A SIMPLE FINITE ELEMENT METHOD OF THE CAUCHY PROBLEM FOR POISSON EQUATION. We consider the Cauchy problem for Poisson equation

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1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volum 14, Numbr 4-5, Pags c 2017 Institut for Scintific Computing and Information A SIMPLE FINITE ELEMENT METHOD OF THE CAUCHY PROBLEM FOR POISSON EQUATION XIAOZHE HU, LIN MU, AND XIU YE Abstract. In this papr, w introduc a simpl mthod for th Cauchy problm. This nw finit lmnt mthod is basd on last squars mthodology with discontinuous approximations which can b implmntd and analyzd asily. This discontinuous Galrkin finit lmnt mthod is flxibl to work with gnral unstructurd mshs. Error stimats of th finit lmnt solution ar drivd. Th numrical xampls ar prsntd to dmonstrat th robustnss and flxibility of th proposd mthod. Ky words. Finit lmnt mthods, Cauchy problm, polyhdral mshs. 1. Introduction (1) W considr th Cauchy problm for Poisson quation u =f, in Ω, u =0, on Γ 1, u n =g, on Γ 1, whrωisabounddconvxpolytopaldomaininr d withd = 2,3and Ω = Γ 1 Γ 2. Assum that Γ 1 is simply connctd. Th Cauchy problm (1) is wll-known to b ill-posd [1, 4]. It has applications in many diffrnt aras such as plasma physic, lctrocardiography, and corrosion non-dstructiv valuation (.g., [5, 12, 16, 21]). Du to th ill-posdnss, th numrical approximation of th Cauchy problm is vry difficult and challnging. Traditionally, rgularization tchniqus, such as Tikhonov rgularization [26] and th quasi-rvrsibility approach [23], wr usd to provid robust numrical schms. Many diffrnt finit lmnt mthods hav also bn dvlopd for solving th Cauchy problm (1). In [15, 24, 25], Galrkin typ approachs ar proposd basd on structurd grids or spcial formulation of th continuous problm. Th rgularization tchniqus ar also usd in finit lmnt sttings,.g., [6, 3, 7, 13]. In [2, 11, 19, 18], th Cauchy problm (1) is rformulatd as minimization problms and thn solvd numrically with possibl rgularizations. Mor rcntly, primaldual formulation is proposd and solvd by discontinuous Galrkin (DG) finit lmnt mthods with suitabl stabilization/rgularization, s [9, 10]. Th purpos of this papr is to dvlop a simpl finit lmnt mthod to approximat th solution of th Cauchy problm (1) whn it xists and is uniqu. This mthod is dsignd aiming on asy implmntation and asy rror analysis. Th mthodology of th schm is combining th last squars tchniqu with discontinuous approximations. Suitabl stabilization trms ar addd to nsur th Rcivd by th ditors on Dcmbr 31, 2016, accptd on April 21, Mathmatics Subjct Classification. Primary, 65N15, 65N30, 76D07; Scondary, 35B45, 35J

2 592 X. HU, L. MU, AND X. YE stability of th discrtization. As a rsult, our mthod lads to a symmtric and positiv dfinit linar systm of quations and is flxibl to us on gnral polygonal mshs with hanging nods. W prov that our discontinus finit lmnt solution approachs to th solution of th modl problm (1) whn th msh siz approachs to zro. Convrgnc rat ar studid in both nrgy norm and L 2 -norm basd on th conditional stability of th continus Cauchy problm. Comparing with xisting mthods, our approach is attractiv du to its simplicity. Th numrical rsults also show th fficincy of th proposd approach which confirms our thortical rsults. Th rst of th papr is organizd as follows. In Sction 2, w rcall Cauchy problm and its conditional stability rsults basd on a traditional wak formulation. Our nw simpl discrtization is givn in Sction 3. W study its stability and rror stimats in Sction 4 and 5, rspctivly. Finally, w prsnt som numrical xprimnts to dmonstrat th stability of th WG formulation in Sction