Application of the Mixed Hybrid FiniteElement

WATER RESOURCES RESEARCH,

VOL. 32, NO. 6, PAGES 1905-1909, JUNE 1996

Comment on "Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity?"by R. Mos(, P. Siegel, P. Ackerer, and G. Chavent Christian

Cordes

Departmentof MathematicalMethodsfor Applied Sciences, Universityof Padua,Padua,Italy

Wolfgang Kinzelbach• Institute of EnvironmentalPhysics,Universityof Heidelberg,Heidelberg,Germany

where (xi, Yi) are the coordinatesof the verticesof the triangularelementandA is the elementalarea.Note that if there is In recentyearsa large amountof researchhasbeen devoted no source,sink, or changeof storagein the element, i.e., Q • + to the applicationof mixedfinite elements(MFE) to ground- Q2 + Q3 = 0, v becomesconstantbecausethe terms of wi water flow problems.However, there is still a lack of under- dependingon the coordinatesx andy cancelout. standingof the meritsof the method,of its relationto alternative The drawback of an indefinite stiffness matrix is circumapproaches,and of the effectsof the elementgeometryon the vented by hybridization.Then the systemis solved for one qualityof the results.We try to put the methodinto perspective unknownpotential head TPi on each element edge.After this by provingthat the lowest-order MFE methodyieldsexactlythe is done the correspondingfluxesQi can be obtainedelementsameresults(pressures, fluxes,and velocities)as (1) finite vol- wise by umes(FV) in generaltriangularelements,and (2) finite differences(FD) in triangularelementsobtainedby dividingrectangles into triangles(as doneby Mos• et al. [1994]). B21 B22 B23 Q2 = P•TP2} (3) B31 B32 B33 Q3 PE TP3/l The mixed method requires in case 1 about 1.5 times as manyunknownsand in case2 about3 timesasmanyunknowns thecoefficients Bii arecalculated byan integration over as the standardmethods.The identity in case 2 was already where the element area: noted by Durlofsky[1994], but the proof is carried out here becauseMos• et al. [1994, p. 3004] state that "suchan equivalence is not established". (4) B,j= • w,w• dA For clarity we restrict ourselvesto isotropichydraulicconductivities.Anisotropycan be accommodatedin a straightfork is the hydraulicconductivity.For the analyticalsolutionof ward way. Introduction

(4) we definerii astheedgevectorfromnodei towardnodej,

Liiasits• length (Li.-- ri.),and byapplying thescalar product,

General triangular elements

2

In the lowest-order MFE formulation for triangular elements the unknownsare the potential headsin the elements PE andthe fluxesacrossthe elementedgesQ i (Figure1). Once the fluxeshave been computed,velocitiesin any point can be obtained elementwiseby

2

2

rifik= 5(Li•+ L•k- L•), wefind

1(3L 122 q3L 123 -k223 -3L 122 qk123 qk223 -- 3L2•2 + L2•3+ L223 3L2•2-L2•3+ 3L223 k 212 nI- k 213 - 3L 223

B=48kAL2 12_ 3L 213 q-k 223

2

2

-- L 12+

basis functions

w•= •

•

El2q-LiB- 3L23 / 2

with vectorial

2

L123Li3 + L23 2 2 2

v = QlW• + Q2w2q- Q3w3

3L

2

13+

3L

2

(5)

23/

To understandthe physicalcorrespondence between Q• and TPi underlying(3), let usconsideran elementwith no sources, sinks,or changeof storage,i.e., Q• + Q2 + Q3 = 0. In this casethe summationof the three equationsin (3) yieldsPE --

Yl

(2) • (TP• + TP2+ TP3),andTP1,TP2,TP3span alinear head W3= •-

distribution(i.e., a constantheadgradient)insidethe element. Equation (3) can alsobe solvedfor the corresponding fluxes acrossthe element edges,e.g.,

Y3

k

•Now at Instituteof Hydromechanics andWater Resources Management,SwissFederal Institute of Technology,ETH-Hoenggerberg, Zfirich, Switzerland.

