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1 SIAM J. NUMER. ANAL. Vol. 4, No., pp c 23 Society fo Industial and Applied Mathematics Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see VARIATIONAL MESH ADAPTATION METHODS FOR AXISYMMETRICAL PROBLEMS WEIMING CAO, RICARDO CARRETERO-GONZÁLEZ, WEIZHANG HUANG, AND ROBERT D. RUSSELL Abstact. We study vaiational mesh adaptation fo axially symmetic solutions to twodimensional poblems. The study is focused on the elationship between the mesh density distibution and the monito function and is caied out fo a taditional functional that includes seveal widely used vaiational methods as special cases and a ecently poposed functional that allows fo a weighting between mesh isotopy (o egulaity) and global equidistibution of the monito function. The main esults ae stated in Theoems 4. and 4.2. Fo axially symmetic poblems, it is natual to choose axially symmetic mesh adaptation. To this end, it is easonable to use the monito function in the fom G = λ ()e e T + λ 2()e θ e T θ, whee e and e θ ae the adial and angula unit vectos. It is shown that when highe mesh concentation at the oigin is desied, a choice of λ and λ 2 satisfying λ () <λ 2 () will make the mesh dense at = than in the suounding aea whethe o not λ has a maximum value at =. The pupose can also be seved by choosing λ to have a local maximum at = when a Winslow-type monito function with λ () =λ 2 () is employed. On the othe hand, it is shown that the taditional functional povides little contol ove mesh concentation aound a ing = λ > by choosing λ and λ 2. In contast, numeical esults show that the new functional povides bette contol of the mesh concentation though the monito function. Two-dimensional numeical esults ae pesented to suppot the analysis. Key wods. mesh adaptation, vaiational method, mesh egulaity, equidistibution AMS subject classifications. 65M5, 65M6 PII. S Intoduction. Mesh adaptation has become an indispensable tool fo use in the numeical solution of PDEs. One of the most widely used appoaches fo geneating adaptive meshes is a vaiational method. With such a method, meshes ae geneated as images of a efeence mesh though a coodinate tansfomation between the physical and computational (o logical)domains. The tansfomation is detemined as the minimize of a functional fomulated to measue difficulties in the numeical appoximation of the physical solution, typically though a so-called monito function pescibed by the use to contol the mesh adaptation. A vaiational method often esults in an elliptic (PDE)mesh geneation system. Such a system geneates smooth meshes, allows fo full specification of mesh behavio at the bounday, does not popagate bounday singulaities into the domain, has less dange of poducing mesh ovelappings, and can be solved efficiently using many well-developed Received by the editos Januay 28, 22; accepted fo publication (in evised fom) Septembe 23, 22; published electonically Mach 9, 23. This wok was suppoted in pat by NSF (USA) gant DMS-7424, and NSERC (Canada) gant OGP Depatment of Mathematics, The Univesity of Texas at San Antonio, San Antonio, TX (wcao@math.utsa.edu). Depatment of Mathematics, Simon Fase Univesity, Bunaby, BC V5A S6, Canada. Cuent addess: Nonlinea Dynamical Systems Goup, Depatment of Mathematics and Statistics, San Diego State Univesity, San Diego, CA 9282 (caete@math.sdsu.edu). Depatment of Mathematics, the Univesity of Kansas, Lawence, KS 6645 (huang@math. ukans.edu). Depatment of Mathematics, Simon Fase Univesity, Bunaby, BC V5A S6, Canada (d@cs. sfu.ca). 235

2 236 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see algoithms. Moeove, the equidistibution pinciple, a concept which has been used successfully in one-dimensional mesh adaptation [3], can be natually extended to multidimensions in the vaiational famewok. Finally, many mesh featues, such as othogonality, smoothness, and concentation, can be incopoated explicitly into the mesh adaptation functional. A numbe of vaiational methods have been developed in the past. Fo example, Winslow [5] poposes the vaiable diffusion method fo which the mesh lines play the ole of equipotentials of a potential poblem [4]. Backbill and Saltzman [] develop a popula method combining mesh concentation, smoothness, and othogonality. Seveal functionals ae fomulated by Steinbeg and Roache [3] to contol mesh popeties such as the spacing of the points, aeas o volumes of the cells, and the angles between mesh lines. Dvinsky [4] uses the enegy of hamonic mappings as his mesh adaptation functional. Knupp [9, ] and Knupp and Robidoux [] develop functionals based on the idea of conditioning the Jacobian matix of the coodinate tansfomation. A functional balancing mesh egulaity and adaptivity is poposed by Huang [6]. Some theoetical wok has been devoted to bette undestanding the existing methods. Cao, Huang, and Russell [2] study the qualitative effect of monito functions on the esulting mesh fo a geneal class of vaiational methods that includes Winslow s method [5] and the method using hamonic mappings [4] as special cases. In the ecent wok of Huang and Sun [8], the monito function fo the functional of [6] is defined based on intepolation eo estimates, and asymptotic eo bounds ae obtained fo intepolation on the esulting adaptive meshes satisfying the so-called isotopy and equidistibution conditions. The ability of the esulting method to geneate adaptive meshes satisfying these conditions is also demonstated numeically. Nevetheless, moe wok emains to be done on bette undestanding the existing vaiational methods, especially on pecisely how the monito function contols the concentation of the geneated mesh. In this pape we pesent such a study fo two functionals, the taditional one studied in [2] and the new one poposed in [6], fo the simple but impotant case of two-dimensional poblems with axisymmetical solutions. These types of poblems aise in many pactical situations, paticulaly fo poblems with blowup o quenching solutions. Thee has been consideable ecent inteest in solving highe-dimensional blowup poblems such as the Schödinge equation, and this wok was motivated by the obsevation that the standad moving mesh pocedues geneally pefom inadequately on such poblems (e.g., see [2, 2]). Let (x, y)be the coodinates in the physical domain Ω, and let (ξ,η)be the coodinates in the computational domain Ω c. The taditional functional is ( (.) I tad [ξ,η] = ξ T G ξ + η T G η ) dxdy Ω and the new functional in [6] has the fom ( I new [ξ,η] = γ g ξ T G ξ + η T G η ) q dxdy (.2) Ω +( 2γ)2 q Ω g (J g) q dxdy, whee J = x ξ y η x η y ξ is the Jacobian of the coodinate tansfomation, G is the (matix)monito function with deteminant g, and q and γ (, /2] ae pa-

