ), uh is the solution of (4), and fh, gh, and bh are orthogonal L2 projections off , g, and b onto the spaces of piecewise constant functions corresponding to Th or Eh.

The function uK := vK + uh|ωKis a finite element solution of the local problem

−ε∆u+ b · ∇u = fh in ωK ,

u = uh on ∂ωK \ (∂K ∩ ΓN ) ,

ε∂u

∂n∂K= gh on ∂K ∩ ΓN

in the space VK ⊕ (Vh|ωK). Therefore, the difference uK − uh|ωK

= vK can be expected to providesome information on the error (u− uh)|ωK

. We introduce the following error indicators:

ηK,1 := ‖vK‖20,ωK, (L2-NORM)

ηK,2 := ε |vK |21,ωK, (H10-NORM)

ηK,3 :=|K|ε‖πVK

(b · ∇vK)‖20,ωK, (CONV-NORM)

ηK,4 := ε |vK |21,ωK+|K|ε‖πVK

(b · ∇vK)‖20,ωK, (H10-CONV-NORM)

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where πVKis the orthogonal L2 projection of L2(ωK) onto VK . Now, we define the error estimators

ηi :=

( ∑K∈Th

ηK,i

)1/2

, i = 1, 2, 3, 4 . (7)

For shape-regular meshes, robust lower and upper bounds were obtained for the estimator η4 in[18]. Nevertheless, it is interesting and important to study computationally also the properties of thequantities η1, η2, η3, in particular, for anisotropic meshes.

4 Optimisation of node positionsWe use the opt-adapt algorithm developed in [13–15] to achieve anisotropic refinement by opti-misation of the node positions of a suitable coarse mesh, keeping the connectivity of the mesh fixed,combining this with standard isotropic refinement of the optimised coarse mesh.

As performance function for the minimisation procedure one of the error estimators (7) is used,J := η2

i . A discussion which error estimator is suitable will follow in Section 5. Regardless which ofthe four variants is used, it is obvious that J depends on the finite element solution uh and the nodepositions in the mesh s, J = J(uh, s).

Of course it is necessary to place some constraints on this minimisation to guarantee that theresulting mesh is appropriate for finite element calculations. Here we use the same constraints as in[13], summarised in the following statement of the optimisation problem.

Problem 1. Minimise J(uh, s), with respect to the node positions s, subject to:

1. uh is the solution of the FE discretisation (4) on the mesh with node positions s,

2. the boundary of the mesh matches the boundary of the domain Ω,

3. the mesh is non-self-overlapping,

4. interior angles of the triangles stay bounded well below π,

5. the aspect ratio of triangles varies smoothly (i.e. changes in the aspect ratio of neighbouringelements must be bounded).

Constraints 1 to 4 are obvious or standard geometric restrictions, while condition 5 is motivatedby [13], where it was found to improve reliability of a (different) error estimator. The precise defini-tion and treatment of constraints 2–5 is as in [13, Section 3.3.5].

Overall Problem 1 comprises a highly nonlinear optimisation problem, with nonlinear constraints,and especially the discretised PDE (4) posing a challenge.

For the purpose of anisotropic mesh refinement it is not necessary to compute a solution of thisoptimisation problem to high accuracy, if such a solution even exists. The practical algorithm limitsthe effort spent in the optimisation procedure, allowing only a fixed number of iterations. The mainpurpose of the optimisation is to introduce suitable anisotropy into the mesh. After this optimisationof the coarse mesh, standard isotropic adaptive refinement (red-green refinement) is used to furtherreduce the error to achieve the desired accuracy. However, as the numerical experiments in Section 6and our previous work demonstrate, the anisotropy introduced by the optimisation of the coarse meshallows a far more accurate solution than it is possible to obtain by applying red-green refinementalone at comparable computational cost, if the solution shows anisotropic behaviour.

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move nodes

adaptive isotropic refinement

loop

solve

estimate

refine mesh

define initial mesh (coarse)

adaptive solver

adaption by node movement

optimisation method loop

solve adjoint

solve

estimate

evaluate derivatives

Figure 1: Overall adaption strategy opt-adapt

The overall refinement algorithm is summarised in Figure 1, which is adapted from [13], wheredifferent combinations of the optimisation with mesh refinement were also considered, but this ap-proach performed best.

In the following two subsections we briefly summarise the main components of the optimisationpart of this algorithm. A more detailed discussion can be found in [13].

