ERROR ESTIMATION AND ADAPTIVITY FOR FINITE-VOLUME METHODS ON UNSTRUCTURED TRIANGULAR MESHES: ELLIPTIC HEAT TRANSFER PROBLEMS

ERROR ESTIMATION AND ADAPTIVITYFOR FINITE-VOLUME METHODS ON UNSTRUCTUREDTRIANGULAR MESHES: ELLIPTIC HEAT TRANSFERPROBLEMS

Marcio A. Martins and Ramon M. ValleMechanical Engineering Department, UFMG, Campus Pampulha,

Belo Horizonte, Brazil

Leandro S. OliveiraChemical Engineering Department, UFMG, Belo Horizonte, Brazil

Denise BurgarelliDepartment of Mathematics, ICEX=UFMG, Campus Pampulha,

Belo Horizonte, Brazil

In this article, a simple and reliable a posteriori error estimate methodology for the finite-volume method on triangular meshes and an adaptive mesh refinement procedure arepresented. The proposed error estimate employs a high-order approximation for the scalarat the triangles faces. The estimate technique does not demand expressive computationalefforts and memory storage. The adaptive procedure is based on the equal distribution of theerror over all the triangles, allowing for suitable local mesh refinements. The error ismeasured by an H1 norm, and its convergence behavior is evaluated using four ellipticproblems for which the analytical solutions are known. The error differences using ana-lytical and estimate solutions are compared for those problems, and good performance ofthe adaptive procedure is verified.

1. INTRODUCTION

The ®nite volume method is a discretization method suitable for the solution ofconservation laws described by means of elliptic, parabolic, or hyperbolic partialdi� erential equations. This method has been extensively used in several engineering®elds, mainly in computational ¯uid dynamics [1, 2]. The local conservation of thenumerical ¯uxes between each pair of neighboring control volumes makes themethod attractive in simulation problems in which the ¯ux is of relevance, such asproblems of ¯uid mechanics, heat, and mass transfer.

Received 16 November 2001; accepted 4 May 2002.

Marcio A. Martins is grateful for the support of the Brazilian government agency CAPES for a

scholarship to carry out this study.

Address correspondence to Leandro S. Oliveira, Chemical Engineering Department, UFMG, Rua

EspõÂ rito Santo, 35, 6 Ê andar, Belo Horizonte, MG, Brazil, CEP: 30160-030 . E-mail: [email protected]

Numerical Heat Transfer, Part B, 42: 461±483, 2002

Copyright # 2002 Taylor & Francis

1040-7790 /02 $12.00 + .00

DOI: 10.1080/1040779019005403 0

461

In the last decade, the ®nite-volume method in unstructured meshes has seensigni®cant developments, mainly in the discretization methods. The development ofgeneric mathematical formulations for the gradient in the vector form makes itpossible for the implementation of several classes of problems with just a fewalterations in the computational codes. The convergence theory of the ®nite-volumemethod in several space dimensions has only recently been undertaken. As a result ofsuch theory, the complete discrete spaces and the respective norms are known, soerror estimates can now be formulated. A mathematical framework of the ®nite-volume method is presented and thoroughly discussed by Galouet and Herbin [3]and Herbin [4].

Methodologies for error estimation and mesh adaptation have been proposedin order to improve the accuracy of numerical solution of partial di� erential equa-tions. A way to reduce the numerical error is by reducing the characteristic size of themesh polyhedrals, and the resulting procedure is called adaptive h re®nement. Thisclass of re®nement can easily handle the occurrence of high local gradients andirregular geometries. Both error estimate and h-re®nement methodologies are wellestablished for the ®nite-element method [5, 6]. However, for the ®nite-volumemethod, there are only a few methodologies in the literature [7±10]. This latedevelopment in ®nite volumes can be justi®ed by a previous lack of mathematicalsupport, which is required for developing the error estimate formulations.

In the ®nite-volume error estimate framework, Richardson extrapolation [11] isthe most popular methodology, and it has been used to solve adaptively a fewengineering problems [10, 12]. This estimate is quite reliable on re®ned meshes, sinceit takes into account the smoothness of the spatial variation. However, the method