Copyright1996by the AmericanGeophysicalUnion. Paper number96WR00567. 0043-1397/96/96 WR-00567 $09.00

Q•= -•-[TP•(r23r23 ) + TP2(r23r31) + TP3(r23r•2)] (6) However, there is a much more straightforwardway to compute the fluxesacrosselementedges,requiringaboutone third fewerunknowns,the methodof finitevolumes(FV). Insteadof solvingthe systemfor headsTP i at element edges,it can be

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CORDES AND KINZELBACH: COMMENTARY

QI Q2

TP2

PE

TP

TP3 i

2

Figure 3.

Additional elements at fixed head boundaries.

Q3 Figure 1.

Unknowns for the mixed finite element formula-

L

: kl(TPolQol Hi)L1

tion.

(10)

Equations(9) and (10) yield solvedfor headsHe at the circumcentersof the elements

Lo

(pointswherethethreeperpendicular bisectors intersect). To

L1

U••oo + Uo• ; TPm = Lo L•

provethe identityof the results,He is expressed in termsof TP1, TP2, and TP3. Linearinterpolation yields

4-

(11)

ko

1

He= 4-•2[TPl(r13r23) (r12r32) + TP2(rl2rl3 ) (r13r23) + TP3(rl2rl3)(r12r32)]

and

(7)

Q01=L L0 L1

Inserting(7) into (6) leadsto

+

k0 L 23

Q• = k(H•r- TPO

r12r13

(8)

L234A

(12)

k•

Equation(12) shows thatthefluxesacross theelementedges obtained byMFE using(3) canbe expressed equivalently just by the headsin the circumcenters. (Actually,headvaluesat

elementedgesand circumcenters provideidenticalinformation, sinceeitherof themcanbe transformed into the other, tweenthecircumcenter andtheedgemidpointcorresponding i.e. by (7) and (11) respectively.) Furthermore, (12) represents to TP1. Considernowthe twoelements shownin Figure2. L the formulation used in the method of finite volumes. The is the lengthof the commonedge,L o andL 1 the distances correspondingFV stiffnesscoefficientis from the commonedgeto the two circumcenters with heads L Ho andH1, respectively. Using(8) anddenoting by TPol the Note that the denominatorin (8) expresses the distancebe-

= - Lo L1 Sol

headvalueat thecommon edge,thefluxacross thisedgecan be expressed by both

(13)

+

k0 L

- TPoi) Qoi = ko(Ho Lo

(9)

kl

FluxesQs corresponding to elementalsources,sinks,or a change of storage in thetransient casearealsoappliedin the sameway in FV and MFE, i.e., in the elementalwater balance:

and

Qs = Q, + Q2 + Q3

(14)

The equivalence of thetwoapproaches extends furthermore to the calculation of the flow velocities which in both casescan be

carriedout elementwise by (1). Qo3 ÷

Lo

.

•_............... •"'Ho Qo2

Differences mightonlyoccurat boundaries withprescribed headbecause in theFV methodtheheadvalues cannotalways be applieddirectlyto theelementedges.FD modelers circumvent thisproblemby addinga layerof verythin cellsat such boundaries. At headboundaries in FV models,elementswith a rightangleasshown in Figure3 canbeaddedto applya head valuedirectlyat thecenterof eachboundary edge.The other

angles andtheconductivity of theadditional elements playno role.

Figure 2. Triangular finite volumeswith head valuesat the circumcenters of the elements.

Besidesthe fact that the FV approachleadsto identical results withlessunknowns thanMFE (a meshcomposed ofN triangular elements contains at least1.5N elementedges), it

CORDES

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1907

Lx

Lx

Lx

Lx

Ly

Cy

Figure 4. A right triangularelement. Ly

also allowsidentificationof mesh geometrieswhich lead to nonphysicalresults.For example,if a rhombusis dividedinto Figure 5. Ten mixed finite elementscorrespondingto five two trianglesby its longerdiagonal,the circumcenters lie out- finite difference cells. sideof their respectiveelementsandLo andL • in (13) become negative.Thus the stiffnesscoefficientSo• becomespositive and the matrix is no longeran M matrix.A positiveSo• means that a head decreasefrom point 0 to point 1 correspondsto a koTPo + ksTPs

flux from

1 toward

0. This can result

in local minima

and

maxima in the head field and a nonphysicalorientation of fluxesand velocities.To avoid.suchcasesthe triangulationfor FV (and of coursealsofor MFE) hasto have the properties formulatedby Forsyth[1991] for standardtriangularelements. For example,in the homogeneousand isotropiccasethe triangulationhas to be a Delaunay triangulation(the circumcircle of each triangle must containno other node in its interior).

