
echpi-phem

variable valippathilulla gadakangal upayogikkunna pease vais-polynomial ekadesharekhakale adisthaanamaakkiyulla bhagika deferential samavaakyangal pariharikkunnathinulla oru samkhyaa reethiyaaya phainit elament reethiyude (phem) oru pothu pathippaanu echpi-phem.
"(echu)" ", polynomial digri" "(pii)". burna eyude pioneering kruthiyil ninnaan echpi-feminte uthbhavam. parimithamaaya moolaka reethi "athivegam" othucherunnuvennu kandethiya sabo ivo babuka
echu-refinementukalude anuyojyamaaya samyojanam upayogichaanu mesh parishkarikkunnath
(gadakangale cheruthaayi vibhajikkunnu), pii-refinementukal (avayude vardhanavu)
polynomial digri). beejaganita nirakkumayi maathram othucherunna mattu parimitha ghadaka reethikalumaayi thaarathamyappeduthumbol exponential kanverjans eee reethiye valare akarshakamaakkunnu. exponential kanverjans
echpi-feminte pravachikkappettath saidhaanthikamaayi mathramalla, nireekshikkappedukayum cheythu.
niravadhi swatantra gaveshakar.
saadhaarana sthreekalil ninnulla vyathyaasangal.
echpi-phem pala kaaryangalilum staanderd (lovast-oordar) femil ninnu vyathyasthamaan.
fitchera prashnam.
fitchera prashnam (fitchera corner prashnam ennum ariyappedunnu) adaptive phem kodukalude oru staanderd benchmark prasnamaanu. staanderd feminteyum echpi-feminteyum prakadanatthile naadakeeyamaaya vyathyasam kaanikkan oraalkk ithu upayogikkaam. prashna jyamiti oru corner illaatha oru cube aanu. krithyamaaya parihaarathinu kendratthil oru eaka gradient (ananthamaaya sammarddathinte saamyam) undu. krithyamaaya parihaarathekkurichulla arivu ekadesha pishak krithyamaayi kanakkakkaanum vividha samkhyaa reethikal thaarathamyam cheyyaanum sahayikkunnu. udaaharanathinu, adaptive feminte moonnu vyathyastha pathippukal upayogichaanu prashnam pariharichathu leanier gadakangal, quadratic gadakangal, echpi-phem enniva upayogichu.
kanverjans graafukal ekadesha pishak digri oof freedam (dof) yude ennathinte pravartthanamaayi kaanikkunnu. doff ennathu ekadesha kanakk nirvachikkaan aavashyamaaya anjjaathamaaya paaraamettarukaleyaanu soochippikkunnathu, koodaathe doffinte ennam stiphness maatriksinte valuppathin thulyamaanu. mattu rund reethikaludeyum samyojanathekkaal valare vegathilaanu echpi-feminte samyojanam ennu graafukalil vaayanakkaaran kaanaan kazhiyum. perfomence gapp valare valuthaanu, leanier phem ottum othucherilla (nyayamaaya samayat) koodaathe quadratic feminu ekadesham 17,000 doff upayogichu echpi-phem nediya krithyathayil ethaan lakshakkanakkin allengil dasalakshakkanakkinu doff aavasyamaanu. thaarathamyena kurachu digri oof freedam upayogichu valare krithyamaaya phalangal neduka ennathaanu echpi-feminte pradhana shakti.
echpi-feminte karyakshamatha.
cheria pease vais-leanier gadakangale apekshichu valia hye-oordar gadakangal upayogichu sugamamaya pravartthanangal valare kaaryakshamamaayi kanakkakkaan kazhiyum. poojyamaaya diriclet athirthi vyavasthakalulla oru dimensional poyson samavaakyam rund vyathyastha meshukalil pariharikkappedunna chuvadeyulla chithratthil ithu chithreekarichirikkunnu. krithyamaaya pariharam saine functionaanu.