4.1 Discrete Adjoint Technique, Constraint 1The most crucial part to treat this optimisation Problem 1 in an efficient manner is Constraint 1, thediscretised PDE (4). This is indeed the key to make this whole approach feasible. We treat thisby considering the reduced performance function, where the solution operator of (4) is inserted intothe performance function. The discrete adjoint technique [6, 16] allows very efficient evaluation ofthe gradient of this reduced performance function, thus providing everything that is required for anefficient numerical optimisation solver (SQP-type) for this reduced problem.

9

We briefly state the ingredients of the discrete adjoint technique. Consider the discretised PDE (4)expressed as equation system,

R(u(s), s) = 0, (8)

where u is the vector of finite element coefficients and R(·, ·) = 0 represents the FE equations on themesh defined by the vector s containing all coordinates of the nodes of the mesh. Let J(u(s), s) be ascalar valued function which depends upon u(s), in our case this will be the error estimator (7). Thereduced performance function then is

I(s) := J(u(s), s), (9)

so long as (8) holds.The discrete adjoint technique is to compute the gradient DI

Ds of I(s) by solving an adjoint equa-tion system [

∂R

∂u

]TΨ =

[∂J

∂u

]T(10)

for Ψ, which is of the same dimension as u. Note that ∂R∂u is the stiffness matrix of (4). Once the

so-called adjoint solution vector Ψ is computed, the whole gradient DIDs can be evaluated by

DIDs

=∂J

∂s−ΨT ∂R

∂s. (11)

Note that (11) consists mainly of the partial derivatives of the finite element equation system anderror estimator with respect to the node positions. These derivatives are not typically used in afinite element software. In [16] a method is described in detail (for the example of the Poissonequation) how these terms can be computed from terms that are typically used in the assembly of theequation system. Generalisation to the equation (4) and the error estimators (7) is straightforward.The evaluation of (11) costs roughly the same as assembly plus error estimation.

The advantage of the discrete adjoint technique is that, once the original equation system (8)is solved and I(s) evaluated from (9), the gradient DI/Ds is evaluated for essentially the cost ofonly one more solve (10) of the discretised PDE and evaluation of (11). This is in contrast to othermethods, which typically require a number of solves of the discretised PDE (8) that is proportionalto the size of vector s.

4.2 Numerical optimisation solverThe discrete adjoint technique removes Constraint 1 from Problem 1 by treating the reduced per-formance function. With the remaining constraints Problem 1 still is a smooth nonlinear inequalityconstrained optimisation problem.

For problems of this class sequential quadratic programming (SQP) is established as a reliableand efficient technique [10, 17]. SQP requires a quadratic model of the (reduced) performance func-tion, which in this work is defined using the BFGS update, thus in the end the gradient informationprovided with the discrete adjoint technique is sufficient, no second order derivatives are required.The quadratic sub-problems with linear constraints, which appear in the SQP method are solved withan interior point method.

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5 Suitability of error estimatorsBefore we attempt optimisation in algorithm opt-adapt with the error estimators (7), we willassess their behaviour on a family of structured anisotropic meshes. To this end we use the followingexample, whose solution has typical exponential and parabolic boundary layers.

Example 1. [5, Problem I]: We consider the problem (1) with Ω = (0, 1)2, ΓD = Γ, ΓN = ∅,b = (−1, 0)T , ub = 0, and a right-hand side f such that, given ε > 0, the function

u(x, y) =

(cos

π x

2− e−x/ε − e−1/ε

1− e−1/ε

)(1− e−y/

√ε) (1− e−(1−y)/

√ε)

1− e−1/√ε

is the solution of (1).

As the exact solution u is known, for any given mesh and approximate solution uh the error esti-mators (7) (which will be denoted by ||eest|| in the figures below) can be compared to the respectivenorms of the error e := u− uh, i.e.,

‖u− uh‖0,Ω , ε1/2 |u− uh|1,Ω ,

( ∑K∈Th

|K|ε‖b · ∇(u− uh)‖20,K

)1/2

, (12)

and (ε |u− uh|21,Ω +

∑K∈Th

|K|ε‖b · ∇(u− uh)‖20,K

)1/2

. (13)

Each of these norms can be expressed by means of L2 norms on the triangles K. In our implementa-tion, the arguments of these local L2 norms are replaced by quadratic interpolates and then the normsare evaluated exactly. The resulting norms will be denoted ||e|| in the figures below. Even thoughthis approximation of the norms (12), (13) produces some error as well, this error is of higher orderthan the norms (12), (13) and is thus ignored in the comparisons.