3 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 237 Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see ametes. Hee, q is equied in ode fo the fist integal of (.2)to be convex. The featues of these functionals and the oles of the paametes will be discussed in sections 2 and 3. Axisymmetical poblems. Fo simplicity, we assume that the physical domain is Ω = {(x, y) x 2 + y 2 < } and the computational domain is Ω c = {(ξ,η) ξ 2 + η 2 < }. Let the pola coodinate systems fo the physical and computational domains be { x = cos θ, y = sin θ, { ξ = R cos Θ, η = R sin Θ. Conside the case whee the solution u(x, y)is axially symmetic; i.e., u is invaiant unde otation about the cente (, ). It is natual to choose an axially symmetic coodinate tansfomation (.3) R = R(), Θ=θ fo mesh adaptation. To this end, it is easonable to use the monito function in the fom (.4) G = λ ()e e T + λ 2 ()e θ e T θ, whee e and e θ ae unit vectos in the adial and angula diections, espectively. Thus, G is detemined by its adial and angula components λ > and λ 2 >. We ae inteested in the elationship between the monito function and the mesh distibution. In paticula, we focus on the mesh density D(). The Jacobian of the coodinate tansfomation J is easily seen to satisfy ( ) (ξ,η) (.5) J det = R dr (x, y) d, and thus the mesh density is given by (.6) D() = R dr d. The cental aim of this pape is to gain insight into how much contol one has on the mesh density D()by appopiately choosing λ and λ 2. In ode fo the vaiational method to be successful one needs that the solution to the vaiational poblem gives a mesh distibution compatible with the chosen monito function. Fo example, it is natual to choose one o moe of the eigenvalues of the monito function to have a highe value (a maximum)in the egion whee a physical solution needs a high concentation of mesh points; e.g., see [2]. It will become clea below that this is not always achievable and that if one is not caeful in choosing the appopiate elation between λ and λ 2 it is possible fo the mesh density maximum to occu at a diffeent location than that of the maximum of the eigenvalue. This can in tun lead to a lage eo in the numeical appoximation of the physical solution. An outline of the pape is as follows. In sections 2 and 3 basic popeties of the taditional and new functionals fo adially symmetic poblems ae pesented. In section 4 we cay out an in-depth analysis on the contol of the mesh density via the monito function. In paticula, we find that the elationship between the adial (λ )and the angula (λ 2 )components of the monito function is cucial fo a good contol of the mesh density. Section 5 pesents some two-dimensional numeical esults highlighting in pat the lack of contol of the mesh concentation fo a wide choice of monito functions. A bief analysis is given in section 6 fo the taditional functional applied to spheically symmetic poblems in thee dimensions. Finally, section 7 contains conclusions and comments.

4 238 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see 2. The taditional functional. In this section we conside the taditional functional (.)fo axisymmetical poblems and give some of its basic popeties. It is a genealization of the functionals fo Winslow s method and Dvinsky s method of hamonic mappings. The monito function G can be defined by abitaily choosing λ and λ 2. Howeve, it is woth pointing out that a numbe of commonly used monito functions can be obtained though the intedependent elationship (2.) λ 2 = λ p fo some powe p. Fo example, we have (HM) p = : hamonic mapping monito function; (Al) p = : aclength monito function; (2.2) (Ws) p = : Winslow s monito function; (St) p = 2 : stong concentation monito function. In pola coodinates the gadient opeato eads as = e + e θ θ, and it follows fom (.3)and (.4)that ξ T G ξ + η T G η = ( dr (2.3) λ d Substituting (2.3)into (.)gives I tad [R] =2π [ λ ( ) 2 dr + d λ 2 ) 2 + λ 2 ( ) 2 R. ( ) ] 2 R d, and its Eule Lagange equation is d ( ) dr (2.4) + R =. d λ d λ 2 This equation is supplemented with the bounday conditions (2.5) R()=, R()=. Fo a given monito function (i.e., fo given λ and λ 2 ), solving (2.4) detemines the esulting mesh tansfomation R(). 2.. Nonnegativeness and mesh cossing. We have R() fo (, ). To see this, we note that the minimum of R()occus at the left end and/o an inteio point due to the bounday conditions (2.5). If R()= min R(), then we have R() fom (2.5). If instead the minimum point is (, ), then R ( )=and R ( ). Fom (2.4) R( )= d ( )( ) dr + d2 R λ 2 d λ d λ d 2. Hence, in eithe case R() R( ). Futhemoe, (2.4)gives dr d = λ R(x) xλ 2 (x) dx, so it follows that dr d > fo (, ); i.e., the mesh tansfomation is guaanteed to be nonsingula and poduce no mesh cossing.

