High accuracy analysis of anisotropic finite element method for a class of nonlinear degenerate wave equation

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1 Aalable onlne wwwjocprcom Jornal o emcal an Parmacecal Researc 67:-5 Researc Arcle SSN : ODENSA : JPR5 H accrac analss o ansoropc ne elemen meo or a class o nonlnear eenerae wae eqaon Zan n nan Zao an Sao Jan Scool o Maemacs an Pscs Hanan ollee Hanan na ABSTRAT Te conerence analss o e blnear ne elemen meo o a class o non-lnear eenerae wae eqaon on ansoropc meses s consere n s paper Moreoer e lobal sperconerence or semscree sceme s propose ro nerpolaon nsea o e Rz olerra projecon o e eac solon e wors: Nonlnear eenerae wae eqaon; ansoropc meses spercon- erence NTRODTON s wellknow a wae eqaons can arse rom man pscal process an a lo o em are nonlnear an wel se Sc as e perrke Sne-Goron eqaon wc occr n qanm mecancs a moel n e elaso plascmcrosrcre wc escrbe e lonnal moon o an elaso-plasc bar an an-plane searn n e case o spaces mensonntereoreman maemacans an pscss ocs er aenon o s e nonlnear wae eqaons an ere ae been a lo o mpresse lerares[-] Te sperconerence s o e ne elemen meos s one o e mos ace researc sbjecs bo n eorecal an n praccal compaons Howeer seems a ere are ew ses ocsn on e accrac analss o ne elemen meos or e nonlnear wae eqaons w sspaon especall on e ansoropc meses As a maer o ac n man cases e relar or qas-norm assmpone ρ T were T s a aml o ranlaon o Ω ρ ma T are e ameer o an e bes crcle conane n respecel an s a pose nmber nepenen o an escrbe n [] are rea cences n applcaon o ne elemen meos For eample e solon o some ellpc problems ma ae ansoropc beaor n pars o e ene oman Ts means a e solon onl ares sncanl n ceran recons s an obos ea o relec s ansorop n e screon b sn ansoropc meses w a ner mes sze n e recon o e rap araon o e solon an a coarser mes sze n e perpenclar recon Beses some problems ma be ene n narrow oman or eample n moeln a ap beween roer an saor n an elecrcal macne we emplo e relar paron o e oman e cos o calclaon wll be er Tereore o emplo ansoropc meses w ewer erees o reeom s a beer coce o oercome ese cles Howeer ansoropc elemens are caracerze b were e lm can be consere as n s case e Bramble-Hlber emma can no be se n e esmae o e nerpolaon error We ae o appl e ansoropc ne elemen meo as n [5-7] n s paper we conser e conerence analss o e blnear ne elemen meo o a kn o nonlnear eenerae wae eqaon on ansoropc meses Te sperclose an sperconerence properes or semcree ρ

2 Zan e al J em Parm Res 67:-5 sceme s obane base on e ansoropc nerpolaon eorem propose n [5-7] an e neral enes eelope n [8-] w e elp o nerpolaon o solon o e problem cosere nsea o reerrn o e Rz olerra projecon o e eac solon wc makes e proo raer smpler an e preos ses MODE PROBEM AND TS ARATONA FORMATON onser e ollown nonlnear wae eqaon w sspaon: Ω [ T ] Ω [ T ] Ω were Ω s an open bone se n R w smoo bonar For smplc we assme: an are pscz connos w respec o w pscz consan ; Ω [ T ] s a nqe solon o Ω an are all known ncons s r W r Ω Ω We enoe b e sanar Sobole space o s-erenal ncons n b s r an s r H Ω W s Ω an s s S r s norm an semnorm Troo e paper ncaes a pose consan possbl eren accrrenceswc s nepenen o e mes parameers b ma epen on an T [ ] Ten e weak orm o s o n : T H Ω sc a H Ω Ω ONSTRTON OF THE FNTE EEMENT ˆ [ ] [ ] aˆ ˆ ˆ e be e reerence elemen w or erces a a aˆ l ˆ aˆ ˆ l ˆ aˆ aˆ lˆ aˆ ˆ l ˆ aˆ a ˆ s ene as: an or ees a a an an Te sape ncon space on ˆ Σ ˆ { ˆ ˆ ˆ ˆ } Pˆ { ξ η ξη} span ˆ ˆ a were can be easl cecke a e nerpolaons ene aboe are well-pose an e nerpolaon ncons be epresse as: ˆ ˆ can ˆ ˆ ξ η ˆ ξ η ˆ ξ η ˆ ξ η ˆ as been proe n [9] a e aboe nerpolaon operaor as an ansoropc nerpolaon properes e or an α α α α ere ols Dˆ α Dˆ α ˆ ˆˆ ˆ ˆ ˆ