We consider parametric Shishkin-like meshes, see figures 2 and 3. The meshes are defined suchthat parameters a = 1/2, c = 1/4 result in a uniform mesh of triangles, with 9 by 9 nodes. For all0 < a ≤ 1/2, 0 < c ≤ 1/4, within each of the sub-domains Ω11, Ω12, Ω21, Ω22, Ω31, Ω32 the nodesare equidistantly spaced in x and y direction.

In [5] meshes of this type have been analysed, with the result that the parameters

a = aS := min1/2, 5

2ε log(n),

c = min1/4, 5

2

√ε log(n),

with n = 9 in our case, are suitable for the given setting. We call aS the Shishkin parameter. Wecompare the behaviour of the different error estimators with respect to variation of the parameter a,keeping c fixed at the value as given above, see again Figure 3 for an example.

In order that the optimisation of the node positions in the mesh, starting from a uniform mesh,can produce a mesh of similar quality as the a priori analysis, the error estimators ||eest|| should have

11

x

y

1

10 a

c

1−c

11Ω Ω

Ω

Ω

12

21Ω

Ω

22

3231

Figure 2: Parametric Shishkin-like meshes, definition.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shishkin−like mesh, eps=1.0e−03, a=9.4e−02

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shishkin mesh, eps=1.0e−03, a=a

S=5.5e−03

a) b)

Figure 3: Parametric Shishkin-like meshes, a) a = 0.094 6= aS , b) a = aS as derived in a priori analysisin [5]

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the same qualitative behaviour as the exact error ||e|| for 0 < a ≤ 1/2, and in particular should havelocal extrema only very close to the extrema of ||e||.

Figure 4 shows the results for all four norms considered for the perturbation parameter ε = 10−3,for which the value aS ≈ 0.0055 is suitable according to the a priori analysis in [5]. One can see thatall four norms produce a global minimum near aS , and that these global minima of ||eest|| and ||e||match reasonably well.

However, further away from aS qualitative differences in the behaviour of the error estimatorsoccur. All three of (H10-NORM), (CONV-NORM) and (H10-CONV-NORM) estimate the error ata ≈ 0.15 slightly higher than at a = 0.5, even though the exact error is lower at a ≈ 0.15. Thus,a gradient based optimisation of ||eest|| starting at a = 0.5 would essentially be unable to find theglobal minimum, due to insufficient robustness of the estimators. In contrast, for the (L2-NORM) theestimator decreases all the way from a = 0.5 to the global minimum.

The same behaviour, but even more pronounced is observed for ε = 10−4 in Figure 5.

6 Numerical ExperimentsIn order to investigate properties of the refinement algorithm opt-adapt we present results forthree example problems of (1), each for several values of the diffusion parameter ε > 0. BesidesExample 1 we will use the following two examples, for which the exact solution u is unknown.

Example 2. : We consider the problem (1) with the L-shape domain Ω = (−1, 1)2\( [0, 1]×[−1, 0] ),ΓN = ( 1 × (0, 1) ) ∪ ( (−1, 1)× 1 ), ΓD = Γ \ ΓN , b = (1, 1

2 )T , f = 1, g = 0, and ub = 0.

Example 3. [8, Example 6.2]: We consider the problem (1) with Ω = (0, 1)2, ΓN = 0 × (0, 1),ΓD = Γ \ ΓN , b = (−y, x)T , f = 0, g = 0, and

ub(x, y) =

1 for (x, y) ∈ (1/3, 2/3)× 0,0 else on ΓD.

The simple structure of the problem data b, f , g and ub is chosen in order to demonstrate featuresof the solution of (1) that are typical, not implied by unusual choices of the problem data.

In comparison to Example 1, these two examples add difficulty because their solution has inte-rior layers. Treatment of interior layers with a priori theory is more difficult, Shishkin meshes arenot known for these cases. This becomes even more involved, as the interior layer in Example 2originates from the reentrant corner of Ω at (0, 0), and the exponential boundary layer along the edge0 × (−1, 0) also interacts with the corner singularity which is to be expected there.

Further, for Example 3 the interior layer is curved, due to the variable convection field b (vortexaround point (0, 0)). We chose this example to investigate if the optimised coarse meshes provide anadvantage for such curved layers.