5 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 239 Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see Mesh tansfomation fo hamonic mappings. Fo the case of hamonic mappings (p = oλ 2 =/λ )it is possible to find an analytical fom fo the mesh tansfomation R(). We explicitly constuct R()hee since it then seves as the basis of study fo othe cases. Using the change of coodinates (2.6) λ (x) s() = x dx, (2.4)eads d2 R ds 2 + R =. Its solution satisfying the bounday conditions (2.5)is R() =e s. In section 4., using a tansfomation based on (2.6), we study moe geneal monito functions (including aclength and Winslow)in detail. 3. The new functional. The fomulation of the new functional (.2)is based on the so-called isotopy (o egulaity)and equidistibution (o adaptation)equiements fo an eo distibution [6]. Specifically, the fist integal tem coesponds to the egulaity equiement, while the second is associated with equidistibution. These two equiements ae balanced by adjusting the value of the paamete γ. When q = o γ =/2, the second integal becomes constant o simply vanishes, and only the isotopy plays a ole. When q = the functional gives ise to the enegy functional of a hamonic mapping. The elation between the new and taditional functionals will be addessed late in section 3.3. Fom (.4)the deteminant of G is g = det(g) =λ λ 2. Let (3.) Λ= λ λ 2, (3.2) µ () = λ Λ, µ 2() = λ 2 /q Λ. /q Using the symmety assumption, we can ewite (.2)as I new [R] =γ [ µ ( ) 2 dr + d µ 2 ( R Its Eule Lagange equation is given by γ ( d β q R ) (3.3) β q + γr d µ µ 2 whee ( 2γ)2q (q )R β q ) ] 2 q [ ] RR d +( 2γ)2 q q gd. g ( RR g β = ( ) 2 dr + µ d µ 2 ) q 2 d d ( ) 2 R. ( ) RR =, g The highly nonlinea fom of the new functional does not lend itself to a staightfowad analytical teatment of its basic popeties. Nonetheless, we devote the est of this section to the study of seveal special cases of (3.3)subject to the bounday conditions (2.5). These cases ae impotant because they help to bette undestand the functional and link it to the taditional one.

6 24 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see The exact equidistibution case (γ = ). We fist conside the case γ = which coesponds to exact equidistibution. Assuming that R() > fo (, ), (3.3) implies (3.4) R dr g d = α, whee α is a constant. Fom (.5), this is equivalent to J g = α, which is a multidimensional genealization of the well-known equidistibution pinciple in one dimension. This equation guaantees that J, the Jacobian of the coodinate tansfomation, does not change sign in the domain The pue isotopycase (γ = /2). Fo γ = /2 the mesh equation (3.3) educes to [ d β q R ] = βq R. d µ µ 2 As in section 2., it is easy to show that R() and R (). Thus, fo this case the mesh is also guaanteed not to coss. In Figue we depict R ()fo the taditional functional and the new functional with γ =/2 and seveal values of q. As can be seen, the mesh tansfomation fo the new functional with diffeent values of q is quite simila to the taditional functional. This is not supising since fo γ =/2 the new functional shifts all the weight towads isotopy and thus esembles the taditional functional The case q =. When q =, the second integal in (.2)becomes constant. Fom (3.)the mesh equation (3.3)educes to d ( ) Λ dr (3.5) + ΛR =. d λ d λ 2 Once again it is easy to pove that mesh cossing will not occu. Note that the mesh equation (3.5)is independent of the paamete γ and vey simila to (2.4)fo the taditional functional. In fact, fo the hamonic mapping case whee Λ =, the mesh equations (3.5)and (2.4)ae identical. Fo the Winslow monito function case (Λ = λ = λ 2 )the mesh equation is ( d dr ) (3.6) = R d d. The solution of (3.6)compatible with the bounday conditions is R() =. Theefoe, the case q = of the new functional method gives a tivial coodinate tansfomation R = and does not allow fo any contol of the mesh concentation when a Winslowtype monito function is used. Finally, fo the aclength monito function (λ 2 = )the mesh equation is d ( ) dr + R d λ d =. λ This mesh equation is equivalent to that fo the taditional functional (2.4)using a Winslow-type monito function with λ instead of λ. In summay, except fo the Winslow case, fo q = the new functional coesponds to the taditional functional with a suitable choice of the monito function.

7 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 24 Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see R () [HM] R () [Al] R () [Ws] Fig.. Compaison of R () fo the taditional functional (solid line) and the new functional with γ = /2 (q = 2: dashed, q = 3: dotted-dashed, q = 4: dotted). The plots coespond to the thee popula choices of monito function (hamonic mapping, aclength, and Winslow) with λ () = + exp( 2 /a)/a (a =.). 4. Contol of mesh densityvia λ and λ The taditional functional. The Eule Lagange equation (2.4)fo the taditional functional elates the coodinate tansfomation to the monito function fo a given choice of λ and λ 2. The pupose of this section is to use this to show that pecise contol of the mesh density D()cannot be achieved fom the choice of λ and λ 2. In fact, we pove that the maximum fo the mesh concentation does not occu at the maximum of λ, esulting in misplacement of mesh concentation. Let us then take (2.4)and solve fo the mesh density D()in (.6). Motivated by the tansfomation (2.6)leading to the exact solution of (2.4)fo the hamonic mapping monito function (λ 2 =/λ ), we conside the change of dependent vaiable (4.) R() =e s() with s() = λ (x) v(x) dx x fo a to-be-detemined and bounded function v. Substituting this into (2.4)yields the ODE fo v dv d = λ ( ) (4.2) Λ 2 v2. It satisfies (4.3) v()= Λ(), since any othe initial value (at = )poduces an unbounded solution v. The choice (4.3)is compatible with the special case of the hamonic mapping whee v() =.

