j

COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 70 (1988) 203-235NORTH-HOLLAND

AN hop ADAPTIVE FINITE ELEMENT METHOD FOR THENUMERICAL SIMULATION OF COMPRESSIBLE FLOW*

Philippe DEVLOO. J. Tinsley aDEN and Paresh PATIANITexas Illstitllte for Compwatio/la/ Mechanics. The University of Texas at Austin.

Austin, TX 78712-1085. U.S.A.

An h-p adaptive finite element method is presented for the numerical simulation of compressiblefluid flow. In addition to refining the mesh in regions where the estimated error is large. the polynomialdegree of the elements is increased. It is shown that by using high-order polynomials highly accurateapproximations are obtained to a regularized form of the equations governing viscous compressibleflow.

1. Introduction

Adaptive finite element techniques provide attractive approaches to many classes ofproblems in computational fluid dynamics. Foremost is their possible use in optimizing thecomputational process: the production of the best numerical simulation of a complex flow forthe least computational effort. Moreover, such approaches provide a high level of flexibility inmodeling and algorithm development and can, in theory, also provide quantitative estimatesof the accuracy of computed solutions to practical problcms. A survey of the literature onadaptive finite clement methods has been recently compilcd by aden and Demkowicz [30];see also the survey on adaptive grid schemes by EiserT1an [14] .

The basic idea underlying most adaptive methods is to assess the quality of an initial coarsemesh solution of a given problem by employing some type of a posteriori error estimate and tothen change the structure of the approximation in some systematic way so as to improve thequality of the solution. Among methods used for improving the quality of solutions areh-methods, in which the mesh is refined, p-methods, in which the local degree of thepolynomial shapefunction is increased, and moving grid methods, in which the number ofnodes is held constant and the mesh is distorted to relocate grid points in areas of high error.In earlier papers, we have presented effective II-methods for compressible flow problems [8,12, 28, 29}; p-methods are very closely related to spectral element methods, which have beenapplied to incompressible, viscous steady flow in our paper [29), and by Patera and hisassociates [26, 31]. Combined hop methods have been proposed for linear elliptic boundaryvalue problems by Guo and Babuska [16], but to date have not be~n employed for complexproblems in fluid dynamics. However, the judicious use of hop methodologies may well helpresolve some of the difficult computational issues encountered in modern computational fluid

• The support of this work by the Office of Navat Research under contract NOOOI4-84-K-0409 is gratefullyack nowledged.

0045-7825/88/$3.50 © 1988. Elsevier Science Publishers B.Y. (North-Holland)

204 Ph. Devloo et al., hop adaptive FEM for compressible flow simulation

dynamics: the simultancous rcsolution of shocks alld singularities and thill boundary layerswith models having a manageable number of degrees of freedom.

In this paper, we present a combined hop adaptive strategy for accurately modeling theNavier-Stokes equations of compressible fluid flow in two dimensions. The strategy used inthe adaptive h-p procedure is similar to the one used for adaptive h-relinemcnt proceuures.First, -the mesh is refined to a specified level in both h- and p-parameters for a regionsurrounding the no-slip boundary. This provides a first approximation of the viscous boundarylayer'region where higher resolution of flow features is required. Next, the optimal h-mesh isobtained adaptively for the rest of the domain using the linear elements distributed so as toequidistribute the error measures over all elements. This pass generally resolves the shock-type region or the region of discontinuity within the limits of the available computer resources.In the present study, an upwindcd-diffusion Petrov-Galerkin scheme is used to producc arelaxed (or regularized) model of the compressible Navier-Stokes equations for two-dimen-sional domains.

The paper is divided in four sections. Following this introduction, notations and pre-liminaries are introduced and the Petrov-Galerkin modef of compressible flow is derived.Next, some details of the hop adaptive scheme are discussed, including the specific algorithmused in this implementation. Next, the implementation of the adaptive hop refinementtechnique is described. In the fourth section, several examples are discussed to demonstratethe use of the h-p adaptive method.

2. Preliminaries and governing equations

2. J. The Navier-Stokes equations for compressible flow

Throughout this work, we shall be concerned with the numerical analysis of high-speed,viscous, compressible flow in arbitrary two-dimensional domains. It is gencrally viewed that anacceptable mathematical model for such flow phenomena is embodied in the Navier-Stokcsequations, which can be written in the compact conservation form,

a,v +divlF=O \I(x,t)EDx(O,I/.),dt .

with boundary and initial conditions, e.g.,

Vex, 0) = Vo(x) \Ix ED,

vex, t)lro = VD(x) \Ix E TD '

II • Il = ° \Ix E TN

u=O }e = elJTs(x) (2.1)

where D is the llow domain, (0. t F) is a time interval of interest, x = (x, y) is a point in 1R2, t isthe time parameter, t ~ 0, V is the vector of conservation variables, V = [p, pu, pv, pe]\ p isthe fluid density, u=(u,u) is the flow velocity. pe is the total energy, pe=(lI(y-l))P+

Ph. Devloo el til .. h-p adaptive FEM for compressible flow simulatioll 205

!p(U2 + v2), p is the thermodynamic pressure, 'Y is the ideal gas constant (ratio of specificheats), 11 is the outward normal to the boundary of fl, an is the boundary of n, ro c an is thepart of an on which Dirichlet conditions are applied. TN c'an is the part of an on whichno-Ilow boundcuy conditions are applicd, r;, C an is the part of an on which no-slip boundaryconditions are applied, and IF is the two-dimensional flux tensor with four components.

The flux can be expressed as a pair of flux vectors,

IF = (E, F) ,

[

pu ]2E = P u + P - 'Tn

puu - Tn '

(pC + p)1I - IITxx':"" UTx)' + lfx

[

pu ]puu - Tx)'F= 2 .

pV + P - Tyy

(pe + p)U - UTx)' - VT)')' + qy

(2.2)

Here, we use the notation Tu' Txy, Tyy for components of the viscous stresses, and qx' qy forcomponents of conductive heat flux.