NOMENCLATURE

a constant

ak polynomial coe� cients

A area

b characteristic constant

C characteristic constant

e error

e unit directional vector

f(x) source term

g(x) source term

hc characteristic constant

hi diameter of triangle i

Hm

Sobolev space

k(x) generic function

L2

space of square integrable functions

n unit normal vector

nt number of triangular cells

p convergence order

r relative error

R2

regression coe� cient

S length of the shared triangle face

t unit tangent vector

u(x) continuous scalar

ui discrete scalar

V volume, m3

x coordinate vector

ak eigenvalues

qOi boundary of the triangle OI

qO boundary of the calculation

domain OD distance

Z local coordinate axis along the

triangle face

z dimensionless coordinate along localx axis

x local coordinate axis along the line

joining the triangles centroid

f local error indicatorO calculation domain

Oi triangle i

Subscripts

f face value

Z local axis along the triangle facex local axis along the line joining the

triangles centroid

462 M. A. MARTINS ET AL.

requires solutions on at least two meshes, di� ering in size by a factor of 2, just toestimate the error throughout the computational domain. An error estimate basedon solution reconstruction using a weighted least-square interpolation scheme wasalso proposed in the literature [13]. This estimate reconstructs the solution by meansof piecewise linear or quadratic functions, used to calculate the derivatives at thediscrete points. Although this estimate is inherently simple, it was veri®ed in [9, 10]that it is less e� ective than other estimates [11, 13] when strong nonlinearities arepresent, so it is suitable only to solve problems in which suitably re®ned initialmeshes can be generated and smooth solutions are present. A third class of errorestimate presented in the literature is based on an error equation derived from theoriginal system of equations. In this estimate, error source terms are obtained by ananalysis of the numerical scheme. This error estimate technique is compared to theprevious one in [9±14], where good e� ectiveness of estimate could be veri®ed. Again,this last error estimate requires the solution of a system of equations, and its for-mulation leads to a speci®c equation for the error, which is fully dependent on theprimary problem equations. Further details on this error estimate technique can befound in [9, 10, 14].

Based on the review presented, it is clearly noticeable that most of the errorestimation procedures for ®nite-volume methods require the solution of a system ofequations. As a conclusion, extensive computational e� ort is necessary to solve theproblem adaptively. In the present work, an a posteriori error estimate, based onlocal solution reconstruction, that can be expressed as an algebraic equation isproposed. Hence, the need for solutions of systems of equations in the error esti-mation procedure is eliminated. The proposed estimate can be classi®ed as a trun-cation-error type. As discussed previously, the main objective consists on thedevelopment of a fast and reliable a posteriori error estimate and an adaptive pro-cedure for ®nite-volume methods on unstructured triangular meshes.

The article is organized as follows. The following section describes a ®nite-volume discretization scheme for triangular meshes. The error estimate andthe adaptive procedure are presented in Section 3. In Section 4, a study of theconvergence of the error estimate and the performance of the proposed adaptivesolution are presented for three generic elliptic heat transfer problems. The problemswere chosen in order to allow for complete control of the gradients distributions.

2. FINITE-VOLUME DISCRETIZATION SCHEME

Consider the generic elliptic boundary-value problem:

H ¢ ¡k…x†Hu…x†‰ Š ˆ f…x† in O …1a†

u…x† ˆ u0…x† in qO …1b†

¡ k…x†Hu…x† ¢ n ˆ g…x† in qO …1c†

¡ k…x†Hu…x† ¢ n ˆ h…x† u…x† ¡ u1‰ Š in qO …1d †

where u(x) is a scalar ®eld (e.g., temperature), k(x), f(x), g(x), and h(x) are functions,u0(x) is the scalar ®eld at the boundary qO of the domain (O), and u1 is the scalar

ERROR ESTIMATION FOR FINITE-VOLUME METHODS 463

value surrounding qO. Equations (1b), (1c), and (1d) represent the boundary con-ditions of Dirichlet, Neumann, and Robin, respectively. The integral form of Eq.(1a) is given by

¡Z

qOi

k…x†Hu…x† ¢ n dS ˆZ

Oi

f…x† dx in Oi 2 O …2†

where Oi is a set of triangular control volumes (i.e., cells) in which O ˆ [Oi. In the®nite-volume discretization scheme, the di� usion ¯ux k…x†Hu…x† on the shared tri-angular faces needs to be approximated. In order to deal with general meshes (e.g.,any triangular, quadrilateral, other meshes) and di� usion matrices, an approxima-tion of the entire gradient should be carried out. There are two classes of approx-imation in the literature, the Green-Gauss type [1±4] and the polynomial Lagrangianinterpolation [15]. The diamond cell method is of the Green-Gauss type and will beused in this work due to the inherent simplicity and existent mathematical frame-work. The diamond cell de®nition is depicted in Figure 1, where xi and xj denote thecoordinates of the centroids of two neighbor triangles, xn and xs are the verticescoordinates such that j…xi ¡ xj† £ …xn ¡ xs†j > 0; xf is the coordinate at the sharedface, and x and Z constitute a nonorthogonal basis. Using the diamond cell method,Eq. (2) is approximated by means of the discrete gradient at each face belonging tocell i (Figure 1), resulting in