Cauchy Problm W dnot th standard Lbsgu sapcs by L 2 (D) and D R d, d = 2,3, with corrsponding norms L2 (D) (or D ). H s (D) dnot th standard Sobolv spac of indx s 0 along with th corrsponding norm and smi-norm H s (D) (or s,d ) and H s (D) (or s,d ), rspctivly. For th Cauchy problm (1), if th (d 1)-masur of Γ 2 is nonmpty, it is an ill-conditiond problm. In practic, as shown in [4], such Cauchy problm is not wll-posd du to masurmnt rrors. Howvr, following th traditional argumnts, if th undrlying physical procss is stabl, i.., if th boundary data ar known on th whol boundary, thn th problm is wll-posd, it is natural to assumthat th Cauchyproblm(1) hasauniqu solutionin th idalizd caswith unprturbd data. Thrfor, w assum that f L 2 (Ω), g H 1 2(Γ 1 ), and that thr is a uniqu solution u H 2 (Ω) satisfis (1). Our analysiswill b basd on this assumption and th so-calld conditional stability dscribd latr in Sction A Traditional Wak Formulation. In ordr to introduc th conditional stability of th Cauchy problm (1), w nd to first look at th wak formulation of th Cauchy problm (1). Following [1], w introduc two Sobolv spacs and H 1 Γ 1 (Ω) := {v H 1 (Ω) : v Γ1 = 0}, H 1 Γ 2 (Ω) := {v H 1 (Ω) : v Ω\Γ1 = 0}. Th wak formulation for (1) is: find u H 1 Γ 1 (Ω) such that (2) a 0 (u,v) = l(v), v H 1 Γ 2 (Ω), whr a 0 (u,v) := Ω u v dx, and l(v) := fvdx+ gvds. Ω Γ 1 Again, w ar not assuming this wak formulation of Cauchy problm is wll-posd sinc inf-sup stability dos not hold in gnral [4].

3 SIMPLE FEM OF THE CAUCHY PROBLEM 593 Rmark 2.1. Although th wak formulation (2) can b naturally drivd from th Cauchy problm (1), which maks it a usual choic whn dvloping numrical schms (s,.g., [9, 10]). Our numrical schm is not dsignd basd on this wak formulation Conditional Stability. It is wll-known that th Cauchy problm (1) and th wak formulation (2) is ill-conditiond in gnral. Thrfor, w us a conditional stability rsult [1, 10] in our rror stimats, which can b statd as th following thorm as prsntd in [10]. Thorm 2.1 ([10], Sction 2.1, quation (2.5)). Lt Ω b a connctd opn st of Lipschitz class. Lt u H 1 Γ 1 (Ω) b a wak solution to th Cauchy problm (2), whr f L 2 (Ω), g H 1 2 (Γ1 ), and l (H 1 Γ2 ) ǫ. If u satisfis u H 1 (Ω) E, thn u L 2 (Ω) ω(ǫ) whr ω(t) = C 1 (E)( log(t) +C 2 (E)) µ, for C 1 (E),C 2 (E) > 0, 0 < t < 1 and 0 < µ < 1. As w shall s latr, this conditional stability of th Cauchy problm (2) hlps us to driv a L 2 rror stimat for our proposd schm. 3. A Discontinuous Finit Elmnt Mthods In this sction, w propos our nw discontinuous Galrkin finit lmnt mthod for solving th Cauchy problm (1). In ordr to do that, w first introduc som notations that ar ndd for th discrtizations. Lt T h b a shap rgular partition of th domain Ω consisting of polygons in two dimnsion or polyhdrons in thr dimnsion satisfying a st of conditions spcifid in [27]. Dnot by E h th st of all dgs or flat facs in T h and lt E 0 h = E h\ Ω b th st of all intrior dgs or flat facs. For vry lmnt T T h, w dnot by h T its diamtr and th msh siz of th whol partition T h is dfind as h = max T Th h T. Morovr, w us h to dnot th diamtr of an dg or a flat fac E h. Nxt, basd on th partition T h, w can dfin a discontinuous finit lmnt spac V h as follows for k 2, (3) V h = {v L 2 (Ω) : v T P k (T), T T }. Sinc w us discontinuous functions, w nd dfin jumps as usual. Lt two nighboring lmnts T 1 and T 2 hav as a common dg/fac and unit normal vctors n 1 and n 2 on pointing xtrior to T 1 and T 2, rspctivly. W thn dfin jumps [φ] and [ φ n] on Eh 0 as following [φ] := φ T1 n 1 +φ T2 n 2, [ φ n] := φ T1 n 1 + φ T2 n 2, and for Γ 1, w dfin [φ] := φ, [ φ n] := φ n.