8 242 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see Lemma 4.. v() > fo all [, ]. Poof. This is an immediate esult of the initial condition v()= /Λ() > and the fact that v > on the line v =. The oveall behavio of v is detemined by the nullcline (4.4) v null () = Λ(). Lemma 4.2. v min v() v max fo all [, ], whee v min = min {/Λ()} and v max = max {/Λ()}. Thus, the solution v() is bounded by the minimum and maximum of the nullcline. Poof. Note that v > below the nullcline and v < above it. Since v min v null (), we have v. This and v()= /Λ() v min imply that v() v min. Similaly, we have v() v max. Define λ as the point whee λ attains its maximum, i.e., λ ( λ )= max λ (). [,] We have the following lemma. Lemma 4.3. Let λ be a stict maximum point of λ (so λ ( λ ) < ), and let λ 2 = cλ p fo some powe p> and some constant c>. Then, v( λ) > Λ( λ ). Poof. Fo this paticula choice of λ 2,wehave Λ ( λ )=, Λ ( λ ), v min = Λ( λ ). We pove the lemma by contadiction. Fom Lemma 4.2, we can assume only v( λ )= /Λ( λ ). By diffeentiating (4.2) twice and using the fact that λ ( λ )=,weget v ( λ )=v ( λ )=, v ( λ )= 2λ ( λ )Λ ( λ ) λ Λ( λ ) 3. This implies that v() =v( λ )+ ( λ) 3 v ( λ )+O(( λ ) 4 ) 6 = v min + ( λ) 3 v ( λ )+O(( λ ) 4 ). 6 Hence, v() < v min at some points in the neighbohood of λ, which contadicts Lemma 4.2. Figue 2 shows a typical vecto field fo v. To study the mesh density, note that in tems of s, D() = R dr d = es s e s = s e 2s, and its ate of change dd d = d ( s e 2s ) d ) = (s e2s +2s 2 s (4.5) = e2s 2 ( λ v + λ v 2 λ v +2 λ2 v 2 ).

9 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 243 v max Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see v v min Fig. 2. Typical vecto field fo (4.2). The total vaiation [v min,v max] of the nullcline (4.4) (dashed line) bounds the behavio of the solution with v() =/Λ() (solid line). Using (4.2)one obtains (4.6) dd d = e2s 2 { [ ( λ v + λ2 v λ ) 2 ( + Λ 2 ) ]} λ 2. Equation (4.6)detemines whee the mesh density eaches an extemum in tems of λ and λ 2. In geneal, it is desied that the mesh has a highe concentation of points at the maximum location of λ so that the mesh concentation can be contolled by choosing λ. We fist conside mesh concentation at the oigin =. Theoem 4.. (i) If λ () λ 2 (), then D () has the same sign as λ () λ 2 () whethe λ has a maximum at =o not. Specifically, if λ () >λ 2 (), then D () > (i.e., the mesh at the oigin is coase than in the suounding aea), and if λ () <λ 2 (), then D () < (i.e., the mesh at the oigin is dense than in the suounding aea). (ii) Let λ 2 () =λ (). If λ (), then D ()λ () >. If λ ()= but λ (), then D ()λ () >. Poof. Let y() =λ ()v(). Note that y()= λ ()/Λ()= λ ()/λ 2 (). Equation (4.5)can be ewitten as dd d = e2s 2 ( y 2y ) + 2y2. Expanding the backeted tems on the ight-hand side about =,weget { } dd d = e2s 2 2 y()(y() )+ y ()(4y() )+ O() { ( ) ( ) } = e2s 2 λ () λ () 2 λ 2 () λ 2 () + y λ () (4.7) () 4 λ 2 () + O(). Thus, if λ () λ 2 (), the fist tem in the backet dominates. In this case, D () has the same sign as λ () λ 2 (). The esult in (i) follows.

10 244 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see We now pove pat (ii) using (4.6). This can also be done though (4.5), but highe ode tems must be used. Using the assumption λ () =λ 2 ()and expanding the backeted tems of (4.6)about =,weget dd d = e2s 2 {v()λ ()+ [λ ()v ()+ λ ()( + v ()) + λ ()v()] + O( 2 )}. The esults in (ii)follow since λ ()= implies v ()=. We now conside the case whee mesh concentation away fom the oigin is desied, i.e., when λ >. Let D : D( D )= max [,] D(). The following theoem shows the elative positioning of D with espect to λ. Theoem 4.2. Let λ >. (i) If λ ( λ ) >λ 2 ( λ ), then D ( λ ) > and thus D > λ. (ii) Futhe, if we assume that λ 2 () =λ () (Winslow s method) and λ is a stict maximum point of λ (i.e., λ ( λ ) < ), then D ( λ ) > o again D > λ. Poof. (i)the esult is an immediate consequence of (4.6)and the assumptions. (ii)when λ 2 () = λ (), we have Λ() = λ (). Lemma 4.3, the fact that λ ( λ )=, and (4.6)imply that D ( λ ) >. We note that a esult can be obtained fo the geneal choice of λ 2 satisfying λ 2 () =λ ()only at = λ. Moeove, numeical expeiments (see below)show that the mismatch between D and λ fo the Winslow monito function is elatively small. The situation with λ ( λ ) <λ 2 ( λ )is much moe complex. Note that the last tem in (4.6)is now negative. In ode to detemine the elative positions of D and λ it is necessay to compae all the tems on the ight-hand side of (4.6). It is possible fo D and λ to coincide. Howeve, numeical esults (see section 4..4)also show that fo λ 2 = λ p with p>, D can be located on eithe side of λ. It is emphasized that pat (i)of both Theoems 4. and 4.2 equies no explicit elationship between λ and λ 2, although we typically apply them to the monito functions defined in (2.2) The hamonic mapping case (p = ). In this case, λ 2 =/λ. Assuming that λ ( λ ) >, we have λ ( λ ) > λ 2 ( λ ). Theoem 4.2 implies that D > λ, o the location of maximum mesh density is to the ight of the maximum fo λ. When a (local)highe mesh concentation at the oigin is desied, Theoem 4. implies that if (a) λ () >, the mesh at the oigin is coase than in the suounding aea, whethe λ has a maximum at = o not. This effect is clealy depicted in Figue 3 (left column, second plot)whee D > λ = implies a failue to concentate the points at the oigin. If instead (b) <λ (), then λ () λ 2 ()and the mesh will be dense in the cente than in the suounding aea. We now conside conditions unde which λ > and D coincide. Thee is a tight estiction on the choice of λ since D = must hold whee λ =. Notice that we have Λ = fo the cuent case. Equation (4.6)implies that this can be achieved if and only if λ ( λ )=. Howeve, this cannot hold in geneal unless the high mesh concentation is desied only at the global maximum point and λ ( λ )=canbe achieved by escaling.