For an isotropic heat-conducting Stokesian tluid, the stresses are given in terms of velocitygradients by

(ali au)-2 2---,~u: - 3 µ ax ax , (au ali)

Tyy = 5 µ 2 ay - ax ' (au au)Txy = µ ay + ax ' (2.3)

while the heat flux satisfies Fourier's law,

(2.4)

with µ the fluid viscosity and k the coefficient of thermal conductivity.Let rg denote thc space-timc domain, C{j = n x (0, I,..). For our purpose, we choose to

rewrite the governing equations in the following weak form.Find V E V(W) such that Vcp E cP(W),

- L (:~)IV dx dl - f<J} Vcp: IFdx dl + fro cpllF' 11 dw = 0 , (2.5)

togethcr with conditions

206 Ph. Dev/oo et al.. hop adaptive FEM for compressible flow simulation

u(x, 0) = 1l1l(X)

u(x, t) = 1i[)(X)

d Ii' R= 0

VxEfl,

Vx E l~) ,

VxEJ~,

VxEr.<;.

where V(cg) is the space of trial functi~nsii'hd cJ>(cg) is the space of test functions, and it isunderstood that 'Yep: IF= trace(VepllF). -

We are also interested in studying several special cases of (2.5). These include the inviscidcase (i.e., the Euler equations). Then (2.5) still holds with appropriate definitions of V(W) andcJ>(cg) but it is understo<?d that

It is well known that for hyperbolic systems such as the Euler equations, discontinuoussolutions U can exist. Moreover, without additional assumptions, solutions need not be unique(see [24]). We shall attempt to limit ourselves to seeking solutions to the weak form of theEuler equation that satisfy an entropy condition. In one space dimension, it can be shown thatsuch a solution is unique.

We are also interested in wnvection-diffusion problems in which we seck a scalar-valucdfunction g satisfying

g, + f(g)x + sgxx = 0g(x. 0) = gll(x)

g(x, r) = gll(x)

VxEfl, Vt>O,

VxE12 .

V x E I~) ,

(2.0)

where go(x) is thc initial value and gn(x) is the Dirichlet value of g on rD'

2.2. Numerical algorithm

The above equations will be approximated by a version of the SUPG algorithm, theimplementation of which follows closely earlier work by Mallet [27].

Let Vir denote an appropriate approximation space of trial and test functions. Generally, Virwill be spanned by a set of piecewise polynomial basis functions defined over a partition L1offl into finite elements {flJ,_2 ~ e ~ E(L1):

If

- -il = U{l~ ,

L1x. = dia(fl,,) ,

fl" n ill = 0, e ~ f .

we may introduce the piecewise constant mesh function

h = hex) , hex) = L1x. for x E fl" .

Ph. Devloo et al .. hop adaptive FEM for compressible flow simulatioll 207

Wilh lhese not;llions ill force, the sle<ldy-st<lte <lpproximatL: solution of <I two-dimensionalquasilinear hyperbolic system of conservation laws using the above algorithms will satisfy thefollowing algorithm.

SU PC J1/gorithmFind V" E V" sllch Ih:11Vcp E V".

- in Vtp: f(U") dx

CFL i all+ _..Ix II atp A /, , div F(U") dxn x \ A- + B

CFL..I t 1 a tp I 1 . (") 1 I ( " )+ 2' -a B V 2 2 dlV F U dx + tp F U . 11 dwn Y A +B ro

CFL..I t 1 (I 1 . (" ) .+ -' htp'A v' , dlV F U flrro A-+B .

+ htpl

8 VA2~ n2 div F(U")ny) dw

= 0 + appropriate boundary conditions, (2.7)

where CFL..\x is an upwind parameter (the spatial CFL number), and (VA2 + B2)-1 is thematrix whose inverse square is A2 + B2

• Here A = aE/au = A(Uh), B = aF/au = B(Vh

).

The underlying relaxed equation is

, CFL..IX ( 1 .) CFL..\x ( .1 .)dlv IF - - hA / 2 , dlv IF - 2 hB V' , dlV IF = 0 .

'\ A + B- x . A- + B- y

The remainder of our investigation shall focus on the numerical solution of (2.7).

3. Adaptive procedures

(2.8)

3.1. Introduction

In this section wc discllss tbe implementation of both the adaptive h- and hop refinementprocedures. It is shown that the entire adaptive method implementation (both h- andp-refinement) hinges on the gcneration and use of one device: an array that keeps track ofelement neighbors. The adaptive p-refinement is implemented using tensor products ofone-dimensional Chebyshev polynomials.

3.1.1. Adaptive refinement strategiesThe numcrical viscosity terms in (2.7) (those involving the parameter CFL.1x) provide an

indication of the local approximation crror. These tcrms, whicb are similar to a truncation

208 Ph. Devloo et al .. hop adaptive FEM for compressible flow simulation

error in the Petrov-Galerkin approximation, provide a conveniently calculated local errorindicator and arc used as a basis for mesh refincment in this study. Thus, at node i in a mesh·,we have

I 1 .It = CFL.h 1 h alp; A 2 _ dlv f dxIlee"II; 2 n ax VA +B

1 1CFL.:1x ( "alp; B _ ~ divlFdx+bc.

+ - JJI ay VA2 + B~(3.1 )

Here IIe~SlIl; is the nodal value of the error measure and A, B, and F are computed using thefinite element solution Ult of (2.7).

The underlying idea is that as the mesh size is decreased, the magnitude of this viscousterms will also decrease. Roughly, this gives an estimate of second derivatives at the nodes.The element error indicator is then taken to be the maximum of the nodal values of error forelement K.

3.1.2. Steady state, maximum level imposedOur adaptive procedure is initiated by first generating a coarse initial finite element grid on

the flow domain fl. Elements in this coarsest grid are assigned a level zero, and subsequentrefinements produce elements of higher level. The particular h-method strategy employedhere is essentially the same as that we described in our earlier work [12]: the initial mesh isbisected so that the number of level-one elements is (at most) four times the number oflevel-zero elements. When a particular element is refined, the four new elements have alevel-one unit higher than their parent clement. This implies that if we impose a restriction onthe highcst Icvcl of any clcment, then we also deline the maximum -number of elements in thegrid. This procedure is especially suitable when the size of the smallest element in the grid isrestricted. The refinement algorithm then consists of the following steps.

Step 1. Build a coarse mesh, compute an initial solution.Step 2. Estimate the error ovcr the mesh.Step 3. Compute emnx' the maximum element error over the mesh.Step 4. Refine all elements K for which

• eK ~ Oemax (0 is the assigned tolerance, 0 < 0 < 1), and• LEVEL(K) < MAXLEVEL.

Step 5. Compute the new solution on the refined mesh.Step 6. If:

• the number of elements that were refined in Step 4> 0, go to Step 2;• the number of elements that were refined in Step 4 = 0, stop.

This algorithm ensures that the area of the finest element in the grid is bounded below andthat the maximum numher of elements is limitcd. Moreover, if e K > Oem".' then Icvcl(K) = maxlevel.