Due to the observations of Section 5 most of the experiments are performed with the (L2-NORM)error estimator. Figures 6, 8 and 10 show the estimated error over the number of mesh nodes for thetested refinement methods: uniform refinement, isotropic adaptive refinement (red-green) and algo-rithm opt-adapt. For Example 1 the exact error and Shishkin-meshes are also available, thus theseare included in Figure 6. Note that only the coarse mesh is constructed as the usual Shishkin meshas described in Section 5, then this mesh is subsequently refined with red-green refinement. How-ever, the anisotropy introduced with the coarse-Shishkin mesh is a huge advantage, thus producingfavourable results.

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0 0.1 0.2 0.3 0.4 0.510

−2

10−1

100

a

parametric meshes, eps=1.0e−03, L2−NORM

||e||||e

est||

ref a

0 0.01 0.02 0.03 0.04 0.0510

−2

10−1

100

a


||e||||e

est||

ref a

0 0.1 0.2 0.3 0.4 0.5

100

a

parametric meshes, eps=1.0e−03, H10−NORM

||e||||e

est||

ref a

0 0.01 0.02 0.03 0.04 0.05

100

a


||e||||e

est||

ref a

0 0.1 0.2 0.3 0.4 0.510

−1

100

101

102

103

a

parametric meshes, eps=1.0e−03, CONV−NORM

||e||||e

est||

ref a

0 0.01 0.02 0.03 0.04 0.0510

−1

100

101

102

103

a


||e||||e

est||

ref a

0 0.1 0.2 0.3 0.4 0.510

0

101

102

103

a

parametric meshes, eps=1.0e−03, H10−CONV−NORM

||e||||e

est||

ref a

0 0.01 0.02 0.03 0.04 0.0510

0

101

102

103

a


||e||||e

est||

ref a

a) b)

Figure 4: Comparing robustness of different error estimators for ε = 10−3, a) a ≤ 0.5, b) closeup fora ≤ 10aS .

14

0 0.1 0.2 0.3 0.4 0.510

−2

10−1

100

101

a


||e||||e

est||

ref a

0 1 2 3 4 5

x 10−3

10−2

10−1

100

101

a


||e||||e

est||

ref a

0 0.1 0.2 0.3 0.4 0.5

100

a


||e||||e

est||

ref a

0 1 2 3 4 5

x 10−3

100

a


||e||||e

est||

ref a

0 0.1 0.2 0.3 0.4 0.510

0

101

102

103

104

a


||e||||e

est||

ref a

0 1 2 3 4 5

x 10−3

100

101

102

103

104

a


||e||||e

est||

ref a

0 0.1 0.2 0.3 0.4 0.510

0

101

102

103

104

a


||e||||e

est||

ref a

0 1 2 3 4 5

x 10−3

100

101

102

103

104

a


||e||||e

est||

ref a

a) b)

Figure 5: Comparing robustness of different error estimators for ε = 10−4, a) a ≤ 0.5, b) closeup fora ≤ 10aS .

15

We observe that with decreasing ε the advantage of the refinement strategy opt-adapt becomesmore pronounced, which is expected since the anisotropy in the solution also becomes more pro-nounced. However, at a certain problem dependent value of ε this good performance of opt-adaptis lost, and eventually the advantage vanishes. The most significant advantage is obtained at ε = 10−3

for examples 1 and 2 and at ε = 10−4 for Example 3. Our study of an even larger set of test problemssuggests two reasons for this loss of improvement. The first is that for very low values of ε spuriousoscillations occur even with the stabilised method, which increases the severity of the non-linear be-haviour of the optimisation problem. The second apparent reason is that the optimisation problemsalso become more and more ill-conditioned, since length-scales of order ε and lower are expected tobe important in the node position vectors at the same time as length-scales of order one.

The first reason can be overcome to a certain degree by running the finite element computa-tions on a finer mesh, utilising the original coarse mesh as parameterisation of the fine mesh for theoptimisation. However this also increases the computational cost.

Figures 7, 9 and 11 show examples of the resulting meshes and solution. The example solutionplot is always provided on uniform mesh and moderate ε, because this is more suitable for visual pre-sentation. The example meshes are provided for the value of ε, where advantage of the opt-adaptalgorithm is the largest.