11 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 245 λ = λ =.2 Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see λ D (HM) D (Al) D (Ws) D (St) Fig. 3. Nomalized mesh densities obtained with the taditional functional fo diffeent monito functions defined in (2.2) and with λ () = +exp( ( λ ) 2 /a)/a (a =.) that has its maximum located at λ = (left column) and λ =.2 (ight column). The top plot depicts λ (). Fo guidance, we plot, along with the nomalized densities (solid lines), the nomalized cuve fo λ () (dashed lines). Fom the above analysis we see that if λ ( λ ) >, D will be located to the ight of λ. This failue to place the highe concentation of points in the desied aea is depicted in Figue 3 (ight column, second plot) The aclength case (p = ). Fo the aclength case λ 2 ()=, a simila analysis as the one fo the hamonic mapping case can be caied out. We assume λ ( λ ) > since this is the one commonly used in the liteatue. If λ >, Theoem 4.2 and λ ( λ ) > =λ 2 imply that the maximum of the mesh density occus at a location to the ight of that of the maximum of λ. This mismatch is illustated in Figue 3 (ight column, thid plot). The agument fo λ = is simila, and thee is again a mismatch (to the ight) between the locations of the maxima of the mesh density and λ (see Figue 3, left column, thid plot The Winslow case (p = ). If a high mesh concentation is desied at a stict maximum point λ > ofλ, Theoem 4.2 implies that D will be located to the ight of λ. Nevetheless, as Figue 3 (ight column, fouth plot)shows, the mismatch between the maxima fo D and λ can be vey small (compaed with the othe cases). On the othe hand, Theoem 4. implies that the mesh has highe concentation

12 246 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see λ D (St) a ) a 2 ) λ D (St) b ) b 2 ) Fig. 4. Nomalized mesh densities obtained with the taditional functional fo the stong concentation case (p =2) with λ () =+A exp( ( λ ) 2 /a)/a (a =.2, λ =.5) and (a) A =. and (b) A =. The top plots show λ (). Fo guidance, in a 2 and b 2 we plot the nomalized cuve fo λ () (dashed lines) along with the nomalized densities (solid lines). at = if eithe λ () < o = is a local maximum point of λ. Figue 3 (left column, fouth plot)shows good ageement between the shape of λ and the mesh density The stong concentation case (p = 2). Conside fist the case whee a highe mesh concentation at the oigin is desied. Theoem 4. implies that the maximum fo the mesh density is located at the oigin if λ () > (see the last plot in Figue 3, left column). Howeve, the ate of change of the density may be a vey lage negative value popotional to lim e 2s / 3. This effect is obseved in Figue 3 (last plot, left column)whee the mesh density is vey steep at the oigin, giving an oveconcentation of points at =. Incidentally, ou use of the tem stong concentation fo the p = 2 case eflects this behavio. Fo λ >, the cuent situation is moe complex than the pevious cases and Theoem 4.2 does not apply if λ ( λ ) >. Figue 4 shows that D can be located to eithe side of λ. Since in this case we have λ () <λ 2 (), Theoem 4. implies that the mesh concentation has a maximum at the oigin. Thus, it is possible fo the mesh concentation to have two (o moe)maxima, one nea the desied location λ and a spuious (and steep)maximum at = (see the last plot in Figue 3, ight column) The new functional. The Eule Lagange equation (3.3)coesponding to the new functional is too complex to cay out an analysis simila to the one fo the taditional functional, and we instead pefom a numeical study of the elation between the monito function (λ and λ 2 )and the mesh density D(). In paticula, we show that by appopiate contol of the weighting γ between isotopy and equidistibution it is possible to educe the mismatch between the location of the maximum fo the monito function and that of the maximum fo D(). As fo the taditional functional, we concentate ou attention on monito functions of the type (2.)and use the same notation as in (2.2)to designate the most popula choices of p. Note that fo the hamonic mapping monito function g =, and equidistibution eads as J = constant, giving no mesh contol in the new functional. As a esult, it is expected