4 594 X. HU, L. MU, AND X. YE Basd on thos jumps, now w introduc two bilinar forms as following s(v,w) := (h 3 [v][w]+h 1 [ v n][ w n])ds, a(v,w) := ( v, w) T +s(v,w), whr (, ) T is th usual L 2 -innr product on a lmnt T T h. Now w ar in th position to propos our discontinuous finit lmnt discrtization for th Cauchy problm (1) as follows, Algorithm 1. A numrical approximation for (1) can b obtaind by sking u h V h satisfying th following quation: (4) a(u h,v) = (f, v)+ h 1 g v nds, v V h. Γ 1 Rmark 3.1. Hr, for th sak of th simplicity, w only considr th cas that both th Dirichlt and Numann boundary conditions ar dfind on th sam subst Γ 1 of th boundary Ω. Howvr, th discontinuous finit lmnt mthod and th rsults w will prsnt in th following sctions can b simply adjustd so that thy still hold for th gnral cas whn Dirichlt and Numann boundary condition parts ar diffrnt. 4. Stability Estimats Bcaus th continuous problm (1) is ill-posd, it is important to study th stability of th discrtization (4), i.., th wll-posdnss of th proposd discontinuous Galrkin finit lmnt schm. This is th main focus of this sction. For studying th stability, w nd to choos a suitabl norm and show that th DG discrtization (4) is wll-posd with rspct to this norm. Lt us first dfin a smi-norm inducd by th bilinar form a(, ) as following (5) v 2 := a(v,v) = v 2 T +s(v,v) = v 2 T + h 3 [v]2 ds+ and anothr smi-norm inducd by s(, ) as v 2 s := s(v,v). Nxt Lmma shows that th smi-norm actually is a norm. h 1 [ v n]2 ds. Lmma 4.1. Undr th assumption that th Cauchy problm (1) has a uniqu solution, th smi-norm dfins a norm. Proof. It is sufficint to show that v = 0 implis v = 0. Not that whn v = 0, whav v = 0 onachlmnt T and v and v n arcontinuousacrossth dgs. Morovr, v = 0 and v n = 0 on Γ 1. Thrfor, v is a solution of th Cauchy problm (1) with f = g = 0. By th assumption of uniqunss of th solution, w hav v = 0 which shows that is in fact a norm.