Ph. Devloo e( al., h-p adalnil'(' FEM for compressible Jlow simlllation 209

3.1.3. Steady-state, h-p adaptive refinement procedureAs a lirsl approach 10 adaptive Ii-p n:lillcmcllt, wc proposc a prm.:edure which attcmpts a

p-cnrichment only in the region surrounding the no-slip boundary. In this sense. we hope toconcentrate the higher-order elements in the boundary layer region where we expect theyprovide the most effective resolutions of the flow. As some of the advantages the p-enrichmcnt has ovcr the h-rellncment. wc list:• Thc convcrgencc ratc of mcthods cmpl_oyipg higher-order clements is gcner;lIly µ,realn tll:11l

Iltal or relincd lilll;;lr Clcllll':llls. . .,.~;• Enrichcd c1emcnts need kss integration points per computed degrce of freedom:

- A quadratic' element requires 9 integration points, an equivalent composite of 4 linearclements requires 16 integration points.

-A cubic element requires 16 integration points, an equivalent composite of 9 linearelements requires 36 integration points; etc.

We propose the following h-p refinement procedure.

Step 1. Build a coarse mesh.Step 2. Refine the mesh to a maximum level in both h- and p-fashion for a region

surrounding the no-slip boundary.Slep 1. Compute an initial solutionStep 4. Derive an error estimatc over ·the mesh.Step 5. Compute emax' the maximum error over the mesh.Step 6. Refine all linear elements for which

• eK~Bem"x·O~B~1.• LEVEL(K) < MAXLEVEL.

Step 7. Compute the new solution.Step 8. If:

• the number of elements refined in Step 6> O. go to Step 4:• the number of elements modified = 0, stop.

This refinement procedure ensures that the area of the finest element in the grid is boundedbelow and that the maximum number of elements is limited. Moreover, if eK > Bem•x, then:(1) LEVEL(K) = MAXLEVEL. and pol degr K = max(pol degr) in the region surroundingthe no-slip boundary, or (2) LEVEL(K) = MAXLEVEL.

3.2. An h-method

The data structure and details about practical implementation of the h-method have beendiscussed in our previous paper [12]. Here we present a brief overview. The regular finiteclement codes generally cmploy the following data structure:

• NODES(J, NEL) is the global node number of node J of element NEL;• XCO(JCO, NODG) is the JCO coordinate of global node NODG;• NNODE, NELEM are the numbcr of nodes and number of elements in the grid.

The following additional arrays are required for refinement:

2tO Ph. Devloo et III., h-p adaptive FEM for compressible flow simlllmiofl

• NELCON(NC, NEL) is the NOh connection of element NEL;• LEVEL(NEL) is the lcvel of element NEL (only if maxlev is imposed) .

The NELCON:array refers to neighboring elements of a particular element. Figure 1 showsthe connectivity' numbering of a particular element. As one side of an element may beconnected to two different elements. the dimension of NELCON is 8 instead of 4.

This rather simple data structure is c..:asilycimplemcntcd in a vcctorizable working code. Thestorage requirements are comparable with otheri~chemes [4, 25]. It also has the advantage thatit does not require any type of tree structure Un general, tree structures are used to defineconnectivity). As the classical data structure of the finite element program is still used, thisadaptivc refinement scheme can bc easily inscrtcd into existing finitc c1cmcnt codes.

As the solution marches toward the steady state it may be necessary to unrefine the part ofthe grid which was refined earlier. This is done to equidistribute the error indicator. In orderto guarantee that the lInrefinement occurs (e.g. see Fig. 2) in the same way as the refinement,we add an array which reflects the refinement history of the mesh:

NELGRP(I,IGR): the /th element of group IGR:if > 0, refers to an element,if < 0, refers to a group.

In this sense, we build a data structure of groups which contain elements or groups. When anelement is refined, it is transformed into a group which contains the four new sllbelements. Allthe references to the original element are changed to the new group.

7 3

46

A

8_2

1

-~

5

Fig. I. COllllcl:livity of all clement. Fig. 2. Example of a bad refinement.

Ph. Devloo et al., hop adaptive FEM for compressible flow simulation 211

Tn order to keep track of the deleted clements, nodes. and p,roups, we cre<ltc lists of freeelcnH.:nts, free nodes, and free groups. In this way we cun avoid compressing the dutastructure each time a group of elements is unrefined. A free element is also characterized byzero NODES entrics and a free group is characterized by zero NELGRP entries.

3.3. A pCfnethod for viscous flow

This section will describe an implemcntation procedure for elements of arbitrary polynomialciegree. No additional global data structure is needed for the p-refinement procedure. Thereare also no limitations on the difference in polynomial degree between two adjacent elements.In this sense, p-refinements are easier to implement than h-refinements. Denote: MXNDF asthe number of degrees of freedom considered in the problem, MXPL as the maximum degreeof polynomial implemented in the program.

Then only the dimensions of the NODES and GF arrays need to be augmented to

NODES(MXPL *MXPL,MAXEL) ,

GF(MXNDF, MXPL *MXPL,MAXEL) ,

where GF(J, J. NEL) contains the entry for the right-hand side of element NEL for Ith degreeof freedom for the 1th node (including the higher-order nodes).

3.3.1. Element shape functionsThe element shape functions are computed as the tensor product of one-dimensional

Chebyshev polynomials.The Chebyshev polynomials are defined recursively as

To = 1 , TI = x,

T"+1(X)=2xT,,(x)- T"_I(x) , n~2.

In this work, we modify the Chebyshev polynomial slightly such that T~(-1) = T:,(1) = 0,n>2,

T* = I - x T* = 1+ xo 2' I 2

T:'(x) = T,,(x) - T~(x) - (-1)"T~(x) .

The Chebyshev polynomial were chosen for the following reasons:(1) They arc economical to compute. As they are computed recursively, each additional

polynomial requires only three floating point operations.(2) They are bounded, i.e., II T~ (x)lI.s; 2, 'Vx E [-1,1], 'Vn > O.(3) They are hierarchical in the sense that element matrices corresponding to higher-degree

polynomials contain as submatrices those corresponding to polynomials for any lowerdegree. Hierarchical shapefunctions generally result in better conditioned matrices,better convergence rates, and will facilitate the implementation of adaptive p-refinement.