In all three examples anisotropic elements can be observed, which are characterised by highaspect-ratios (AR)

AR :=1

2

H2

|T |=H

h,

where H is the length of the longest edge of triangle T and |T | = 12Hh its surface area, with h the

height of T perpendicular to the longest edge. Aspect ratios up to 440 are observed in the optimisedcoarse meshes (Figure 9), demonstrating that significant anisotropy can be introduced this way.

A particularly interesting result is that the mesh is aligned with the interior layers in examples 2and 3. Further, Figure 10 shows a significant advantage of the optimised coarse mesh even in the caseof a curved interior layer. In this example most of the elements along the layer have very moderateaspect ratios, but the alignment of the mesh alone already poses an advantage. Note that the interiorlayers are characteristic layers, for which the convergence on uniform meshes is already faster thanfor the exponential layers at outflow boundaries.

Finally, in Figure 7 the robustness of the error estimates on the refined meshes can be assessed. Incontrast to the observations in [13] (for different error estimators), the efficiency of the error estimatedecreases during the locally isotropic refinement after the optimisation phase (observed for ε = 10−2

and ε = 10−3). This is somewhat surprising, as on the optimised coarse mesh the quality of the errorestimator is at its best. Maybe this can be overcome by using a different error estimator during thered-green refinement phase. Nonetheless, the optimised coarse mesh is a significant improvementcompared to the uniform coarse mesh and even the Shishkin mesh.

16

102

104

106

10−6

10−4

10−2

100

102

104

#nodes

L2−

NO

RM

of err

or

history for eps=1.0e−01

||e||L

2 − opt−adapt

||eest

||L

2 − opt−adapt

||e||L

2 − iso adapt

||eest

||L

2 − iso adapt

||e||L

2 − Shishkin

||eest

||L

2 − Shishkin

||e||L

2 − uniform

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

102

104

106

10−4

10−3

10−2

10−1

100

101

102

103

#nodes

L2−

NO

RM

of err

or


||e||L

2 − opt−adapt

||eest

||L

2 − opt−adapt

||e||L

2 − iso adapt

||eest

||L

2 − iso adapt

||e||L

2 − Shishkin

||eest

||L

2 − Shishkin

||e||L

2 − uniform

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

102

104

106

10−4

10−3

10−2

10−1

100

101

102

103

#nodes

L2−

NO

RM

of err

or


||e||L

2 − opt−adapt

||eest

||L

2 − opt−adapt

||e||L

2 − iso adapt

||eest

||L

2 − iso adapt

||e||L

2 − Shishkin

||eest

||L

2 − Shishkin

||e||L

2 − uniform

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

102

104

106

10−3

10−2

10−1

100

101

102

103

#nodes

L2−

NO

RM

of err

or


||e||L

2 − opt−adapt

||eest

||L

2 − opt−adapt

||e||L

2 − iso adapt

||eest

||L

2 − iso adapt

||e||L

2 − Shishkin

||eest

||L

2 − Shishkin

||e||L

2 − uniform

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

Figure 6: Convergence histories for Example 1, (L2-NORM)

17

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

initial mesh, max(AR)=2.0e+00

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

x

example solution, eps=1.0e−02, uniform, 1089 nodes

y

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

final mesh for eps=1.0e−03, max(AR)=2.0e+02

0 5 10 15 20

x 10−3

0.44

0.445

0.45

0.455

0.46

x

y

final mesh for eps=1.0e−03 (close−up)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

opt−adapt final mesh for eps=1.0e−03, max(AR)=7.8e+02

0 5 10 15 20

x 10−3

0.44

0.445

0.45

0.455

0.46

x

y

opt−adapt final mesh for eps=1.0e−03 (close−up)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

iso adapt final mesh for eps=1.0e−03, max(AR)=5.0e+00

0 5 10 15 20

x 10−3

0.44

0.445

0.45

0.455

0.46

x

y

iso adapt final mesh for eps=1.0e−03 (close−up)

Figure 7: Meshes for Example 1, (L2-NORM)