5 SIMPLE FEM OF THE CAUCHY PROBLEM 595 Basd on th dfinition of th norm, w can show th following fundamntal stability rsult for th DG discrtization (4), i.., th corcivity and continuity of th bilinar form a(, ). Lmma 4.2. Th bilinar form a(, ) satisfis th following continuity proprty, (6) a(v, w) C v w, and th corcivity proprty (7) a(v,v) = v 2. Proof. (7) follows from th dfinition and (6) follows from th Cauchy-Schwarz inquality, a(v,w) = ( v, w) T + (h 3 [v][w]ds+h 1 [ v n][ w n])ds E h \Γ 2 ( v 2 T) 1 2 ( w 2 T) ( h 3 [v] 2 ds) 1 2 ( h 3 [w] 2 ds) ( h 1 [ v n] 2 ds) 1 2 ( h 1 [ w n] 2 ) 1 2 C v w. W hav provd th lmma. Now w naturally hav th wll-posdnss rsult of th DG discrtization (4). Thorm 4.1. Undr th assumption that th Cauchy problm (1) has a uniqu solution, th discontinuous Galrkin finit lmnt schm (4) has a uniqu solution. Proof. This is a dirct rsult of Lmma 4.2 and Lax-Milgram Thorm. Rmark 4.1. Du to th simplicity of our DG schm, and th naturally inducd norm, th stability rsult can b obtaind asily following th standard argumnts. This is diffrnt from th primal dual formulation proposd rcntly in [10], whr a inf-sup condition is ndd. 5. Error Estimat In th prvious sction, w hav shown that th proposd DG schm (4) is wll-posd. In this sction, w focus on th rror stimats and driv th rror analysis for th DG solution u h obtaind from (4). Error stimats in both nrgy norm and L 2 -norm ar considrd. First, w rcall th standard trac inquality which is usd latr in our analysis. For any function ϕ H 1 (T), th following trac inquality holds tru (s [27] for dtails): (8) ϕ 2 C ( h 1 T ϕ 2 T +h T ϕ 2 T).

6 596 X. HU, L. MU, AND X. YE NxtwdfinthL 2 -projctionasusual. Foranyv L 2 (Ω), thl 2 -projction Q h v V h is dfind as following (9) (Q h v,w) = (v,w), w V h. Th following lmma shows th approximation proprty of th L 2 -projction undr crtain rgularity assumption of th functions. Lmma 5.1. Lt v H k+1 (Ω), k 2, and Q h v V h b th L 2 projction of v. Thn thr xist a constant C indpndnt of th msh siz h, such that, (10) v Q h v Ch k 1 v k+1. Proof. Using th dfinition of Q h (9), and th trac inquality (8), w hav v Q h v 2 = (v Q h v) 2 T + [v Q h v] 2 +h 1 [ (v Q h v) n] 2 ) C (h 3 ( v Q h v 2 2,T +h 4 v Q h v 2 T +h 2 (v Q h v) 2 T ) Ch 2k 2 v 2 k+1, which finishs th proof of th lmma. Now w ar rady to driv th rror stimat in th nrgy norm. Th analysis follows dirctly from th corcivity (7) and continuity (6) of th bilinar from a(, ) and th approximation proprty of th L 2 -projction (10). Thorm 5.1. Lt u h V h b th finit lmnt solution of th problm (1) obtaind by th DG discrtization (4). Undr th assumption that th Cauchy problm (1) has a uniqu solution u and furthr assum that u H k+1 (Ω) H 1 Γ 1 (Ω), k 2, thn thr xists a constant C indpndnt of th msh siz h, such that, (11) u u h Ch k 1 u k+1. Proof. Obviously, th tru solution u of (1) satisfis a(u,v) = (f, v)+ h 1 g v nds. Γ 1 Subtracting (4) from th abov quation implis Thn w hav, for any v V h, a(u u h,v) = 0, v V h. u u h 2 = a(u u h,u u h ) = a(u u h,u v)+a(u u h,v u h ) = a(u u h,u v), which, by th continuity of th bilinar form a(, ), implis u u h C inf v V h u v. Now,bychoosingv = Q h uandusingthapproximationrsults(10), wobtain(11), which complts th proof.