212 Ph. Devloo et al .. hop adaptive FEM for compressible flow simulation

The shapefunctions for a rectangular element are of the form

The computation of the values of a shapefunction at an integration point ({, 1]) cansymbolically be written in the following fashion:

P = degree of the polynomialPSIK = shapefunction values in the {-direction

DPS1K= derivatives of PSIKPSIL = shapefunction values in the 1J-direction

DPSIL = derivatives of PSILSHAPE = subroutine that computes the values of the shapefunction and its deriva-

tives at a (one-dimensional) integration point

CALL SHAPE ({, P, PSIK, DPSIK)CALL SHAPE (1], P. PS1L, OPSIL)DO 2 L = 1, P + 1002 K= 1, P + 1

C1= shapefunction index corresponding to the

Lth shapefunction in the (-directionand the Kth shapefunction in the 1]-direction

CPSI(I) = PSIL(L )*PSIL(K)DPSI(l, !) = DPSIL(L )*PSI K(K)DPSI(2.!) = PSIL(L)*DPSIK(K)

2 CONTINUE

where

PSI(J) = SHAPEFUNCT10N VALUE OF NODE I AT ({, 71)DPSI(1, I) = (-derivative of PS1(I)DPSI(2, I) = 1]-DERIvATIvE OF PSI(I) .

3.3.2. Nodal numbering schemeAs the polynomial degree of an element can be arbitrary, a nodel numbering convention

must be established. We use the following convention:(1) nodes 1 through 4 are numbered according to the usual element convention (see, e.g.,

Fig. 3);(2) for polynomial degrees quadratic to highest order,

• the first four nodes are assigned to the sides in an antic10ckwise direction,• the remaining nodes are assigned to internal degrees of freedom.

It is important to /lote thal. as the shapdunclions arc hierarchical, the shapcfunctiuncorresponding to a particular node number will be the same for every type of element,

Ph. Dc:v/oo /!t al .. hop adaptil'/! FEM for compressible flow simulatiotJ 213

4

813

121

, - .. -~o··.

14,15;16

510

3

611

2

Fig. 3. NoLial numbering for a cubic clement.

regardless of the polynomial degree of the element. Figure 3 illustrates the nodel numberingfor a cubic element. Figure 4 illustrates a sample quadratic shape function.

The connection between the one-dimensional shapefunctions and the two-dimensionalshapcfulll;tiolls is estahlished hy straightforward Boulean transfurmations.

3.4. lmplemelllation of an adaptive p-enrichmelll

Consider a finite element grid with elements of arbitrary degrees p, as shown in Fig. 5. Weimpose the following constraints: the degree of the polynomial on the side of any element is

'--/-------~._----. -.. -..---- /

j--------.--.-J1 ___

-- --:----.-- 7- _

2

Fig. 4. Example of a shape function for a quadratic element.

3

214 Ph. Devloo et al., hop adaptive FEM for compressible flow simulation

p=:J p=2 - - - -

p=l p=4 p=2 - -- --- -.

p =3 p=2 p=1 - -

p=l - - - - - -

Fig. 5. Sample p-enriched mesh.

equal to the minimal polynomial degree of the element on either side, i.e., if a lincar elcmentis adjacent to a cubic element, then the polynomial degree on the side between both is linear.

If the polynomial degree of the side of an element is lower than the polynomial degree ofthe element itself. then thc highcr-order nodcs of the element on that side 'need to beconstrained. This is where the hierarchical shape functions are most beneficial.

Due to the hicrarchical naturc of the polynomials, it is sufficicnt to constrain all higher-order nodes to zero. This constraint can be imposed in two ways:

(1) Construct a list of constrained nodes, and constrain the nodal values and right-handside aftcr cach iteration.

(2) Assign a 'special' node number (e.g., zero) to the constrained nodes. When loopingover the nodes of the element, check whether the node-number is equal to the specialnode number. If so, ignore the values associated to that node.

In this study, the second method was employed beca.use its implementation is simpler andrequires less storage.

The implementation of a p-enrichment routine can be described in thc following steps:Step 1. Determine the polynomial degree of the enriched element.Step 2. Generate new node numbers for the additional internal degrees of freedom .Step 3. Loop over the sides of the element:

• if the polynomial degrce of the neighboring element is greater than or equal tothe new polynomi'tl degree of the enriched element, then generate a new sidenode for the enriched element and for the side element; otherwise, set the newside node number equal to the 'special' node number.

3.5. Higher-order constraints for hop refinement procedures

A differcnt issue that must be overcome in implementing hop methods is the handling of thetransition between elements of different h-level or different polynomial degree. For example,in the situation illustrated in Fig. 6, one must develop 'constraints' on various polynomial

Ph. Dev/oo et al" hop adaptive FEM for compressible flow simulation

p=l p.. 1

p=4 p=1

p=l p .. 1

Fig. 6. A transition from a p-Ievel of 4 to 1 across elements which have different h-tevels.

215

degrees to effect a transition or interface between polynomials of degree 4 and polynomials ofdegree 1. This must be done in a way that preserves continuity of the global finite elementbasis function across interclcment boundaries.

Here we describe a simple procedure for constructing a higher-order constraint at interele-ment boundary. The method is easily implemented on meshes on which tensor products ofChebyshev polynomials are used as shape functions, as is done here. It provides for continuityacross element boundaries without loss in polynomial degree. The effect of this process isillustrated in Fig. 7, where it is observed that for the same refinement objective shown in Fig.6 quartic transitions are lIscd ovcr a rcgion of smaller clements between c1emcnts of the sameh-level but of differing local polynomial degree. Now only polynomials on interfaces involvinglinear (or bilinear) elements are constrained to be linear polynomials (such as those elementswith boundaries on line AB in Fig. 6), while elsewhere full continuity and full degree ispreserved on element interfaccs.

3.5.1. Problem statemetllGiven a function which is defined as a linear combination of Chebyshev polynomials over

an intcrval [0, 1]. what arc the coefficients of the Chebyshev polynomials for the same functionwhen these Chebyshev polynomials are orthogonal with respect to an interval [a, b].

Concisely. the problem statement can be rewritten as: given f(x) = E7=0 a;'T;(x), find 13;suchthat f(x) = E~'=O 13;L;(x), where T;(x) is the Chebyshev polynomial of order i defined over[0,1], and L;(x) is the Chebyshev polynomial of order i defined over [a, b].

A

p..4 p-1

p=4 p=1

p=4 p .. 1

B

Fig. 7. Higher-order transition.

216 Ph. Devloo et al .. hop adaptive FEM for compressible flow simulation

The solution can be found by wfltmg each Chebyshev polynomial T;(x) as a linearcombination of the Chebyshev polynomials L;(x):

;

T;(x) = L X;kLk(X) ;k=O

then

" " i

f(x) = L Cl;T;(x) = L L Cl;X;kLk(X) .;=\ ;=0 k=O

Therefore

"13k= L Cl;X;k .