3 Zan e al J em Parm Res 67:-5 Ω R be a cone polon n - plane compose b a aml o recanlar For e sake o conenence le meses wc oes no nee o sas e relar an qas-norm assmpons[] For an enoe e barcener o elemen b ; e len o ees parallel o -as or -as b respecel am ma e F : ˆ be an ane mappn ene b ξ η Ten e assocae ne elemen space s { ˆ o F Pˆ } Ω : H Dene e nerpolaon operaor Ω Pˆ o F ˆˆ o F Ω : H as THE SEM-DSRETE SHEME AND ONERGENE ANAYSS : T Te appromaon problem corresponn reas as: Fn [ ] Ω sc a W e smlar wa o e proo n []we can e e sem-scree as a nqe solon From an e Scaln ecnqes we can easl e e ollown lemma H [ T ] emma : Ω we ae Teorem : On e poeses o emma an e assmpons are sase ere ols Proo : e rom or an e we e e error eqaon sbracs n 5 en 5

4 Zan e al J em Parm Res 67:-5 { } 6 neran 6 w respec o an non a we ae { } 5 7 sn Yon neqal an emma ollows a 8 B emma assmpon an ac neqal els 5 9 ombnn 8 an 9 an appln Poncar e neqal we see a From Gronwall's emma ere ols Ts B e ranle neqal e esre resl en ollows Te proo s complee THE SPERONERGENE ANAYSS Now we wll rn or aenon o e sperclose proper A rs we sall nroce e ollown mporan lemmas emma []: For an we ae e ollown neqales 5 emma : ] [ T H Ω or an ; we ae O 5 O 5 Proo : Appln emma w e smlar wa o e proo n [8] ollows a

5 Zan e al J em Parm Res 67:-5 O O 5 55 ombnn 5 an 55 we e 5 W e same ecnqes aboe we can ere 5Te proo s complee Appln e smlar armens as ose n Teorem an sn e abae lemmas we can easl oban e ollown sperclose proper H Ω H Ω Teorem : Assme an are e solon o an respecel n orer o s e spercoerence resls o e blnear elemen or or problem we consrc e nerpolaon en 56 posprocessn operaor as ollows: ombnn or ajacen small elemens a 9 e corresponn paron s enoe b an no one b elemen E e erces o E are enoe b For an H Ω we ene H Ω e bqarac arane nerpolaon e or an Q an a a js 9 From e ansoropc eorem propose n [567] we know e nerpolaon operaor proper Also we can easl e e ollown lemma Ω emma 5 : For an H e nerpolaon operaor sas sases e ansoropc Teorem : ner e assmpon o Teorem ere ols Proo : Noe a Te proo s complee orm emma 5 an Teorem we e 5 REFERENES [] Xao-lan na Spor Scence an Tecnolo []o Yan-cn Jornal o Sana Pscal Ecaon nse []Wan Ha-ze jornal O Nancan Jnor ollee 5-56

6 Zan e al J em Parm Res 67:-5 [] e Jornal o Senan Spor ners - [5]Zan S-e Jornal o Nanjn nse o Pscal Ecaon [6]Pan Jornal o nanjn nse o pscal ecaonnaral scence [7] Y-e; n Wen-ao Jornal o Ganzo Pscal Ecaon nse [8] X Go-qn Jornal O Hebe nse O Pscal Ecaon [9] en Qn-on na Spor Scence an Tecnolo [] Tan Jn-nn Jornal o Nanjn nse o Pscal Ecaon 9-5 5