X3

jˆ1

¡k…xf† DZuj ¡ ui

Dx…tx ¢ nZ† ¡ un ¡ us

DZtx ¢ tZ

tx ¢ nZ

³ ´µ ¶ˆ f…xi†Ai in Oi 2 O …3†

Figure 1. Notation and diamond cell de®nition for triangular meshes.


where n and t are the unit normal and the unit tangential vector at the basis x and Z,respectively, Dx and DZ are the distances between neighboring centroids and thenodes n and s at the shared face, respectively, and Ai is the area of the triangular cellOi. The nodal values un and us are approximated by a least-square interpolationscheme, using the centroid of the triangle that share the respective nodes [2, 16].Thus, the implicit scheme results in a set of algebraic equations based on the cellcentroids only. The application of the boundary conditions given by Eqs. (1b)±(1d) ispresented and thoroughly discussed in [16].

The ®nite volume discrete form of Eq. (1), given by Eq. (3), is attractive for thesolution of problems in which highly irregular geometries are present and a localnonorthogonal basis cannot be de®ned. Although the discretization scheme usedhere yields a more complex scheme than those that use a local orthogonal basis, thisscheme can be employed for any triangular mesh (e.g., triangulations in whichmaximum angles are greater or equal than p/2), as is veri®ed in the results section ofthis article.

3. ERROR ESTIMATE AND ADAPTIVITY

Whenever a numerical method is used to solve a di� erential equation, a dif-ference will exist between the numerical and the exact solutions. If the exact solutionis known, the error in each mesh cell (i.e., local error), as well as in the whole domain(i.e., global error), can be calculated using several norms. In most engineering pro-blems, it is not possible to obtain an exact solution, and therefore the solution errorcannot be calculated. Hence, it should be estimated. When an estimate is usedinstead of the exact solution, it can be used to control both local and global solutionerrors. This control is made by means of an adaptive procedure. The most popularadaptive procedures are of the h type, which are based on establishing a relationshipbetween the size of the mesh cells and the solution error. The procedures usuallyconsist of distributing the global error equally over the mesh, increasing ordecreasing the size of the mesh cells as needed.

The error ei for the triangle Oi is de®ned in this work as

ei ˆ u…xi† ¡ ui …4†

where u(xi) and ui are the estimated and the ®nite-volume solutions, respectively, atthe centroid of the triangle Oi. The norms used to measure the error are usually theL2 and the H1 norms. For ®nite-volume methods, these norms are written in thediscrete form [17] and are given by

jjeijjL2 ˆ hi u…xi† ¡ ui‰ Š2n o

1=2…5†

jjeijjH10

ˆX3

jˆ1

Dx DZu…xi† ¡ u…xj†

Dx¡ ui ¡ uj

Dx

µ ¶2

( )1=2

…6†

where hi is a measure of the size of the triangle Oi. In Eq. (6), the H10 seminorm

becomes a norm in H1, as proved in [16].


The selection of one of these error norms depends on the problem ofinterest. It is well established for the ®nite-element method that the H1

0 norm ismore suitable to measure the estimate error when elliptic problems are solved,because the entire formulation is dependent on the gradient. The same situation isobserved in the ®nite-volume method, where the basic principle of thediscretization scheme is the local conservation of the numerical ¯ux at the controlvolume. Hence, the H1

0 norm is chosen as a measure of the error in this work,since it follows the same principle used in the discretization scheme. Notice thatthe continuous form of Eq. (6) consists of a measure of the di� erence between thegradients of the numerical and the estimate solutions. Thus, this norm is similarto the well-known energy norm commonly employed in works dealing with ®nite-element methods [18].

The error estimate proposed in this work belongs to the class of truncationerror estimates, which are based on one-dimensional Taylor series. Along the x axis(Figure 1), the solution at any x coordinate can be evaluated by

u…x† ˆ ui ‡ qui

qxdx

1!‡ q2ui

qx2

d 2x

2!‡ q3ui

qx3

d 3x

3!‡ ¢ ¢ ¢ ‡ qnui

qxn

d nx

n!‡ ¢ ¢ ¢ …7†

where u(x) is the continuous solution along x, and dx is the distance from x ˆ 0.Recall that the approximate gradient in Eqs. (3) and (6), between the shared cen-troids i and j obey Eq. (7) truncated in the second term, for x ˆ Dx and u(Dx) ˆ uj,thus

qui

qxˆ qu…xf†

qx’ uj ¡ ui

Dx…8†

The estimate consists of an a posteriori evaluation of the ®rst four terms of theTaylor series, so a more reliable approximation for the scalar u at the cell face can beobtained. Thus, a function belonging to C