;=0

Thus, the problem is reduced to determining X;k'

3.5.2. Recursive relations for Chebyshev polynomialsIn order to determine X;k' we will relate both recursive relations for T;(x) and L;(x). The

recursive relations can be found in [1]. For T/(x) , the recursion relation is given by

1'_\(x) = -1 + 2x , To(x) = 1,

TII+I(x)=(-2+4x)T,,- T"_I(x) Vll~O.

The recursive relation for L;(x) can be found by substituting the identity

L.(x) = j'.(~), , b - a

in the recursive relation for T;(x):

L_1(x) = -1 + 2 x - ab-a'

(x-a)L,,+,(X)= -2+4 b-a L,,(x)-L,,_,(x) VIl ~ 0 .

3.5.3. Derivation of X;k

From the recursive relation for T;(x) and L;(x) we will derive a recursive relation for X;k'

Define .1 = b - a,

1'n(x) = Ln(x) ,

1'1(x) = .1LI (x) + (-1 + L1+ 2a)Lo(x) ;

therefore

Ph. Devloo e/ al .. hop adap/i1'e FEM for compressihle flow simula/io/l

Xlln = 1,

XUi = U Vi >0,

- XI\l =(-1 + L1+ 2a) ,

XII = L1,

XI;=O Vi>O.

Assume we know

i

Ti(x) = L XikL,,(X) Vi ~ n ,"cO

we will derive X,,+I.k VO~k~n + 1:

T,,+,(X) = (-2+ 4x)T,,(x) - Tn_l(x),

This can be rewritten as:

Tn+,(X) =(-2+4 (x-a»)' l' ( )_tn_I(X) +(-2 +2+ 4a)t ()L1 L1 "x L1 L1 L1 n

X

( (x-a»).s Tn_l(x) (-2 4a)-= -2+4 L1 ~,Xn"L,,(x)- L1 + T+2+"L\ T,,(x)

.s {( (x - a)) }= ,,£:nX"k -2+4 L1 L,,(x)-L,,_,(x)

~ () T,,_,(x) (-2 4a) - ( )+ k~1I X"kLk-1 x - L1 + T + 2 + L1 Tn X

n

= 2: X""Lk+I(X)k-ll

+ ~ _ T"_I(x) (-2 4a)-k~IXnkLk-I(X)- L1 + Ll'+2+"L\ Tn(x),

where we substituted the recursive value for L" + I(x).Therefore

II 11

Tn+,(x) = L1 2: X"kLk+I(X) + L1 L X"kL"_I(X)k-n "-0

" II- L X,,_uLI;(x)+(-2+2L1+4a) L X""L,,(x),k-ll k-n

which fully determines X" + 1.k; k = 1- n + 1.

217

218 Ph. Devloo et al.. hop adaptive FEM for compressible flow simulation

Four cases need to be distinguished:

for k = n + 1: X,,+l.k = LlX,III ;

for n-:> k> 1: X,,+ I.k = LlX".k-' + LlX".k+' - X,,-I.k + (-2 + 2.1 + 4a)X"k ;

for k = 1: X,,+J.k = 2LlX".k_1 + LlX",k+l - X,,-l ,k + (-2 + 2.1 + 4a)X"k ;

for k = 0: X,d I.k = L\X".ki'; :-'X~{1.k + (-2 + 2.1 + 4a)Xllk •'co,

3.5.4. Extension to finite element shapefunctionsIn finite element computations, we use modified Chebyshev polynomials. Therefore, the

transformation matrices from Chebyshev polynomials to finite element shapefunctions and itsinverse need to be derived.

Denote by T. the ith shapefunction defined over [0,1], and by Ti the ith shapefunctionI ~ _

defined over [a, b], then the relation between Ti and Ti can be written as

To(x) = To(x) - T, (x)2 '

Til (x) = '/'II(X) - I ~ll(X)lTI (x)

n even,

n odd.

In matrix form, this relation can be written as

1 _12 21 12 2

-}[T] = I, -1 } I[T] = Till T] .

-1

On the other hand

Lo(x) = To + TI , L. (x) = - To + Tl '

( ) = - { + To + 1'1 n even,L" x Til (x) + - -

- Tn + T, n odd

which call alsu be written ill matrix furm as

1 1-1 1

1 1 1[LJ=/ -1 1 1 I[t] = T;I[T].

1 1 1

(3.2)

(3.3)

Ph. Devloo et al .. h-p adaptive FEM for compressible flow simulation

Therefore

4. Numcrical rcsults

219

In this section, we discuss numerical examples used to demonstrate the features and theimplementation of the algorithms presented earlier. It is not practical to test every feature.Some of the large-scale two-dimensional runs took several days (weeks) to complete on theMicro-Vax computer available. During calculations, the program was regularly interrupted toimplement improvements on the algorithm, improve graphics output, etc.

4.1. One-dimensional steady shock problem

The problem statement is depicted graphically in Fig. 8. The exact solution is equal to twoconstant solution ordinates separated by a discontinuity at an arbitrary point in the domain.The algorithm used corresponds to the supa algorithm described in Section 2.2 foronc-dimcnsional quasilincar systems of cquations. Thc stcady-state finite clcmcnt solutionsatisfied

CFL (!!. oifl; I AI oUr. dx - ( o~: IF(Ur.) dx = 0 V . E V"(n) .Ax In 2 ax ax In ax ifl, (4.1)

No flux interpolation was used in this case. Figure 9 shows an initial solution to this problemusing 20 linear elements and CFL.1X = 1. This figure presents the computed density. momen-tum, and energy. The quality of the approximation is excellent: the discontinuity is capturedover one element and there is hardly any oscillation. Thespike in the momentum equation isonly 9 percent higher than the exact solution.

U

[1 ]U = 1

J. ~.9463

oFig. 8. One-dimensional steady shoek problem.

[

2,666 ]Un= 1

~.19642

220 Ph. Devloo et al., h-p adaptive FEM for compressible flow simu/atioll

2.4 ii' I 3.0, iii 1.12 i , i i

1.20.8x

1

0.4

2.5 1.08

P Ipu

2.0 1.04

1.5 1.0

A I ,~

I I 1.0' I , I 0.960.4 0.8 1.2 0.0 0.4 0.8 1.2 0.0

x x

2.0

0.8' I , I

0.0

1.2

pc1.6

Fig. I). Finite clement approximation of 1D shock, 20 linc~r clements.