18

102

104

106

10−4

10−3

10−2

10−1

100

101

102

#nodes

L2−

NO

RM

of err

or


||eest

||L

2 − opt−adapt

||eest

||L

2 − iso adapt

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

102

104

106

10−4

10−3

10−2

10−1

100

101

102

#nodes

L2−

NO

RM

of err

or


||eest

||L

2 − opt−adapt

||eest

||L

2 − iso adapt

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

102

104

106

10−4

10−3

10−2

10−1

100

101

102

#nodes

L2−

NO

RM

of err

or


||eest

||L

2 − opt−adapt

||eest

||L

2 − iso adapt

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

102

104

106

10−3

10−2

10−1

100

101

102

#nodes

L2−

NO

RM

of err

or


||eest

||L

2 − opt−adapt

||eest

||L

2 − iso adapt

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)


19

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y


−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

0.5

1

1.5

2

x


y

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y


−0.1 −0.05 0 0.05−0.6

−0.55

−0.5

−0.45

x

y


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y


−0.1 −0.05 0 0.05−0.6

−0.55

−0.5

−0.45

x

y


−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y


−0.1 −0.05 0 0.05−0.6

−0.55

−0.5

−0.45

x

y



20

102

103

104

105

106

107

10−4

10−3

10−2

10−1

100

101

102

#nodes

L2−

NO

RM

of err

or


||eest

||L

2 − opt−adapt

||eest

||L

2 − iso adapt

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

102

103

104

105

106

107

10−4

10−3

10−2

10−1

100

101

102

#nodes

L2−

NO

RM

of err

or


||eest

||L

2 − opt−adapt

||eest

||L

2 − iso adapt

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

102

103

104

105

106

107

10−4

10−3

10−2

10−1

100

101

102

#nodes

L2−

NO

RM

of err

or


||eest

||L

2 − opt−adapt

||eest

||L

2 − iso adapt

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)

102

103

104

105

106

107

10−4

10−3

10−2

10−1

100

101

102

#nodes

L2−

NO

RM

of err

or


||eest

||L

2 − opt−adapt

||eest

||L

2 − iso adapt

||eest

||L

2 − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h2)=o(1/n)


21

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y


0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x


y

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y


0.62 0.64 0.66 0.68 0.70

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x

y


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y


0.62 0.64 0.66 0.68 0.70

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x

y


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y


0.62 0.64 0.66 0.68 0.70

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x

y



22

102

104

106

10−2

10−1

100

101

102

103

104

#nodes

H10−

CO

NV

−N

OR

M o

f err

or


||e||X − opt−adapt

||eest

||X − opt−adapt

||e||X − iso adapt

||eest

||X − iso adapt

||e||X − Shishkin

||eest

||X − Shishkin

||e||X − uniform

||eest

||X − uniform

ref o(h(1/2)

)=o(1/n(1/4))

)

ref o(h)=o(1/sqrt(n))

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y


Figure 12: Convergence histories (left) for Example 1 at ε = 10−3 with (H10-CONV-NORM) and corre-sponding optimised coarse mesh (right)

For completeness we also provide analogous results for the (H10-CONV-NORM), but since theseare as expected less favourable, we provide this only for Example 1 at ε = 10−3, see Figure 12.As expected the mesh is coarsened near the layer, instead of refined as would be desirable, whichconfirms the observations of Section 5. Correspondingly the actual error is increased during theminimisation of the error estimator in this norm. We conclude that the (H10-CONV-NORM) estimatoris not sufficiently robust for this approach.

7 ConclusionsFour different variants of the local problem error estimator from [18] for convection dominatedconvection-diffusion problems have been tested on a set of parametric meshes. Out of the fourvariants only that which uses only the L2-norm of the solutions of the local problems appears to besuitable to allow to find a good anisotropic mesh by starting from a uniform mesh and minimising theerror estimator. The tests indicate that close to an optimal mesh it may be possible or even beneficialto switch over to minimise the error estimator for a stronger norm.

The minimisation of the (L2-NORM) error estimator by moving all node positions in the mesh al-lows to generate suitably anisotropic coarse meshes, if required by the solution of the PDE. However,this only works reliably for down to a certain problem dependent value of the parameter ε.

An interesting future research opportunity might also be to combine this optimisation with aparameter continuation strategy, reducing ε at the same time as optimising the coarse mesh. Thismight enable to extend the range of parameters for which this approach works well.

Acknowledgements The work of P. Knobloch was supported by the Grant Agency of the CzechRepublic under the grant No. P201/11/1304.

23

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[3] I. Christie, D. F. Griffiths, A. R. Mitchell, and O. C. Zienkiewicz. Finite element methods forsecond order differential equations with significant first derivatives. Int. J. Numer. MethodsEng., 10:1389–1396, 1976.

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