rund kesukalilum anjjaatharude ennam thulyamaanenkilum (1 doff), anubandha maanadandatthile pishakukal yathaakramam 0.68,0.20 aanu. ithinartham chathuraakrithiyilulla ekadesha kanakk pease vais-leaniorinekkaal ekadesham 3.5 madangu kooduthal kaaryakshamamaayirunnu ennaanu. nammal oru padi koodi munnott pooyi (eiii) naalu rekheeya gadakangale (bii) oru quartic moolakavumaayi (pii = 4) thaarathamyam cheyyumbol, rund vyathiriktha prasnangalkkum moonnu doff undaakum, ennaal quartic ekadesha kanakk ekadesham 40 madangu kooduthal kaaryakshamamayirikkum. 
neremarichu, cheria loow-oordar gadakangalkku valia hye-oordarukalekkaal mikacha reethiyil singulaarittikal polulla cheria thothilulla savisheshathakal pidichedukkaan kazhiyum. exponential kanverjansilekku nayikkunna eee rund sameepanangaludeyum opttimal kombineshane adisthaanamaakkiyullathaanu echpi-phem. eee exponential kanverjans error vezhsas digri oof freedam enna achuthandil prakadippikkunnu ennathu shradhikkuka. yadhaartha jeevita aaplikkeshanukalkkaayi, ore thalatthilulla krithyathayil ethaan aavashyamaaya computational samayam njangal saadhaaranayaayi pariganikkunnu. eee prakadana soochakathinu echu-, echpi-refinementinu samanamaaya phalangal nalkaan kazhiyum, udha.jii. (web archive link) ennathil anthima chithram kaanuka. echu-femmumaayi thaarathamyappeduthumbol echpi-femine prograam cheyyaanum samaantharamaakkaanum budhimuttaayikkazhinjal, echpi-refinamentinte kanverjans exalance apraayogikamaayekkaam.
echpi-adaptivity.
chila phem sittukal echpi-adaptivitye echu-adaptivity (polynomial digri sthiramaayi nilanirthikkondu bahiraakaashath gadakangale vibhajikkunnath), pii-adaptivity (avarude polynomial digri vardhippikkuka maathram) ennivayude samyojanamaayi visheshippikkunnu. oru moolakatthinte echpi-refinement palavidhatthil cheyyaan kazhiyunnathinaal echpi-adaptivity echu-, pii-adaptivityil ninnu ganyamaayi vyathyasthamaayathinaal ithu poornnamaayum krithyamalla. oru pii-refinementinu purame, moolakathe spacil (echu-adaptivittiyilenne) upavibhagamaakkaam, ennaal upa-gadakangalil polynomial digrikalkku niravadhi kombineshanukalundu. valathuvashathulla chithratthil ithu chithreekarichirikkunnu. udaaharanathinu, oru trikona allengil chatushkona moolakathe naalu upaghadakangalaayi vibhajichaal, athil polynomial digri paramavadhi randaayi vyathyaasappedaan anuvadikkunnu, ithu 3-4 = 81 refinement kaandidettukal nalkunnu (polynomially anisotropic kaandidettukal pariganikkunnilla). samaanamaayi, oru hexahedronine ettu upaghadakangalaayi vibhajikkukayum avayude polynomial digri paramavadhi randaayi maattukayum cheiyunnathu 3 ^ 8 = 6,561 refinement kaandidettukal nalkunnu. oru moolakathinu oru sthirasamkhya nalkunna staanderd phem error esttimettukal automatic echpi-adaptivitye nayikkaan paryaapthamalla.
uyarnna kramatthilulla aakruthiyilulla pravartthanangal.
saadhaarana phem vann gridumaayi bandhapetta aakruthi pravarthanangalil maathrame pravartthikkoo.
sheershakangal ("sheershaka pravartthanangal" ennu vilikkappedunnava). ithinu vipareethamaayi, echpi-phem upayogikkumbol, onnu "edju functionukal" (bandhappettirikkunnu) ennathinekkurichum koodiyaanu.
moolakatthinte arikukal), "mukhatthinte pravartthanangal" (moolakatthinte mukhangalkk anusrithamaayi-3d maathram),
koodaathe "bubble functionukal" (uyarnna kramatthilulla polynomialukal aprathyakshamaavunnu)
moolakangalude athirthikal). thaazhepparayunna chithrangal eee pravartthanangal kaanikkunnu
(orotta gadakathil maathram parimithappedutthiyirikku):
shradhikkuka eee pravarthanangalellaam interior gadakathil nirvachichirikkunnu.