3(O) with nonzero second and third

derivatives can be used. De®ning the polynomial

u…x† ˆX3

kˆ0

akxk …9†

satisfying the previous restrictions, the ak coe� cients can be evaluated by

x ˆ 0 ! u…x† ˆ ui

x ˆ 0 ! qu…x†qx

ˆ Hui ¢ tx

x ˆ Dx ! …x† ˆ uj

x ˆ Dx ! qu…x†qx

ˆ Huj ¢ tx

…10†

where Dui and Duj are the gradients at triangles Oi and Oj, respectively. Using theconditions in Eq. (10), the coe� cients in Eq. (9) become


a0 ˆ ui

a1 ˆ Hui ¢ tx

a2 ˆ 3uj ¡ ui

Dx2¡ Huj ‡ 2Hui

Dx

³ ´¢ tx

a3 ˆ ¡2uj ¡ ui

Dx3‡ Huj ‡ Hui

Dx2

³ ´¢ tx

…11†

Using the coe� cients presented in Eq. (11), the estimate is written as

u…x† ˆ ui ‡ Hui ¢ txjxjDx

‡ 3uj ¡ ui

Dx2¡ Huj ‡ 2Hui

Dx

³ ´¢ tx

µ ¶jxjDx

³ ´2

‡ ¡2uj ¡ ui

Dx3‡ Huj ‡ Hui

Dx2

³ ´¢ tx

µ ¶jxjDx

³ ´3

…12†

The error norm presented in Eq. (6) can be rewritten as [17]

jjeijjH10

ˆX3

jˆ1

Dxf DZu…xi† ¡ u…xf†

Dxf

¡ ui ¡ uf

Dxf

" #2

8<

:

9=

;

1=2

…13†

where uf and u…xf† are the ®nite-volume and the estimate solutions at the shared facebetween the triangles Oi and Oj, respectively, and Dxf is the distance along the x axisbetween the centroid i and the cell face coordinate xf. Introducing the estimate intoEq. (13), and thanks to the ®rst condition in Eq. (10), the ®nal error estimate norm isgiven by

jjeijjH10

ˆX3

jˆ1

D z 3uj ¡ 2zuj ¡ 3ui ‡ 2zui

¡ ¢£(

‡ z2Dx tx ¢ …¡Huj ‡ zHuj ¡ 2Hui ‡ zHui†¤

2

¼1=2

…14†

where z ˆ Dxf=Dx is a dimensionless measure along x.As apposed to error estimates based on residuals [12, 14], the proposed for-

mulation presents not only an error estimate [Eq. (14)], but also an estimate for thesolution, given by Eq. (12). By means of Eq. (14), it is possible to evaluate the localerror at triangle Oi using not only the ®nite-volume solution, but also its gradient.The gradient can be evaluated by taking the three neighboring cell center points andthen applying Gauss’s theorem over a path surrounding the cell center [19].

Having de®ned an error estimate, to obtain an adaptive solution, a metho-dology in which both local and global errors are used to de®ne a new mesh isrequired. An appropriate methodology is the h-type adaptive procedure, and it willbe described in this article. This class of adaptive procedures must obey the followingconditions: the global relative error (rO) must be lower than an imposed value, and


the local error indicator (fi) should be close to unity. The global relative error isde®ned as

rO ˆP

ntiˆ1

jjeijj2H10P

ntiˆ1

jju…xi†jj2H10

0

@

1

A1=2

…15†

where nt is the number of triangles. According to condition (i), the adaptive pro-cedure establish that rO µ ·rO, where ·rO is the imposed relative error. The local errorindicator (fi) establishes a relationship between the error at triangle Oi and the localrequired error (·ei). Following this de®nition, the local indicator is de®ned as

fiˆ

jjeijjH10

jj·eijjH10

…16†

The de®nition of the local required error provides a homogeneous distributionof the error in each mesh triangle. The distributed error for each triangle can bede®ned as the ratio of the global error and the total number of triangles (nt) in themesh. This means that the sum of the square of the local error norms is equallydistributed in the control volumes. Thus,

jj·eijjH10

ˆ ·rOjju…xi†jjH01��

ntp …17†

Whenever fi > 1, the mesh will require re®nement; otherwise, it will be coar-sened. In an optimal mesh, fi will be approximately equal to unity for all the tri-angles and further mesh enhancement is not necessary. It is acceptable that the localerror indicator can be used to de®ne the new size distribution on the mesh, accordingto the equation

hi ˆ h0i

f1=pi

…18†

where hi and h0i are the new and the current size of the triangle Oi, respectively, and

p is the convergence order, which will be assigned the value of 1, since the proposederror estimate formulation does not present any dependence on the convergenceorder [18].