TP1 =

Q1--2

Q2=2

We considerthe right triangle shownin Figure 4. For this elementthe matrixB (5) becomes

] Lx 2 +Ly2 3Lx 22+ 22 22 2 B= 12kLxLy 2 2 L x-Ly

-L x-Ly

Q3 = 2

1

1

ko

ks

•+

Lx TPo-

Ly

TP6

1

1

ko

k6

Ly TPo- TP7 Lx

1

+

ko

(•5)

L x 4- 3Ly /

(16)

TP• correspondsto the head in the circumcenterbecausein a right trianglethe circumcentercoincideswith the midpoint of the hypotenuse. The fluxesin Figure5 are obtainedin a similar

1

k7

Lx TPo-

TP2

Q2- Lyk Lx/2

Ly TPo- TPs Lx

(20)

TP8

Q4= 2L-• 1 1

We insertQ• + Q2 4- Q3 = 0 into (3) and obtainthe fluxes acrossthe element edges,e.g., TP1-

(19)

This showsthat not all TP i are, in fact, independentvariables, as appliedin MFE, but only the onesat the hypotenuses. Thus Q• can be expressedjust by TPo and TPs, and if this procedure is applied analogouslyto Q2, Q3, and Q4, we find

Triangular Elements Obtained by Dividing Rectangles Into Triangles

2 _mx24-my mx_my2 -Lx2+ Ly Ly -Lj-

ko+ ks

+

k0

k8

Denote by Qs the sumof the fluxescorresponding to sources, sinks,and changeof storagein the two center elementsin Figure 5. Then the water balance for the mixed formulation becomes

Qs = Q1 + Q2 + Q3 + Q4

(21)

way, e.g.,

Inserting(20) into (21) yieldsexactlythe balanceequationfor TPo-

TP1

Q1--LykoLx/2

a FD

(17)

To obtain alsoidenticalflow velocities,it is of coursepossible to divide the cell by its diagonaland apply (1) in both triangles.However, it is easierto determinethe velocitycomponentsinsidethe cell just by a linear interpolationbetween

(18)

thevelocity components at thecelledges( Vxbetween-Q •/Ly andQ3/Ly, Vybetween-Q2/Lx andQ4/Lx) asproposed by

and

TP1-

TP•

Qi= Lyks Lx/2

cell.

As in the caseof generaltriangles,the two equationscan be solvedfor the head at the commonedge:

Pollock [1988]. In both casesthe velocitiesare free of divergenceif Qs = 0, but only in the latter case(Pollock'slinear

1908

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interpolation)are theyirrotationaltoo andthereforerepresent the exactsolutionwith respectto the four cell fluxes. Thusthe resultsof all MFE applications givenbyMos• et al. [1994]can equivalentlybe obtainedusingstandardFD requiring aboutthree timeslessunknowns.In the mixedapproachit is not taken into accountthat only the heads at the hypotenusesof the trianglesare degreesof freedom,while the heads at the two shorterelement edgesare linearly dependent.

H2

H3

Idea of Mixed Approach Using standardfinite elements(FE), the flow equation

V (kVH) = 0

COMMENTARY

Figure 6. Two nodal controlvolumes("patches").

(22)