7 SIMPLE FEM OF THE CAUCHY PROBLEM 597 Nxt, w considr th L 2 rror stimat. Du to th lack of rgularity of th Cauchy problm, standard duality argumnt dos not work. Hr, w follow th ida from [10] and dcompos th rror u u h = (u ū h )+(ū h u h ), whr ū h is a C 1 -conforming approximation of u constructd from u h. For th sak of simplicity, w rstrict our discussion on triangular or rctangular mshs in two dimnsion or ttrahdral or hxahdral mshs in thr dimnsion. On thos mshs, such construction can b don on C 1 -conforming spac consisting of macro-lmnts of dgr k +2. As pointd out in [17], this is don by a rcovry oprator which is constructd via avrags of th nodal basis functions (cf., [22, 8, 20]) and is a highordr vrsion of th classical Hish-Clough-Tochr lmnt [14]. Sinc w only nd th xistnc of such a ū h and th construction is not ssntial to our analysis, w do not giv th complt construction hr and rfr intrstd radrs to [17] for a dtaild dscription. Not that, in our cas, w only nd ū h H 1 Γ 1 (Ω). Thrfor, w can us a slightly modification of th construction givn in [17] so that only th nodal valus of ū h on Γ 1 vanish but othr dgrs of frdom ar dfind via avraging. Following th similar argumnt in [17], w hav th following lmma about th approximation proprty of ū h. Lmma 5.2 (Lmma 3.1 [17]). Lt ū h b constructd as shown in Dfinition 3.1 [17], w hav (12) ( ) u h ū h 2 T +h2 u h ū h 2 1,T +h4 u h ū h 2 2,T Ch 4 u h 2 E h. whr v 2 E h := h 3 [v] 2 ds+ Eh 0 h 1 [ v n] 2 ds and C is a constant indpndnt of h and u h. Now w hav (u ū h ) H 1 Γ 1 (Ω) and w can think it satisfis th Cauchy problm in th wak sns, i.. th wak formulation (2) with a diffrnt right hand sid, i.., (13) a 0 (u ū h,v) = l(v), v H 1 Γ 2 (Ω). For th L 2 rror stimats, w nd to stimat l (H 1 Γ2 (Ω)). l (H 1 Γ2 (Ω)) = sup 0 v H 1 Γ 2 (Ω) l(v) v H 1 a 0 (u ū h,v) = sup 0 v HΓ 1 (Ω) v H 1 (Ω) 2 Ω = sup (u ū h)vdx+ Γ 1 (g ū h n)vds 0 v HΓ 1 (Ω) v H 1 (Ω) 2 (u ū h) v + Γ 1 h 1/2 g ū h n h 1/2 v C ( v H1 (Ω) (u ū h ) 2 + Γ 1 h 1 g ū h n 2 ) 1/2 (14) = C u ū h.

8 598 X. HU, L. MU, AND X. YE Not that, sinc a(u u h,v h ) = 0, v h V h, and u h ū h V h, w hav (15) u ū h 2 = u u h 2 + u h ū h 2 = u u h (h 3 u u h 2 +C (u h ū h ) 2 T [u h ū h ] 2 +h 1 [ (u h ū h ) n] 2 ) ( (u h ū h ) 2 T +h 4 u h ū h 2 T +h 2 (u h ū h ) 2 T ) u u h 2 +C u h 2 E h. Basd on th dfinition of Eh and u H 2 (Ω) H 1 Γ 1 (Ω), w hav (16) u h 2 E h = u u h 2 E h u u h 2 s u u h 2. Thrfor, substitut (15) and (16) back into (14) and apply Thorm 5.1, w hav th following stimat l (H 1 Γ2 (Ω)) C u u h Ch k 1 u k+1, and, if h is small nough, w hav l (H 1 Γ2 (Ω)) ǫ, which is ndd for th L2 rror stimat. In ordr to apply Thorm 2.1, w also nd to show that u ū h H1 (Ω) is boundd. To this nd, sinc (u ū h ) H 2 (Ω) H 1 Γ 1 (Ω), thn apply th classical Poincaré-Fridrichs inquality, w hav u ū h H1 (Ω) C( u ū h H2 (Ω) + (u ū h ) n