The complete adaptive solution procedure is presented in Figure 2, and furtherinformation regarding h-type mesh re®nement can be found in [6] and [21]. Theprocedure stores the new size distribution in the appropriate centroids of the originalmesh. Then, a least-squares method is used to interpolate the triangle size from thecentroids to the respective triangle nodes. At the end, a size distribution stored at thenodes of the original mesh is obtained. This mesh is used as a background mesh bythe mesh generator to generate a new one. With an initial mesh and the ®nite-volumesolution, the adaptive procedure automatically generates a new mesh and a new®nite-volume solution, in which the imposed relative error as well as the local errorare both controlled.


Three test cases, representing generic heat transfer problems, are presented inthe next section to evaluate the behavior of the error norm as a function of meshre®nement and to verify the performance of the proposed estimate and adaptiveprocedure. The test cases were selected in a way that the e� ects of essential, natural,and mixed boundary conditions could be evaluated. Also, the selected problemsallowed for complete control of the gradient distributions over the calculationdomains.

4. CONVERGENCE STUDY AND ADAPTIVE SOLUTION

In this section, a study of the convergence of the H10 norm for the error using

both the estimate and the analytical solutions is presented. Also, an error estimatebased on solution reconstruction [13] was implemented and used in the convergencestudy, in order to compare the proposed error estimate performance with that ofanother estimate. Among the properties of the solution error in a given norm, thefollowing must be satis®ed:

jjeOjjN

µ Chp ) log jjeOjjN

µ log…C† ‡ p log…h† …19a†

and

jjeOjjN

!h!0

0 …19b†

where N denotes a given norm, eO is the global error on O, C is a constant, h is ameasure of the cell size, and p is the convergence order.

Figure 2. Procedure used for adaptive ®nite-volume solution.


According to Eq. (19), a reliable error estimate should decrease linearly as thecell size decreases, in log scale. In order to verify the above properties, three test caseswere chosen for which the exact solutions were known. In order to evaluate thesecases, a mesh generator was implemented. It requires only the number of divisions(nv) on unity length as input, and then it generates a set of squares with 1=nv sidelength. At the end, the mesh generator divides each square into four rectangulartriangles. This mesh con®guration avoids anisotropic behavior and also providesequal distribution of triangle sizes.

In this section, the adaptive solution of the test cases is also presented. Thefully adaptive solution requires as input data an initial triangular mesh and animposed relative error, which is the ®nal desired solution error.

Test Case 1: Dirichlet Boundary Condition with Distributed SourceTerm

The ®rst test case is an elliptic problem with Dirichlet boundary condition. Thesource term used in this problem results in gradients along a diagonal line into thedomain. De®ning the calculation domain O ˆ Š0; 1‰ £ Š0; 1‰, this problem is statedmathematically as

H2u…x; y† ˆ f…x; y† ‡ g…x; y† …x; y† 2 O …20a†

u…x; y† ˆ 0 …x; y† 2 qO …20b†

where

f…x; y† ˆ 2a tan¡1 …z†…x2 ‡ y2 ¡ x ¡ y†

g…x; y† ˆ 2a2

1 ‡ z2…x ‡ y ¡ 1†…2xy ¡ x ¡ y† ¡ 4a

1 ‡ z2 xyz…1 ¡ x†…1 ¡ y†µ ¶

z ˆ a…x ‡ y ¡ b†

…21†

where a ˆ 20 and b ˆ 0:8 are constants. The exact solution is given by

u…x; y† ˆ axy…1 ¡ x†…1 ¡ y† tan¡1 a…x ‡ y ¡ b†‰ Š …x; y† 2 O …22†

By means of a sequence of successively re®ned anisotropic meshes, the errornorm was evaluated for each mesh using the exact and the estimate solutions. Forthe estimate solutions, both the proposed error estimate and the implementedsolution reconstruction error estimate (SREE) [13] were used. The behavior of theabsolute and relative error norms, as functions of the size of the mesh triangles, arepresented in Figures 3a and 3b, respectively. The size of the mesh triangles wasde®ned as the diameter of the circle that intercepts the three nodes of the respectivetriangle. The plot of the logarithm of the error norm as a function of the logarithm ofthe triangle diameter allows the evaluation of an approximate order of convergence,according to Eq. (19a). The convergence order (p) as well as the characteristic


Figure 3. Behavior of (a) absolute and (b) relative error norms for test case 1.