13 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 247 γ =. γ =. γ =. Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see λ D (HM) D (Al) D (Ws) D (St) Fig. 5. Nomalized mesh densities (solid lines) obtained with the new functional fo a monito function (dashed lines) such that λ () = + exp( ( λ ) 2 /a)/a (a =., λ =) fo diffeent choices of λ 2 and γ (q =2). that the new functional combined with the hamonic mapping monito function gives no bette esults than those with the taditional functional, even fo a small value of γ Concentation at =. Fo λ =, Figue 5 shows the monito function and the mesh density fo the vaious choices of monito function (2.2)and weighting between isotopy and equidistibution. Fo lage γ (close to /2), the new functional tends to emphasize isotopy, giving simila esults to those fo the taditional functional. Fo γ =. (fist column in Figue 5), the hamonic mapping and the aclength monito functions tend to misplace the position of the maximum fo the density as befoe. Fo the Winslow and stong concentation cases, D()achieves its maximum at =. Deceasing γ puts moe weight on equidistibution, allowing fo a bette distibution of the mesh density. In fact, by deceasing γ (second and thid columns in Figue 5)the maximum fo the mesh density is pulled towads the coect position =. As pointed out above, the hamonic mapping case fails to have its maximum at = even fo vey small γ. Fo the othe cases (p ), as γ tends to zeo, not only the mesh density has its maximum placed coectly, but its shape tends to mimic the shape of λ. This suggests that fo small γ it is possible to contol the position of the maximum mesh concentation as well as the shape of the mesh density fom the choice of monito function. Inteestingly, the Winslow case povides the best contol

14 248 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL γ =. γ =. γ =. Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see λ D (HM) D (Al) D (Ws) D (St) Fig. 6. Nomalized mesh densities (solid lines) obtained with the new functional fo a monito function (dashed lines) such that λ () = +exp( ( λ ) 2 /a)/a (a =., λ =.2) fo diffeent choices of λ 2 and γ (q =2). on the mesh density, and fo small γ (γ <.) D()is almost indistinguishable fom λ () Concentation at >. Fo λ > we obtain simila esults to those fo the taditional functional when using a lage value of γ (see left column in Figue 6). In paticula, the position of the mesh density maximum does not coincide with λ except in the Winslow case. As we decease γ, the new functional weights moe towads equidistibution, and the location of the maximum fo D() tends to appoach λ, again einfocing the obsevation that fo small γ and p it is possible to have a good contol on the mesh density (maximum and shape)fom the monito function. 5. Numeical esults. In this section we pesent some numeical esults obtained with the functionals (.)and (.2). Fo simplicity, squae physical and computational domains and stuctued meshes ae used in the computation. As a consequence, axially symmetic meshes ae not geneated. Nevetheless, the numeical esults ae sufficient to suppot the analysis of the pevious sections and highlight the level of contol of mesh concentation though the monito functions. The (two-dimensional)eule Lagange equations fo functionals (.)and (.2) ae discetized with cental finite diffeences and solved using the moving mesh PDE appoach [5, 7]. With this appoach, a deivative ( x)/( t)(whee x =(x, y) T )with espect to pseudotime t is added to the Eule Lagange equation, and the esulting

15 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 249 Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see paabolic system is integated using a modified backwad Eule scheme with which the coefficients of tems ( x)/( ξ i )and ( 2 x)/( ξ i ξ j )ae calculated at the pevious time level. The linea algebaic system is solved using a peconditioned conjugate gadient method. The conveged mesh is obtained when the oot-mean-squae nom of the esidual is less than 4. All computations stat with a unifom mesh of size 4 4 and use a unifom bounday coespondence between Ω and Ω c. We use q =2 in all cases and, following the common pactice, choose λ to be geate than. Example 5.. The fist example is to geneate adaptive meshes fo the monito function (.4)with (5.) λ =+ a e (.2)2 /a, whee = x 2 + y 2 and a =.. In the (x, y)coodinate system, G has the fom G = λ ( ) x 2 xy x 2 + y 2 xy y 2 + λ ( ) 2 y 2 xy (5.2) x 2 + y 2 xy x 2. The goal is to geneate meshes with highe point concentation aound the cicle x 2 + y 2 =.2 2. The meshes obtained ae shown in Figues 7 and 8. The fist ow coesponds to the taditional functional, while the second, thid, and fouth ows ae fo the new functional with γ =.5,. and., espectively. Each column is associated with a given monito function. The left column of Figue 7 shows that the mesh concentation is badly misplaced fo both the taditional and new functionals using the hamonic mapping monito function (p = ). In this case the taditional functional gives exactly the hamonic mapping method used by Dvinsky [4]. Note that the new functional does not wok well, as expected, since g = and J = constant, giving no contol of mesh concentation. Fo the aclength monito function (p =, the ight column of Figue 7), the taditional functional still poduces the mismatched concentation. Howeve, since g = λ and the equidistibution becomes J λ = constant, the new functional beas the featue of equidistibution and leads to the coect concentation when a small value of γ is used. Inteestingly, with the Winslow-type monito function, both the taditional and new functionals geneate coect mesh concentation see the left column of Figue 8. Fo the case of stong concentation with p = 2 (see the ight column of Figue 8), the new functional poduces the coect esults, wheeas the taditional one seems to oveconcentate mesh points inside the cicle x 2 + y 2 =.2 2, although thee is also concentation aound the cicle. Fom these two figues one can also see that the new functional with γ =.5leads to esults simila to but slightly less adaptive than those obtained with the taditional functional. Example 5.2. The second example is to geneate adaptive meshes fo the monito function (.4)with (5.3) λ =+ a e 2 /a, a =.. The goal is now to geneate adaptive meshes with highe point concentation at the oigin.

16 25 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see (a): old, p = (c): new, gamma =.5, p = (e): new, gamma =., p = - (g): new, gamma =., p = (b): old, p = (d): new, gamma =.5, p = (f): new, gamma =., p = (h): new, gamma =., p = Fig. 7. Adaptive meshes ae obtained fo Example 5. with the hamonic mapping (p = ) and aclength (p =) monito functions. Desiable mesh point concentation is aound the cicle x 2 + y 2 =.2 2 (the bold solid cicle).