is directlysolvedfor potentialheadsH at nodal points.In the can be interpolatedby (1). UsingFD models,water balance conventionalpostprocessing procedure,flow velocitiesare ob- fluxesacrosscell edgesare reconstructedaccordingto (20), tainedelementwiseby insertingthe headgradientinto Darcy's and a bilinear velocitydistributioninsidethe cell can be callaw: culatedby the methodof Pollock[1988].In standardFE models the solutionof (22) impliesa local water balancecorrev = -kVH (23) spondingto each node. In caseof linear triangularelements However,the normalcomponentsof velocitiesobtainedin this the controlvolumescan be interpretedas "patches"[Cordes fashionare not continuouson elementedges.This introduces and Kinzelbach,1992]aroundthe nodes(Figure6). The water divergenceand leadsto streamlineswhich intersectand cross balanceflux Q • in an element with nodal headsH•, H2, H 3 imperviousboundaries.Similarproblemsoccurin FD models, (Figure 6) can be reconstructedby Q• -- S•H• + S•2H 2 qwhen flow velocitiesare postprocessed in the classicalway S•3H3 (So is the elemental Galerkinstiffness coefficient obusing(23). tained by integratingthe scalarproductof the basisfunction Divergencecan be avoidedwhen velocitiesare not derived gradientsover the element area A; since the gradientsare from potentialheadsbut interpolatedbetweencontinuouswa- constant: So = kVN•VNiA). On the basisof thesewater ter balancefluxes.Therefore, in the mixed approach,contin- balancefluxesthe velocityfield insidethe patchcan be calcuuousfluxesacrosselementedgesare appliedand treated as lated accordingto Cordesand Kinzelbach[1992]. In all three additionaldegreesof freedom. cases(FV, FD, and FE) the velocityfield is free of divergence. Actually, the introductionof the mixedmethod for the simIn the specialcase of a two-dimensionalsteadystate flow ulation of groundwaterflow and the computationof stream- problemwith no internal sourcesor sinks,streamlines,which lines was motivatedby the belief that standardhead models provideno traveltime information,canbe constructed without provideno informationon continuousfluxes.However,this is computingthe velocityinsidethe controlvolume.This is posnot true. Continuous fluxes across discrete control volumes are siblebecausein any two-dimensionalflow domainwith known alreadyinherentin FD and FE solutionsof (22), because(22) boundaryfluxesthere is a direct correspondence betweenenis composedof both Darcy'slaw (23) and the water balance trance and exit point independentof flow detailsin the inte(zero divergenceof velocities) rior. In a rectangularFD cell with four boundaryfluxesthe locationof a streamlinecan be computedeasilyby a bilinear Vv=0 (24) streamfunctionformulation[Cordesand Kinzelbach,1992].A Thus FD and FE discretizationsof (22) alwaysimply that closedstreamfunctionformulationis not possiblein arbitrarily discretegroundwaterheadscorrespond,via Darcy'slaw (23), shapedcontrolvolumesof triangularfinite elements.Durlofsky to discretefluxesand that thesefluxessatisfythe water balance [1994]constructedstreamlinesinsidesuchcontrolvolumesby (24) at discretecontrolvolumes.Thereforethe introductionof simplyconnectingentrancepoint and corresponding exit point additionalwater balancefluxesis in somesensea "doublingof by a straightline. Figure 6 of Durlofsky[1994]showsthe draweffort" and leads to a set of unknowns, of which some are backof this approachin heterogeneous models:Insidecontrol linearly dependent, or which can be replaced by a much volumesof nodeswhich lie at the corner of low conductivity smallerset of unknownsyieldingidenticalor similarresults. areas the flow around the corner can of course not be modeled The continuouswater balancefluxesalreadypresentin stan- by a straightline throughthe controlvolume.Thusstreamlines dard FE and FD are usedfor the velocitycomputationsince tend to intersectcornersof low-conductivityareas. Such a behavior does not occur when velocities inside the Pollock [1988] and Cordesand Kinzelbach[1992]. The reason whythe fluxeshavenot beenconsideredbeforemightlie in the controlvolumesare calculatedaccordingto Cordesand Kinfact that usuallyonly the headsare recordedas resultsof the zelbach[1992].This can be seenin Figures5 and 7 of Mos• et numerical calculation. Yet, this makes sense because the al. [1994]wherethe flow aroundthe two low conductivity areas amountof data is smaller(standardFE and FD havelesshead is modeledcorrectly. nodesthan water balancefluxes)and the fluxescan be easily Note that Mos• et al. [1994] claim that our methodfails to reconstructed; i.e., the flux betweentwo neighboringnodesi describethe flow around the low-conductivity zonesproperly andj is equalto their head differencetimesthe globalstiffness becausethe streamlinesin the upper and lower half of Figures coefficient So. 5 and 7 are not symmetricas in the "exact" solution[Mos• et al., 1994,Figure4]. In fact, in the lowerhalf of their Figures5 Computation of Velocity Field and 7 the streamlinesare locatedleft from the onesin Figure In triangularfinite volumeswith headvaluesat the circum- 4; however,Mos6 et al. did not explainwhy: the streamlines centersthe edgefluxesare obtainedby (12) and flowvelocities already start more left in the upper half. This can be seen