constant (C) were evaluated using linear regression, and they are expressed as in Eq.(19a) as

jjeOjjH1

0

ˆ 24:9383 h1:6999 …R2 ˆ 0:9972†

jjrOjjH1

0

ˆ 3:5975 h2:7359 …R2 ˆ 0:9987†…23a†

jjeOjjH1

0

ˆ 3:7753 h0:9307 …R2 ˆ 0:9888†

jjrOjjH1

0

ˆ 0:882 h2:1157 …R2 ˆ 0:9998†…23b†

jjeOjjH1

0

ˆ 25:9097 h1:3009 …R2 ˆ 0:9949†

jjrOjjH1

0

ˆ 4:0289 h2:2814 …R2 ˆ 0:9981†…23c†

where eO and rO are the absolute and the relative errors, respectively, calculated byusing the exact solution, the proposed error estimate, and the SREE, presented inEqs. (23a), (23b), and (23c), respectively. A small deviation from linear behavior forthe error using the exact solution is observed when rough meshes are used, and itsbehavior will be linear as the homogeneous re®nement proceeds. This behavior isjusti®ed by the nonsmoothness of the solution. Despite this deviation, the relativeerror behavior for the proposed estimate can be considered linear in a logarithmscale, as veri®ed by the correlation coe� cients presented in Eqs. (23). Also, it wasobserved that the error using the exact solution converges quite fast when it iscompared to the error with the proposed estimate. Hence, it can be veri®ed in Figure 3that the proposed error estimate is more suitable than the SREE, although theconvergence orders are almost identical. If the SREE is used in an adaptive solutioninstead of the proposed error estimate, a more re®ned mesh will be expected, sincethe relative error values of the SREE (Figure 3b) are higher than those of the pro-posed estimate. The linearity observed for the relative error using the estimatesolution con®rms that the proposed estimate is suitable for use in adaptive proce-dures for ®nite-volume methods on triangular meshes.

A target relative error of 0.25% was imposed. In this problem, the initial meshpresented in Figure 4 was used, and the ®nite-volume relative error using the esti-mate was 3.06% for this mesh. The ®nal mesh and the isolines are presented inFigures 5a and 5b, respectively. It can be veri®ed that the mesh re®nement occurredalong the line where the highest gradients occurred. Also, dere®nements were appliedin the regions where the gradients were less signi®cant.

The relative errors for the ®nal mesh were 0.24% and 0.19% using the esti-mate and the exact solution, respectively, and the e� ectiveness index was 1.2632. Asdiscussed before, a reliable error estimate should tend to the exact error as the meshre®nement proceeds. Thus, the e� ectiveness index will tend to unity as the imposedtarget relative error is reduced. In order to verify the above-mentioned fact, a targetrelative error of 0.125% was also imposed, and the relative error for the ®nal meshwas 0.19% and 0.18% for the estimate and the exact solutions, respectively. Thee� ectiveness index was evaluated as 1.0851, so the above-mentioned assumptionholds.


Test Case 2: Convective and Discontinuous Dirichlet BoundaryConditions

The second test case consisted of solving Laplace’s equation withdiscontinuous Dirichlet boundary conditions. This discontinuity of boundaryconditions is somewhat awkward to handle with the ®nite-element method, due tothe fact that two essential boundary conditions are to be applied in the same node. In®nite-volume methods, this problem is less relevant, since the discrete location is thecentroid and not the nodes of the triangles. Furthermore, a high gradient will beveri®ed near this singular point, so it is a perfect problem to be solved adaptively.The mathematical statement of this test case is given by

H2u…x; y† ˆ 0 …x; y† 2 O …24a†

Hu…x; y† ¢ n ˆ 0 x ˆ 0; y 2 ‰0; 1Š …24b†

Figure 4. Initial mesh for test case 1 (non ˆ 111, not ˆ 188).


Figure 5. Adaptive solution of test case 1: (a) ®nal mesh (non ˆ 879, not ˆ 1670) and (b) isolines for

u…x; y†.


Hu…x; y† ¢ n ˆ 0 y ˆ 1; x 2 ‰0; 1Š …24c†

Hu…x; y† ¢ n ˆ hcu…x; y† x ˆ 1; y 2 ‰0; 1Š …24d †

u…x; y† ˆ 0 y ˆ 0; x 2 ‰0; 0:5‰

1 y ˆ 0; x 2 Š0:5; 0Š

(…24e†

where hc ˆ 3:5 is a constant. The exact solution is given by

u…x; y† ˆX1

kˆ1

2…h2

c‡ a2

k† cos…akx† cosh…ak…1 ¡ y††

£ ¤

…h2c

‡ hc ‡ a2k† cosh…ak†

£ sin…ak† ¡ sin…0:5ak†ak

µ ¶…x; y† 2 O …25†

where ak are the eigenvalues of a tan…a† ˆ hc. The exact solution described by Eq.(25) was implemented for a truncation error of 10