Next, we examine the convergence of the finite element solution to the relaxed equation byconsidering both uniform h- and uniform p-refinement with the same underlying relaxedequation. Figures 10-13 show the sequence of solutions using the uniform h-refinement.Figures 14-17 show the sequence of solutions obtained with the uniform p-refinement. As canbe observed, both series of solutions converge to the same relaxed solution.

We note that the results shown in Figs. 14-17 are of particular interest in that theyrepresent a departure from traditional approaches to shock problems. The results indicate thatshocks can indeed be captured by higher-order finite element methods and that oscillationsnear the shock vanish as p is increased. In contrast to some spectral' methods for suchproblems. no artificial viscosity terms are added to the discrete Euler equation since adequatedissipation is inherent in the upwinded formulations.

2.4 I I i I 3.0, , I I 1.121 I I

2.0 \- r ~ 2.5 1.08

po ~ I 1p pu

1.6 2.0 1.04

I uJ~1.2r I -I 1.51- 1 -I 1.0-'

rlllllll'III"~.I I 1.011l1l11l1ll11llJ I I 0.961 I ,0.8

0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2x x x

Fig. to. Finite elemcnt approximation of 1D shock. 40 linear clements.

Ph. Devloo et 01 .. hop adaptive FEM for compressible flow simulation 2212.4

'orI I ] 1.Uj

.r- :2.0 I-

1.102.5

pc l p

2.ol

~

pu

1.6 1.05

L2~".M") j 1.5 i 1.00

I 1.0 0.950.80.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2

x x x

Fig. II. Finite element approximation of ID shock. 60 linear elements.

2.4

1

I I

I3.01 I I

I1.15

r-'or ] p "[ f ]1.10

pc pu1.6 2.0 1.05

X

Finite clement approximation of 1D shock. SO linear clements.

3.0, I I I

1.2I :

.J0.81 I r

0.0 0.4 0.8x

Fig. 12.

2.4

1.2

1.5

1.00.0

~0.4

-L0.8

1.00

0.951.2 0.0

1.15

0.4x

0.8 1.2

2.0 2.5 1.10

pe1.6

.p

2.0pu

1.05

1.5r- j 1 1.00

1.();L, .J] I I 0.950.8 1.2 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2x xx

Fig. 13. Finite element approximation of lD shock. 100 linear elements.