17 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 25 Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see (a): old, p = (c): new, gamma =.5, p = (e): new, gamma =., p = (g): new, gamma =., p = (b): old, p = (d): new, gamma =.5, p = (f): new, gamma =., p = (h): new, gamma =., p = Fig. 8. Adaptive meshes ae obtained fo Example 5. with the Winslow-type (p =) and stong concentation (p =2) monito functions. Desiable mesh point concentation is aound the cicle x 2 + y 2 =.2 2 (the bold solid cicle).

18 252 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see The meshes obtained ae shown in Figues 9 and. The esults confim the obsevations made in Example 5. and the analysis given in the peceding sections. That is, the taditional functional misplaces meshes fo the hamonic mapping and aclength monito functions and coectly places them fo the Winslow-type and stong concentation monito functions; the new functional with γ =.5leads to meshes simila to but slightly less adaptive than those obtained with the taditional functional; and the new functional with a small value of γ leads to meshes with coect concentation when the aclength, Winslow-type, o stong concentation monito function is used. 6. The taditional functional fo spheicallysymmetic poblems. A simila analysis can be caied out fo the taditional functional applied to spheically symmetic poblems in thee dimensions. Conside ( (6.) I tad [ξ,η,ζ] = ξ T G ξ + η T G η + ζ T G ζ ) dxdydz, Ω whee Ω = {(x, y, z) x 2 + y 2 + z 2 < }. TakeΩ c = {(ξ,η,ζ) ξ 2 + η 2 + ζ 2 < }, and let the spheical coodinates fo Ω and Ω c be x = sin(θ)cos(φ), ξ = R sin(θ)cos(φ), y = sin(θ)sin(φ), η = R sin(θ)sin(φ), z = cos(θ), ζ = R cos(θ). Conside the case whee the physical solution is spheically symmetic. Assume that the coesponding mesh adaptation is also spheically symmetic, i.e., (6.2) R = R(), Θ=θ, Φ=φ. Then it is easonable to use the monito function in the fom (6.3) G = λ ()e e T + λ 2 ()e θ e T θ + λ 3 ()e φ e T φ, whee e, e θ, and e φ ae the unit vectos in the adial, latitudinal, and longitudinal axes. Unde the symmety assumption, (6.)educes to [ ( ) 2 dr I tad [R] = + 2 ( ) ] 2 R 2 d, λ d λ 23 whee λ 23 is defined as 2 λ 23 = λ 2 + λ 3. The coesponding bounday value poblem is given by d ( ) 2 dr + 2 R =, d λ d λ 23 (6.4) R()=, R()=. The tansfomation (4.)can be used fo analyzing (6.4). We obtain the equation fo v v = λ ( 2 v ) (6.5) v 2 λ λ 23 λ

19 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 253 Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see (a): old, p = (c): new, gamma =.5, p = (e): new, gamma =., p = - (g): new, gamma =., p = (b): old, p = (d): new, gamma =.5, p = (f): new, gamma =., p = (h): new, gamma =., p = Fig. 9. Adaptive meshes ae obtained fo Example 5.2 with the hamonic mapping (p = ) and aclength (p =) monito functions. Desiable mesh point concentation is nea the oigin.

20 254 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see (a): old, p = (b): old, p = (c): new, gamma =.5, p = (d): new, gamma =.5, p = (e): new, gamma =., p = (f): new, gamma =., p = (g): new, gamma =., p = (h): new, gamma =., p = Fig.. Adaptive meshes ae obtained fo Example 5.2 with the Winslow-type (p = ) and stong concentation (p =2) monito functions. Desiable mesh point concentation is nea the oigin.

21 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 255 Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see which is subject to the initial condition (6.6) v()= 2 λ () λ ()λ 23 () 2λ (). It is staightfowad to show that the solution v of (6.5)has the popeties stated in Lemmas In the cuent situation, the mesh density has the fom D() = R2 dr 2 d. Its ate of change eads as [ D () = e3s 3 (λ v) 3(λ v) + 3(λ v) 2 ] [ ( ( = e3s 3 λ v + 2λ2 v ) 2 ( + ) )] (6.7) λ λ λ 23 λ 2. We have the following theoems which ae basically identical to Theoems 4. and 4.2. One may notice that in this thee-dimensional case, the elation between λ and λ 23, athe than those between λ and each of λ 2 and λ 3, plays a ole in affecting the coesponding mesh adaptation. Theoem 6.. (i) If λ () λ 23 (), then D () has the same sign as λ () λ 23 (), whethe =is a maximum point of λ o not. (ii) Let λ 23 () =λ (). If λ (), then D ()λ () >. If λ ()= but λ (), then D ()λ () >. Theoem 6.2. Let λ >. (i) If λ ( λ ) >λ 2 ( λ ), then D ( λ ) > and thus D > λ. (ii) Futhe, if we assume that λ 23 () =λ () and λ is a stict maximum point of λ (i.e., λ ( λ ) < ), then D ( λ ) > o D > λ. 7. Conclusions and comments. The question of how vaiational gid geneatos behave when solving poblems with axisymmetic solutions has been investigated. Specifically, two functionals have been analyzed in the pevious sections fo thei abilities to pecisely contol the mesh concentation via monito functions. One is the taditional functional (.)which includes Winslow s method and Dvinsky s method of hamonic mappings as special cases. The othe is the new functional (.2)poposed by Huang in [6] which explicitly includes the isotopy (o egulaity)and equidistibution featues. The analysis is pimaily done fo axisymmetical poblems in two dimensions. Fo axially symmetic mesh adaptation, it is easonable to use a monito function of the fom in (.4). Theoetical esults fo the taditional functional ae given in Theoems 4. and 4.2. Specifically, when highe mesh concentation at the oigin is desied, a choice of the adial and angula components λ and λ 2 of the monito function satisfying λ () <λ 2 ()will make the mesh dense at = than in the suounding aea whethe o not λ has a maximum value at =. The pupose can also be seved by choosing λ to have a local maximum at = when a Winslow-type monito function with λ () =λ 2 ()is employed. Unfotunately, the choice λ 2 () =λ () p with p<, which includes Dvinsky s method of hamonic mappings and the aclength monito function as special cases, will not satisfy the condition λ () <λ 2 ()if λ () > (as

