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[{"key": "dc.contributor.author", "value": "Nokka, Marjaana", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2012-06-06T08:45:42Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2012-06-06T08:45:42Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2012", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1218178", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/37972", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Stokesin yht\u00e4l\u00f6ll\u00e4 voidaan kuvata nesteiden ja kaasujen liikett\u00e4, jos liike on yksiulotteista tai hidasta. Stokesin yht\u00e4l\u00f6 on yksinkertaistettu ja linearisoitu versio Navier-Stokesin yht\u00e4l\u00f6ist\u00e4. T\u00e4ss\u00e4 tutkielmassa keskityt\u00e4\u00e4n kokoonpuristumattomiin ja viskoottisiin nesteisiin ja kaasuihin. Kokoonpuristumattomuus tarkoittaa sit\u00e4, ett\u00e4 nesteen tai kaasun tiheys ei muutu ajan suhteen. Viskoottisuus taas tarkoittaa sit\u00e4, ett\u00e4 nesteill\u00e4 ja kaasuilla on sis\u00e4ist\u00e4 kitkaa, joka muodostuu, kun aineen osaset liikkuvat toistensa suhteen.\n\nVaikka osittaisdifferentiaaliyht\u00e4l\u00f6iden tutkimus on edennyt viime vuosisadalla hyvinkin nopeasti, on analyyttisen ratkaisun l\u00f6yt\u00e4minen vaikeaa tai l\u00e4hes poikkeuksetta mahdotonta. Tapauksissa, joissa analyyttinen ratkaisu l\u00f6ydet\u00e4\u00e4n kohtuullisella vaivalla, on yleens\u00e4 tehty ep\u00e4realistisen suuria yksinkertaistuksia tai yht\u00e4l\u00f6 on alkujaan keinotekoinen. Jotta ep\u00e4realistisilta yksinkertaistuksilta voitaisiin v\u00e4ltty\u00e4, on osittaisdifferentiaaliyht\u00e4l\u00f6ille kehitetty monia numeerisia ratkaisumenetelmi\u00e4, kuten esimerkiksi t\u00e4ss\u00e4 tutkielmassa hy\u00f6dynetty elementtimenetelm\u00e4. Numeeristen menetelmien k\u00e4ytt\u00f6 tuottaa osittaisdifferentiaaliyht\u00e4l\u00f6lle likim\u00e4\u00e4r\u00e4isen ratkaisun, joka hyvin harvoin vastaa tarkkaa ratkaisua. Likim\u00e4\u00e4r\u00e4iseen ratkaisuun sis\u00e4ltyy siis aina jonkunlainen virhe. \n\nFunktionaalisilla a posteriori virhearvioilla voidaan arvioida luotettavasti sit\u00e4 virhett\u00e4, joka tehd\u00e4\u00e4n ratkaistaessa osittaisdifferentiaaliyht\u00e4l\u00f6it\u00e4 numeerisesti. T\u00e4ss\u00e4 tutkielmassa malliongelmana k\u00e4ytet\u00e4\u00e4n yleistetty\u00e4 Stokesin yht\u00e4l\u00f6\u00e4. T\u00e4lle ongelmalle on jo aiemmin johdettu a posteriori virhearviot, t\u00e4ss\u00e4 tutkielmassa johdetaan vastavat virhearviot ottaen huomioon Uzawa-algoritmin erityispiirteet. Uzawa-algoritmi on suosittu menetelm\u00e4 likiarvoratkaisujen l\u00f6yt\u00e4miseen satulapisteongelmille. Lis\u00e4ksi tarkastellaan Uzawa-algoritmin konvergenssia malliongelman tapauksessa.\n\nSeuraavaksi tutkimme approksimaatiovirhett\u00e4, joka muodostuu, kun \u00e4\u00e4ret\u00f6nulotteinen ongelma korvataan \u00e4\u00e4rellisulotteisella ongelmalla. T\u00e4m\u00e4n my\u00f6t\u00e4 virheen yl\u00e4rajaan (majorantiin) tulee vapaita parametreja, jotka tulee minimoida mahdollisimman tarkan yl\u00e4rajan l\u00f6yt\u00e4miseksi. Numeerisessa osuudessa ratkaistaan malliongelma elementtimenetelm\u00e4ll\u00e4 MINI-elementti\u00e4 hy\u00f6dynt\u00e4en ja minimoidaan majorantin vapaat parametrit elementtimenetelm\u00e4ll\u00e4 Raviart-Thomas-elementtej\u00e4 hy\u00f6dynt\u00e4en. Lopuksi vahvistetaan johdettujen majoranttien toimivuus numeerisilla testeill\u00e4. Numeerisia testej\u00e4 varten on koottu molemmille elementtityypeille osittaisdifferentiaaliyht\u00e4l\u00f6ratkaisijat.", "language": null, "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Marjaana Nokka (menokka) on 2012-06-06 08:45:41.562542. Form: Pro gradu -lomake (1 tekij\u00e4) (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake-1-tekijae). JyX data:", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija@noreply.fi) on 2012-06-06T08:45:41Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201206061805.pdf: 657870 bytes, checksum: 17c3f15320577eb1de7f6de7c5b91630 (MD5)\nlicense.html: 5056 bytes, checksum: 98de39bfed6ea2daa3e7f0160c388dad (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2012-06-06T08:45:42Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201206061805.pdf: 657870 bytes, checksum: 17c3f15320577eb1de7f6de7c5b91630 (MD5)\nlicense.html: 5056 bytes, checksum: 98de39bfed6ea2daa3e7f0160c388dad (MD5)\n Previous issue date: 2012", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "65 sivua", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "funktionaalinen", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "a posteriori", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "virhearvio", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "yl\u00e4raja", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "majorantti", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ODY", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Stokes", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "FEM", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Raviart-Thomas", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": 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