1.2

0.81 ! I I0.0 0.4

222 Ph. Del'loo et lIl., hop adaptive FEM for compressible flow simulatioll

2.4. i , I 3.0, i I I 1.12

~~~

2.0 2.5 1.08

pc P pu1.6 2.0 1.04.-

1.2 1.5 1.0

0.8 1.O'Il11IlW)Il11IH' -, I ,0.96

0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2x x x

Fig. 14. Finite element approximation of 1D shock. 20 quadratic elements.

2.4

1

i I 3.01 I j I 1.12

1I

'or ] p "[ 1 j1.08

pe pu1.6 2.0 1.04

1.21- I i 1.5~ II 1 1.0L)\0.8 [

l ".,.I-J I 1.oLClllO(XUXJ/ II I I 0.961 I

0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2x x xFig. 15. Finite clement approximation of tD shock. 20 cubie elements.

2.4

1

i :" I 3.01 i I

I1.15

1

I -----,

2.0 2.51- II -I 1.10

pe p- pu1.6 2.0 1.05

1.00Ll\.-.1.2 1.5 u

0.1\ 1.0 0.!150.0 0.4 O.H 1.2 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2

x x xFig. 16. Finite element approximation of ID shock, 20 quartic elements.

Ph. LJel'/oo el al., h-p adaplive FEM for compressible flolV simulation 223

2.4 I I , I 3.0, i , I 1.15

''[ r ] :'[ J ]1.10

pe pu1.6 2.0 1.0:;

,.

12;......) ~ J.5f I 1 1.00

, I 1.JIUlIllllOUI J I I I 0.950.80.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2

x x x

fig. 17. finite element approximation of ID shock. 20 quintic elements.

4.2. A two-dimensional linear convection problem

In this problem we model the steady-state linear two-dimensional scalar convection problem

iJu- + {3 . 'Vii = 0 'r;f x E f2 . t > 0 .at (4.2)

The finite element approximation of this problem is a Galerkin approximation to the relaxedequation

( .1x ){3.Vii - {3. 'V CFL.l., 2/1{3/1 {3' 'Vii = 0 . (4.3)

where \I {3/1= ({3 . {3)2.Figure 18 shows the domain n with the boundary conditions for u. Note that, in this case,

the relaxed solution and the exact solution coincide. Figures 19-22 show comparisons of finiteelement approximations using different adaptive strategies. Note that comparable results areobtained with the three-level h-approximation using linear clements and the two-Ievel~h-pmethod with quadratic elements in the 'shock' area, but that the latter calculation employssubstantially fewer degrees of freedom.

4.3. Supersonic viscolis flow impending on a fiat plate

4.3.1. General problem descriptionThis problem was chosen because it is well documented and many solutions have been

proposed in the literature [6, 7, 201.In this computation. a Mach 3 supersonic flow impends on a nat plate. The general flow

features and boundary conditions are indicated in Fig. 23.The full Navier-Stokes equations are approximated, as described in Section 2.2. In

addition. the viscosity µ is taken to be a nonlinear function of the temperature of the form [6]

224 I'h. Dcvloo et al .. hop adaptive FEM for compressible flow simulation

(0, I)

(0,0)

Outflow

u=1

Inflow

Outflow

Inflow

(I, I)

(1, 0.75)u=f(y)

(1,0.625)

u=O

( 1,0)

f(y)=3 ((X-O.li~5)" _·':I(X-O.l.i25j")0.125 0.125

Fig. 18. Geometry and bOllllllary conditions of a 2D convection problcm .

...-::: ~

- ./.I;::?:-:10- /

/' ................ ~ r:;:::::....... ~ ~ ......

V~ ~ ~ ~ ~ ::::. ,/---:::: ~ ~ ~ ~ %V v-....... ..........

.........../. ~ ~ ~ ~ ~ v-:/'-~ ~ ~ ~ ~ :::::/

~ :;..--:: ~ / ,/

~ /'

Fig. 19. Coarse mcsh approximation of a linear 2D convection problem.

Ph. Devloo et al .. hop adaptive FEM for compressible flow simulation

-- ~ -- - - -

-

i-i-

~vI I

v V" I..-i

~:v:.v~l"""v ....-r

~ -I 1I1

Fig. 20. Second-level approximation of a linear 2D convection problem.

1

t H-

I

1 i+H-H:

i+i+1* +1 f

I

I

f+ i+

Fig. 21. Third-level approximation of a linear 2D convection prohlem.

225

226 Ph. Devloo et al., h-p adaptive FEM for compressible flow simulation

I III.

I-- :::.v I--- -- =r -r I '" v I IA' _

I .A: I II I II I Iv

,/

I k::::

P- I II II I

v --

I

Fig. 22. Quadratic mesh approximation of a linear 2D convection probtem.

(T)I.S( l+slT )µ. = T", 1'1L + sJXL ' (4.4)

The flow features can be recognized in Fig. 23.• A singularity is generated at the top of the plate.• A curved bow shock is developed from the tip of the plate towards the outflow boundary.• A boundary layer is formed on the plate.

Supersonic Inflow (Dirichlel)Supersonic Inflow (Dirichlel)

Mach:3Ae.IDDDPr:O.72'( = 1.4

y

x

0.1

no slip0;;;;:::::-

L=1.0

outflow 0.75

Fig. 23. Geometry and boundary conditions of Carter's flat plate problem.

Ph. Devloo et al .. It-p adaptive FEM for compressible flow simulation 227

REMARK 4.1. Integration rule. At first, experiments were performed with an eight-nodedelcment and a 2 x 2 intcgration rule for the viscous terms. Whereas this element performs verywell for the elasticity equations, it was unstable in the CFD context. It can be stabilized byusing a 3 x 3 integration rule, but in that case, a nine-node element was the more logicalchoice (the cosCof our computation depends more on the number of integration points thanon the number of degrees of frcedom).

For the computations with exclusively 1!I)ear elements, the convection part of the equationwas computed with a one-point integration ·fl.il!;:.When using adjacent elements of differentorders (e.g. a linear element facing a quadratic), the underintegration of the convection termleads to an anomalous behavior of the solution.

From these experiments, we concluded that full integration for both the convection anddiffusion terms is advisable.

REMARK 4.2. Computation of the bouf/dary fluxes. In the finite element method, boundaryfluxcs arc hcst computed through an intcgral formulation [19, 12, 14J. Considcr thc wcak formof the original steady state Navier-Stokes equations

(4.5)

(4.6)

In general, boundary conditions are imposed by restricting the solution space Vh and bysetting the corresponding test functions equal to zero. .

In this case, the term !F(VI!) . f/ on the wall corresponds to our boundary quanti tics ofinterest

\F(U")' = [(P - 7xx)~/x - 7xyf/y]II -7 f/ +!')Il .xy x y

q"llx + q/ly

The second term corresponds to the normal stress (pressure), the third term corresponds tothe tangential stress (skin friction) and the fourth term corresponds to the heat flux.

Denote IT(s) the skin friction, p(s) thc pressure, q(s) the heat flux, and represent thesebounuary quanti tics by a unc-uimcnsiunal JinitL: ckmcnt approximation

w'" = [0, IT''(S). p"(s), q"(s)] .

Then a weak formulation for these quantities is written as

(4.7)

or

(4.8)

where Mr is thc one-dimensional mass matrix, and the right-hand side is computed in theusual finite element fashion. The same approach was proposed by Douglas, Dupont and

228 Ph. Dev/oo et al., hop adaptive FEM for compressible flow simulation

Wheeler [13] and by Gresho et al. [15], but for a different set of equations. When the massmatrix Mr is lumped, (4.8) corresponds to nodal quadrature as suggested in a recent paper ofHughes.and Mallet [21].

REMARK 4.3 .. The outflow boundary condilion. Considerable difficulties were encounteredwhen modeling the outflow boundary conditions. The more logical approach is to convect out(i.e. include in the outflow boundary calG.ulation) all the fluxes, i.e. the convective flux (Eulerflux), numerical viscosity, and real viSCOSity. Whereas this approach worked satisfactorily forlinear elements. it was unstablc for quadratic 'elements.

The instability is probably generated by the subsonic outllow condition. For this outtlowcondition, one of thc characteristics of the steady-state Euler equations cntcrs the domain,and triggers the instability.

A second approach consists in not convecting out any viscous flux. Fortunately, thisapproach stabilized the solution. but gave an unsatisfactory solution at the outflow part of theboundary layer. Similar problems are discussed in a recent paper by Hughes and Mallet [21].