22 256 CAO, CARRETERO-GONZÁLEZ, HUANG AND RUSSELL Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see commonly taken in the liteatue)and leads to a mesh with coase concentation of points in the cente than in the suounding aea. On the othe hand, when highe mesh concentation aound a ing = λ > is desied, the taditional functional povides fa less contol by choosing λ and λ 2. Indeed, Theoem 4.2 shows that thee suely is a mismatch between the position λ of the maximum of λ and the location D of the maximum of the mesh density if eithe (a) λ ( λ ) >λ 2 ( λ )(which is the case fo the hamonic mapping o the aclength monito function with λ ( λ ) > )o (b)a Winslow-type monito function is used and λ is a stict maximum point of λ. Moeove, a mismatch between D and λ is also possible fo the case λ ( λ ) <λ 2 ( λ ). Indeed, the numeical esults show that D can be located to eithe side of λ when λ ( λ ) > and λ 2 is taken as λ 2 = λ p fo p>. Nevetheless, the numeical esults suggest that D λ is elatively small fo the Winslow case λ 2 = λ. The analysis also shows that fo the hamonic mapping case λ 2 =/λ, D can be made to agee with λ by escaling λ such that λ ( λ )=. Howeve, this can be done if the mesh concentation is needed only at the location of the (global)maximum of λ. Fo axially symmetic poblems, the new functional leads to a nonlinea mesh equation too complex to pemit an analysis like that fo the taditional functional. Nevetheless, numeical esults pesented in sections 4 and 5 show that the new functional offes explicit contol fo mesh concentation by adjusting the value of γ that weights the isotopy and equidistibution. Specifically, when using a lage value of γ (close to /2)we obtain simila esults to those fo the taditional functional cases. Howeve, as we decease γ, the new functional weights moe towads equidistibution, and both the location of the maximum and the pofile of the mesh density tend to coincide with those of λ fo a monito function with a nonconstant deteminant. Fo the case of the hamonic mapping monito function, the deteminant is g = and equidistibution becomes J = constant so no contol of mesh concentation is possible by choosing λ. Thus, as expected, the new functional does not wok in this case even when a small value of γ is used. Analysis has also been caied out fo the taditional functional applied to spheically symmetic poblems in thee dimensions. The esults ae stated in Theoems 6. and 6.2. In the futue we intend to investigate a numbe of highe-dimensional axisymmetical poblems aising in physical applications and show the pacticability of the methods which have pefomed well hee. Acknowledgment. The authos ae gateful to the efeees fo thei valuable comments. REFERENCES [] J. U. Backbill and J. S. Saltzman, Adaptive zoning fo singula poblems in two dimensions, J. Comput. Phys., 46 (982), pp [2] W. Cao, W. Huang, and R. D. Russell, A study of monito functions fo two-dimensional adaptive mesh geneation, SIAM J. Sci. Comput., 2 (999), pp [3] C. de Boo, Good appoximation by splines with vaiable knots II, in Poceedings of the Confeence on the Numeical Solution of Diffeential Equations, Dundee, Scotland, 973, Lectue Notes in Math. 363, G. A. Watson, ed., Spinge-Velag, Belin, 974, pp [4] A. S. Dvinsky, Adaptive gid geneation fom hamonic maps on Riemannian manifolds, J. Comput. Phys., 95 (99), pp [5] W. Huang, Pactical aspects of fomulation and solution of moving mesh patial diffeential equations, J. Comput. Phys., 7 (2), pp

23 ADAPTIVE MESHES FOR AXISYMMETRICAL PROBLEMS 257 Downloaded 9/5/4 to Redistibution subject to SIAM license o copyight; see [6] W. Huang, Vaiational mesh adaptation: Isotopy and equidistibution, J. Comput. Phys., 74 (2), pp [7] W. Huang and R. D. Russell, A high dimensional moving mesh stategy, Appl. Nume. Math., 26 (997), pp [8] W. Huang and W. Sun, Vaiational mesh adaptation II: Eo estimates and monito functions, J. Comput. Phys., to appea. [9] P. Knupp, Mesh geneation using vecto-fields, J. Comput. Phys., 9 (995), pp [] P. M. Knupp, Jacobian-weighted elliptic gid geneation, SIAM J. Sci. Comput., 7 (996), pp [] P. M. Knupp and N. Robidoux, A famewok fo vaiational gid geneation: Conditioning the Jacobian matix with matix noms, SIAM J. Sci. Comput., 2 (2), pp [2] W. Ren and X.-P. Wang, An iteative gid edistibution method fo singula poblems in multiple dimensions, J. Comput. Phys., 59 (2), pp [3] S. Steinbeg and P. J. Roache, Vaiational gid geneation, Nume. Methods Patial Diffeential Equations, 2 (986), pp [4] A. Winslow, Numeical solution of the quasi-linea Poisson equation in a nonunifom tiangle mesh, J. Comput. Phys., (967), pp [5] A. M. Winslow, Adaptive Mesh Zoning by the Equipotential Method, Technical epot UCID- 962, Lawence Livemoe Laboatoy, Livemoe, CA, 98.