¡10.Using the same procedure described in the previous test cases, the error norm

for the absolute and the relative error using the exact and the estimate solutions(proposed error estimate and SREE). The convergence behavior and the con-vergence order results are presented in Figure 6 and in the relations

jjeOjjH1

0

ˆ 1:9163 h1:1887 …R2 ˆ 1†

jjrOjjH1

0

ˆ 3:0184 h2:5920 …R2 ˆ 0:9998†…26a†

jjeOjjH1

0

ˆ 0:8926 h0:8902 …R2 ˆ 1†

jjrOjjH1

0

ˆ 0:9953 h2:0945 …R2 ˆ 0:9998†…26b†

jjeOjjH1

0

ˆ 4:0936 h0:9798 …R2 ˆ 1†

jjrOjjH1

0

ˆ 4:4368 h2:2052 …R2 ˆ 0:9996†…26c†

where the errors were calculated by using the exact solution, the proposed errorestimate, and the SREE, presented in Eqs. (26a), (26b), and (26c), respectively.Figures 6a and 6b show that the error behaviors with respect to mesh re®nement aresimilar for the absolute and the relative error norms. Hence, it can be veri®ed thatboth errors, using the exact and the estimated solutions, converge in agreement toEq. (19a). As presented in Eq. (26), the convergence order for the relative error ishigher due to the previously discussed reasons, and the linearity of the convergence isonce again veri®ed by the close-to-unity regression coe� cient. Again, it can beveri®ed in Figures 6a and 6b that the SREE leads to an overestimation of the error,when it is compared to the proposed error estimate and the exact error.

For this case, the proposed adaptive procedure was also employed, with theinitial mesh presented in Figure 4 and with an imposed target relative error of 0.5%.




The ®nal mesh and the adaptive solution are presented in Figures 7a and 7b,respectively. Notice that the mesh re®nement in Figure 7a was very clustery over thesingular boundary condition coordinates. Although the singular coordinate…x; y† ˆ …0:5; 0:0† has an inde®nite value, both relative local and global estimatederrors presented lower values than the imposed target value, which agrees with theconditions that must be satis®ed according to the proposed adaptive methodology.The proposed adaptive solution works quite well for this problem due to the fact thatthe discrete solution is performed at the triangle centroids and not at the nodes.

For the initial mesh the relative error was 3.99% and, after the mesh re®ne-ment, the relative error with the estimate was 0.483%, against 0.455% obtained withthe exact solution on the same ®nal mesh. Using these ®nal relative errors, thee� ectiveness index was evaluated as 1.0615, indicating again that the proposedadaptive methodology is suitable and reliable for elliptic problems. Also, it wasveri®ed that there is a relationship between the convergence order (which depends onthe problem) and the resulting e� ectiveness index.

Test Case 3: Nonlinear Dirichlet Boundary Conditions

The last test case consisted of solving a nonlinear elliptic equation. Nonlinearheat transfer problems that are governed by elliptic equations are those for which thethermal properties are temperature-dependent and those with a temperature-dependent source term. Evaluation of the performance of both the proposed errorestimate and the adaptive procedure in such problems is of relevance to demonstratethe reliability and applicability of the estimate and adaptive procedure. The selectedproblem has a source term dependent on u…x†, and the mathematical statement ofthis test case is given by

H2u…x; y† ˆ f…x; y; u† …x; y† 2 O …27a†

f…x; y; u† ˆ u3 2a2

b2

³ ´‡ u2 a

br¡ 3

a2

b2

³ ´‡ u

a2

b2¡ a

br

³ ´…27b†

r ˆ

��

x ¡ 1

2

³ ´2

‡ y ¡ 1

2

³ ´2

s

…27c†

The exact solution is given by

u…x; y† ˆ 1

1 ‡ exp ar

b

± ²¡ c

h in o …x; y† 2 O …28†

where a ˆ��2

p=2; b ˆ 5 £ 10

¡3, and c ˆ 30��2

pare constants. Dirichlet boundary

values are computed directly from Eq. (28).The error norm for the absolute and the relative error were calculated using

the exact and the estimate solutions (proposed error estimate and SREE). The


Figure 7. Adaptive solution of test case 2: (a) ®nal mesh (non ˆ 368, notˆ 670) and (b) isolines for u…x; y†.