Γ1 ) C u ū h. Thn us (15) and (16) and apply Thorm (5.1), w hav (17) u ū h H 1 (Ω) C u u h Ch k 1 u k+1 := E h. This mans th H 1 -norm conforming part of th rror is boundd. Basd on th abov stimat and applying Thorm 2.1, w hav th following rror stimat in th L 2 -norm. Thorm 5.2. Lt u H k+1 (Ω) H 1 Γ 1 (Ω) b th solution of th Cauchy problm (1) and u h V h b th finit lmnt solution of th Cauchy problm (1) obtaind by th DG discrtization (4) and E h is dfind in (17). Assum that h is small nough so that l (H 1 Γ2 ) ǫ holds. Hr l is dfind in (13), thn w hav (18) u u h L 2 (Ω) ω(ǫ)+ch 2 u h Eh. whr ω(t) is dfind in Thorm 2.1. Furthrmor, w hav, (19) u u h L 2 (Ω) ω(ǫ)+ch k+1 u k+1. Hr C is a constant indpndnt of th msh siz h. Proof. As shown in (17), u ū h H1 (Ω) E h, thn apply Thorm 2.1, w hav u ū h L2 (Ω) ω(ǫ), whr ω is dfind in Thorm 2.1. Thn apply th triangular inquality and stimat (12), w obtain (18). (19) can b obtaind from (18) and (16) dirctly, which complts th proof.

9 SIMPLE FEM OF THE CAUCHY PROBLEM 599 Rmark 5.1. FromthrrorstimatinthL 2 norm,wcansthatitcontains an optimal part (h k+1 u k+1 ). Howvr, th convrgnc bhavior of th othr trm ω(ǫ) dpnds on th conditional stability of th original Cauchy problm (1). Thrfor, th ovrall convrgnc rat also dpnds on th conditional stability. In th numrical xprimnts, w obsrvsuboptimal convrgncrat whn Γ 1 Ω. Th dtails will b givn in th nxt sction. 6. Numrical Exprimnts In this sction, w shall apply th proposd numrical schms (4) to two tst problms for validating th thortical conclusions in th prvious sctions and dmonstrating th ffctivnss of th proposd DG mthod Tst 1. Considr th Cauchy problm (1) on Ω = (0,1) (0,1) and lt f = 2π 2 (4cos(πx) 2 cos(πy) 2 3cos(πx) 2 3cos(πy) 2 +2) such that th xact solution of (1) is givn as th following, u = sin 2 (πx)sin 2 (πy). W shall considr thr diffrnt choics of th boundary Γ 1, which ar shown in th Figur 1. In this tst, du to th choic of xact solution, it is asy to s that quation (1) is coupld with homognous boundary condition, i.. g = 0. Γ 1 Γ 1 Γ 1 Γ 1 Γ 1 Γ 1 Γ 1 (a) (b) (c) Figur 1. Diffrnt choics of boundary Γ 1 : (a) Cas 1: Γ 1 = {x = 0;x = 1;y = 0;y = 1}; (b) Cas 2: Γ 1 = {x = 1,y = 1}; (c) Cas 3: Γ 1 = {x = 1}. Uniform rctangular msh is mployd in th numrical xprimnt. Dnot th siz of msh as h. W prform th proposd numrical schm with polynomials of dgr two, i.., k = 2. Th rror profils and convrgnc tsts ar rportd in Tabl 1. From Thorm 5.1, it is xpctd that th nrgy norm convrgs with optimal ordr O(h), which is vrifid in th numrical tst. Th rror masurd in th L 2 -norm has also bn rportd in Tabl 1. It can b sn that th convrgnc rat for L 2 -rror ar affctd by th choics of th boundary Γ 1. Whn Γ 1 Ω (cas 2 and 3), th convrgnc rat dtriorats as xpctd according to th conditional stability of th Cauchy problm (1) and rror stimats givn in Thorm 5.2.