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COMMENTARY

1909

clearlyat the right boundaryof the domain.Using a ruler, the taking the harmonicmean. In triangular elements,nodewise reader can easilycheckthat the startingpointsof all stream- conductivitiescorrespondto a constantconductivityvalue in linesin Figure 5 of Mos6 et al. showa displacementof about eachnodalcontrolvolume[e.g.,Narasimhanand Witherspoon, 0.5 mm to the left with respectto the onesin their Figure 4. 1976]. Under equalconditions,i.e., by comparingstreamlineswhich start at the same point (for example,the rightmostline in Acknowledgment.This work was supportedby the EuropeanEnFigure 5 and the secondline from the right in Figure 4), no vironmentalResearchOrganization(EERO). differences can be observed between the solution obtained with our method

and "exact"

one.

References

Mos• et al. [1994]alsocomparedresultsof MFE (whichare identicalto the onesof FD) and FE for heterogeneous cases. Cordes, C., and W. Kinzelbach, Continuous groundwater velocity fieldsandpathlinesin linear,bilinear,andtrilinearfinite elements, They found large differencesbut did not mentionthe reason. WaterResour.Res.,28(11), 2903-2911,1992. The stiffnesscoefficientbetweentwo neighboringFD-cellsis Durlofsky,L. J., Accuracyof mixedand controlvolumefinite element

obtainedby taking the harmonicmean of the two conductivity values.In linear FE the globalstiffnesscoefficientcorresponding to an elementedgeis found by simplyaddingthe contributions of the two adjacent elements,which amountsto an arithmetic weighting. Therefore, in stronglyheterogeneous models,FD approachesgenerallyunderestimatethe effective conductivity,while standardFE overestimateit. Lachassagne et al. [1989,Figure1] showthat in a sufficiently largemodelwith a lognormaldistributionof random conductivities,both approachesFD and FE yield aboutthe sameabsolutedeviation from the theoreticalsolution.If the model is too small (e.g.,

only 10 x 10 zonesasusedby Mos• et al. [1994])or if correlationsoccur,it is of coursepossibleto identify situationsin which

the exact solution

lies closer either

to FD

or to FE

results.

approximations to Darcy velocityand related quantities,WaterResour.Res.,30(4), 965-973, 1994. Forsyth,P. A., A controlvolume finite element approachto NAPL groundwatercontamination,SIAM J. Sci. Stat. Cornput.,12(5), 1029-1057, 1991.

Lachassagne, P., E. Ledoux,and G. de Marsily, Evaluationof hydrogeologicalparametersin heterogeneous porousmedia,in Groundwater Management:Quantityand Quality (Proceedings of the Benidorm Symposium,October1989), IAHS Publ., 188, 3-18, 1989. Mos6, R., P. Siegel,P. Ackerer, and G. Chavent,Applicationof the mixed finite elementapproximationin a groundwaterflow model: Luxuryor necessity?, WaterResour.Res.,30(11), 3001-3012, 1994. Narasimhan,T. N., and P. A. Witherspoon,An integratedfinite difference method for analyzingfluid flow in porous media, Water Resour.Res.,12(1), 57-64, 1976. Pollock, D. W., Semianalyticalcomputationof pathlinesfor finite differencemodels,GroundWater,26(6), 743-750, 1988.

In any case,the different treatment of heterogeneitiesis no criterionto rejecteither of the methods.In fact, the weighting procedureis independentfrom the choiceof FD, FV, and FE. In all three approachesthe stiffnesscoefficientscan be obtained by both harmonic and arithmetic weighting,i.e., by switchingfrom an elementwiseor cellwisedefinitionof conductivitiesto applyingthesevaluesto the verticesof element

C. Cordes,Departmentof MathematicalMethodsfor Applied Sciences,Universityof Padua,35131Padua,Italy. (e-mail:cordes@lewy. dmsa.unipd.it) W. Kinzelbach,Institute of Hydromechanicsand Water Resources Management,SwissFederalInstituteof Technology,ETH-Hoenggerberg, CH-8093 Ziirich, Switzerland. (e-mail: kinzelbach@ihw. baum.ethz.ch)

and cells. In the latter case, FD and FV coefficients are

(ReceivedMay 19, 1995;revisedSeptember14, 1995; acceptedFebruary16, 1996.)

weightedarithmeticallyand FE coefficientsare weightedby