convergence behavior and the convergence order results are presented in Figure 8and in the relations

jjeOjjH1

0

ˆ 1026:8 h2:5008 …R2 ˆ 0:9929†

jjrOjjH1

0

ˆ 7235:1 h2:6059 …R2 ˆ 0:9897†…29a†

jjeOjjH1

0

ˆ 11:295 h1:0400 …R2 ˆ 0:9998†

jjrOjjH1

0

ˆ 313:21 h1:5862 …R2 ˆ 0:9988†…29b†

jjeOjjH1

0

ˆ 124:59 h1:2234 …R2 ˆ 0:8177† …29c†

where the errors were calculated by using the exact solution, the proposed errorestimate, and the SREE, presented in Eqs. (29a), (29b), and (29c), respectively. Therelative error relation for the SREE was not presented since a linear ®t should not beperformed, due to the low correlation coe� cient (R2 ˆ 0:004). Figures 8a and 8bshow that the error behaviors with respect to mesh re®nement are similar for theabsolute and the relative error norms, when the exact solution and the proposederror estimate are compared. Although the convergence order for the proposedestimate is lower than that for the exact solution, both relative and absolute errortend to the exact error as the mesh re®nement proceeds. Hence, the proposed errorestimate converges linearly in log scale, but this same fact is not veri®ed for theSREE, for both the absolute and relative errors. If the SREE is used in the adaptivesolution, an inde®nite number of mesh enhancements will be performed, since thisestimate did not converge for this problem.

For this test case, the proposed adaptive procedure was also employed, withthe initial mesh presented in Figure 4 and with an imposed target relative error of10%. The ®nal mesh and the adaptive solution are presented in Figures 9a, 9b, and9c, respectively. Notice that the mesh re®nement in Figure 9a is very clustery alongthe circle centered in …x; y† ˆ …0:5; 0:5† and radii º 0.2. The results demonstrate thatthe proposed estimate is reliable when applied to nonlinear elliptic problems. Theadaptive solution procedure presented good performance for this nonlinear problem.Due to the very sharp variation of u…x; y† and to allow for a better visualization ofthe solution, both contour and surface plots are presented in Figure 9.

For the initial mesh, the relative error was 50.47% and, after the meshre®nement, the relative error with the estimate was 9.116%, compared to 9.809%obtained with the exact solution on the same ®nal mesh. Using these ®nal relativeerrors, the e� ectiveness index was evaluated as 0.9606, so the proposed adaptivemethodology is also suitable and reliable for nonlinear elliptic problems.

From the three test cases presented in this section, it was veri®ed that theproposed error estimate formulation is suitable for solving a wide class of ellipticproblems. The number of meshes to be generated depends on how many cycles ofre®nement are desired in a given spatial location. In test case 2, the ®nal meshconsisted of the second mesh generated by the adaptive procedure. However, in testcases 1 and 3, intermediate meshes were generated, so the ®nal mesh for test cases 1and 3 were the third and fourth ones, respectively. The di� erences among the


convergence orders for the relative error presented in Eq. (29) are justi®ed, sincethey are dependent on the problem statement. However, convergence orders for theestimate absolute error for all test cases were close to unity, agreeing with the®nite-volume convergence theory, which states that an error estimate measured by



Figure 9. Adaptive solution of test case 3: (a) ®nal mesh (non ˆ 1,603, notˆ 33,182), (b) isolines for

u…x; y†, and (c) surface plot for u…x; y†.


the H10 norm converges with an order close to unity for elliptic problems. Thus, the

assumption of unity order used in the proposed adaptive procedure is adequate.Further details regarding the convergence theory for the ®nite-volume scheme can befound in [16±18].

5. CONCLUSIONS

In this article, a simple and reliable error estimate and adaptive procedure waspresented for ®nite-volume methods on unstructured triangular meshes. The con-vergence studies presented demonstrated that the error estimate formulation is sui-table for elliptic problems since it allows the establishment of relationship between theerror norm and the triangle sizes. Although the H1

0 norm has not been used exten-sively by the ®nite-volume community, it was shown that this norm is reliable tomeasure the error, since the ®nite-volume discretization is based on the gradient (i.e.,¯ux across the cell faces). The convergence orders of the estimate absolute error in alltest cases were close to unity, as predicted by the ®nite-volume theory using the H1

0

norm. Hence, the proposed error estimate holds. The adaptive solution of ellipticproblems in this work demonstrated that both the proposed error estimate and theadaptive procedure were able to handle high local gradients and discontinuities in theboundary conditions. Hence, the proposed methodology could be used to solveproblems with high local gradients and discontinuities. Further studies will be con-cerned with alternative methodologies for improving the triangle’s size distribution.

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ERROR ESTIMATION AND ADAPTIVITY FOR FINITE-VOLUME METHODS ON UNSTRUCTURED TRIANGULAR MESHES: ELLIPTIC HEAT TRANSFER PROBLEMS

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