10 600 X. HU, L. MU, AND X. YE Tabl 1. Tst 1: Error Profils and Convrgnc Tst with Γ 1 for k = 2. 1/h u u h ordr u u h ordr Cas 1 with Γ 1 = Ω E E E E E E E E E E E E E E Cas 2 with Γ 1 = {x = 1,y = 1} E E E E E E E E E E E E E E Cas 3 with Γ 1 = {x = 1} E E E E E E E E E E E E E E Cas 2. In this numrical xprimnt, w shall prform th numrical schm(4) to th Cauchy problm (1) with non-homognous boundary conditions. Lt th domain b Ω = (0,1) (0,1) and choos th right-hand sid function as th following, Th xact solution is f = 2π 2 cos(πx)cos(πy). u = cos(πx)cos(πy). Th diffrnt choics of th boundary Γ 1 ar th sam as th thr choics in th first tst as shown in Figur 1. Again, w us DG schm (4) with polynomials of dgr two, i.., k = 2, and rport th rrors and convrgnc rat in Tabl 2 and w can s that th convrgnc rat for th nrgy norm rror has optimal ordr O(h), which confirms th thortical conclusion givn in Thorm 5.1. For th rror in th L 2 -norm, similar with th first numrical xampl, w obsrv dgnratd rat whn Γ 1 Ω, which confirms Thorm 5.2 du to th conditional stability of th Cauchy problm (1).

11 SIMPLE FEM OF THE CAUCHY PROBLEM 601 Tabl 2. Tst 2: Error Profils and Convrgnc Tst with Γ 1 with k = 2. 1/h u u h ordr u u h ordr Cas 1 with Γ 1 = Ω E E E E E E E E E E E E E E Cas 2 with Γ 1 = {x = 1,y = 1} E E E E E E E E E E E E E E Cas 3 with Γ 1 = {x = 1} E E E E E E E E E E E E E E Acknowldgmnts Th valuabl and carful commnts offrd by anonymous rfrs ar gratfully acknowldgd. This matrial is basd upon work supportd by th U.S. Dpartmnt of Enrgy, Offic of Scinc, Offic of Advancd Scintific Computing Rsarch.This manuscript has bn authord by UT-Battll, LLC undr Contract No. DE-AC05-00OR22725 with th U.S. Dpartmnt of Enrgy. Th Unitd Stats Govrnmnt rtains and th publishr, by accpting th articl for publication, acknowldgs that th Unitd Stats Govrnmnt rtains a non-xclusiv, paid-up, irrvocabl, world-wid licns to publish or rproduc th publishd form of this manuscript, or allow othrs to do so, for Unitd Stats Govrnmnt purposs. Th Dpartmnt of Enrgy will provid public accss to ths rsults of fdrally sponsord rsarch in accordanc with th DOE Public Accss Plan ( work of X. Hu was supportd in part by th National Scinc Foundation undr grant DMS Th work of X. Y was supportd in part by National Scinc Foundation Grant DMS Rfrncs [1] G. Alssandrini, L. Rondi, E. Rosst, and S. Vsslla, Th stability for th Cauchy problm for lliptic quations, Invrs problms, 25 (2009), p

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13 SIMPLE FEM OF THE CAUCHY PROBLEM 603 [26] A. N. Tikhonov and V. Y. Arsnin, Solutions of ill-posd problms, Winston, [27] J. Wang and X. Y, A wak Galrkin mixd finit lmnt mthod for scond ordr lliptic problms, Mathmatics of Computation, 83 (2014), pp Dpartmnt of Mathmatics, Tufts Univrsity, Mdford 02155, Unitd Stats Xiaozh.Hu@tufts.du Computr Scinc and Mathmatics Division, Oak Ridg National Laboratory, Oak Ridg, TN 37831, Unitd Stats mul1@ornl.gov Dpartmnt of Mathmatics, Univrsity of Arkansas at Littl Rock, Littl Rock, AR 72204, Unitd Stats , Corrsponding author: xxy@ualr.du