A third approach consists in convecting out the convection flux and not the numericalviscosity t1ux. This method behaves satisfactorily and is chosen for the present work.

4.3.2. Comparison of different numerical solutionsThe numerical results are obtained for two different finite element grids. Wherever possible

these results are compared with finite difference results of Carter [6] and finite element resultsof Dechaumphai [7].

The details of the finite element grids are given in Table 1. Figures 24 and 25 show thedensity contours and the corresponding finite elemcnt grids. The values for density contoursrange from 0.05 to 2.95 in increments of 0.1. The finite clemcnt grids show higher-orderelements in the boundary layer region and maximum levcl of It-refinement in both thehoundary layt:r ;IIHI shock regioll. The adaptive strategy has allllllllaticdly t:lIllct;lltratt;d tilt;II-refinement in shock region as can be inferred from the corresponding density contours.Figures 26-30 show comparison between the present results and Carter's results. The

Table]The finite element grids

GridTotal number

of elements

Number of Number of Number oflinear quadratic cubic

elements clements elementsNumber of Maximum level

nodes of h-rcfinementMaximum orderof p-enrichment

L2P3UP3

5861345

4631102

4282

81161

13682831

23

3'3

Table 2The comparison of degrees of freedom for Carter's flat plate

Present workfinite elemcnt

Method

Number of nodes

Carter(inite dilTcrenee

2250

[7]finite clement

13231

L2P3 mesh

1368

L3P3 mesh

2831

_a.·_-. I~'._1 I.

a

I'll. Devloo el ill.. hop ildaptive FEM for compressible flow simulation

_.-:-r~rr-r-rll- "'11--[-----i-f-+-- J--{- --·t-L- _. -

-.-

I

b

Fig. 24. (a) Density contours for L2P3 mesh. (b) The L2P3 finite element mcsh.

229

a b

Fig. 25. (a) Density contours for L2P3 mesh. (b) The L2P3 finite etemcnt mesh.

I'h. nt'I'iI}() 1'1 11/.. h-/, lIt/llf/lil'I' FF./l-f for (,oll/f/rI's.\·;I.ft· j/Oll' Sill/lI/III;OI/

0.8-0- L2P3 mesh-+- L3P3 mesha Carter Data

0,6

..J;: 0.4

0.2

0,00.00 0.25 0.50 0.75 1.00 1.25 1.50

0.20

Fig. 26. Exit u velocity profile.

-0- L2P3 mesh-+- L3P3 mesha Caner data

<--<8

0.15

0.10

0.05

0.000.0 0.2 0.4

Y/L

a

a

a

0.6 0.8 1.0

Fig. 27. Exit lJ velocity profile.

Ph. Devloo ct al., hop adaptive FEM for compressible flow siml/lation 231

L2P311lesll-+- L3P3 meshD Carter dala

0.2 0.4 0.6 0.8 1.0

Y/L

Fig. 28. Exit density profile.

0.75

54320.00

o

0.50 I • -0- L2P3 mesh..... L3P3 mesh.. D Carter data-< I-r

0.25

T/Too

Fig. 29. Exit temperature profile.

232 Ph. Devloo et al.. hop adaptive FEM for compressible flow simulation

7.5

5.0

"C

laa

-?2.5

Iaa

-0- L2P3 mesh..... L3P3 mesha Carter data

a a

0.00.0 0.2 0.4 0.6 0.8 1.0

0.20

0.15

0.10

0.05

X/L

Fig. :m Wall pressure distributioll.

-0- L2P3 mesh..... L3P3 mesh

0.000.0 0.2 0.4 0.6 0.8 1.0 1.2

X/LFig. 31. Wall skin friction profile.

Ph. Devloo et al.. h-p adaptive FEM for compressible flow simulation

0.00

233

o -0.01:J

-0- L2P3 mesh.... L3P3 mesh

-0.020.00 0.50

X/L

Fig. 32. Wall heating rates

1.00

agreement is very good for the exit II velocity, the exit v velocity. the exit dcnsity, and thc cxittemperature profiles. Thc result for wall prcssurc distribution shows a slight discrepancy at thcleading cdge. This may be due to Cartcr's grid being not fine enough at the leading-edgesingularity point. The hop rcsults show very good agreement with thOse of Dechaumphai [7].Comparison of results for two finite element grids for wall skin friction and heating rates arepresented in Figs. 31 and 32. The present results agree very well with those of Dechaumphaifor wall skin friction, but there is some discrepancy in the computed (singular) wall heatingrates at the leading edge. We anticipate that the present results may be better for wall heatingrates as higher order polynomials should be able to capture the boundary layer more readilythan fine low-order approximations.

References

III Ivl. I\hr;II11llwil/. ;lIld I. Slq~II11. Ihudl,""" III Malhl:ulalil:al hllll:lillllS wilh hJl'lllulas. liraphs aud Mathclllati-cal Tables (Dover. New York. 1%5).

[2) 1. Babuska and W.e. Rheinboldl. Reliable error estimations and mesh adaption for the finite element method,in: J.T. Oden. ed., Computational Methods in Nonlinear Mechanics (North-Holland. Amsterdam. 1980).

[3] 1. Babuska and W.e. RIH:inboldt. Error estimates for adaptive tlnite element computations, SIAM J. Numer.Anal. 15 (4) (1978).

14) R.E. Bank and A.H. Sherman. A relinement algorithm and dynamic data structure for finite elcment meshes,Rept.. Center for Numerical Analysis, Thc University of Texas. Austin. TX. 1980.

234 Ph. Devloo et al., hop adaptive FEM for compressible flow simulation

[51 K.S. Bcy. E.A. Thornton. P. f)cchallmphai and R. Romakrashnan. /\ ncw ~n.ilc CIcIllC~ll approach ofprediclioll of aerlltherillal loads, I'apcr HS-1533-CI'. 7th AlAI'. Computational HUIIJ DynamiCs Conlerence,.Cincinnati. 011. IlJXS.

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[7] P. Dechaumphai. Mach 3 flow over Il,lt plate: A comparison of finite element rcsults with Carter's solution,Tech', Rept.. Mechanical Engineering and Meehanics Department, Old Dominion University. Norfolk, VA.

[8] L. Demkowicz. Ph. Devloo and J.T. Oden, On an IHype mesh refinement strategy based on minimization ofinterpolation errors, Comput. Meths. Appl. Meeh. Engrg. 35 (1985) 67-90.

. [9] L. Demkowicz and J.T. Oden. An adaptive characteristic Petrov-Galerkin finite element method forconvection-dominated linear and nonlinear parabolic problems in two space variables, Comput. Meths. Appl.Mech. Engrg. 55 (1986) 63-87.

[10] J.E. Dendy, Two methods of Galerkin-type aehieving optimum Lz accuracy for first order hyperbolics, SIAMJ. Numer. Anal. 11 (1974) 637-653.

[111 Ph. Devloo and L.J. Hayes. A fast vector algorithm for a matrix vector multiplication with the finite clemen I

ml.:lhod. T1COM Rep. l-l5-13. University of Tcxas, TX. 1l)l-l5.[12] Ph. Devloo J.T. Oden and T. Strouboulis, Implementation of an adaptive refinement technique for the SUPG

algorithm, Comput. Meths. Appl. Mech. Engrg 61 (1987) 339-358.[13] J. Douglas Jr., T. Dupont and M. Wheeler. A Galcrkin procedure for approximating the flux on the boundary

for elliptic and parabolic boundary value problems. RAIRO (1974) 47-59.[14J P. Eiseman, Adaptive grid generation, Comput. Meths. Appl. Mech. Engrg. 64 (1987) 321-376.[15] P.M. Gresho and R.L. Lee, The consistent Galcrkin FEM for computing derived boundary quantities in

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[19] T.J.R. Hughes L.P. Franca and M. Mallet. A new finite element formulation for computational fluiddynamics: I. Symmetric forms of thc compressible Euler and Navier-Stokes equations and the second law ofthermodynamics. Comput. Meths. Appl. Mechs. Engrg. 54 (1986) 223-234.

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[21} T.J.R. Hughes and M. Mallet, A new finite element formulation for computational fluid dynamics: IY. Adiscontinuity-capturing opcrator for multidimensional advective-diffusive systems, Comput. Mcths. Appl.Mechs. Engrg. 58 (1986) 329-336. .

[22J A. Jameson. Solution of the Euler equations for two-dimensional transonic flow by a multigrid method. Appl.Math. Comput. 13 (1983) 327-355.

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Ph. Dev/oo et al., hop adaptive FEM for compressible flow simulation 235

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