Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations
We are interested in the shape of the homogenized operator F ¯ ( Q ) for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is H a 1 , a 2 ( Q , x ) = a 1 ( x ) λ min ( Q ) + a 2 ( x ) λ max ( Q ) . Linearization of the operator leads to a non-divergence form homogenizat...
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| Veröffentlicht in: | Journal of scientific computing 2018-11, Vol.77 (2), p.936-949 |
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| container_title | Journal of scientific computing |
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| creator | Finlay, Chris Oberman, Adam M. |
| description | We are interested in the shape of the homogenized operator
F
¯
(
Q
)
for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is
H
a
1
,
a
2
(
Q
,
x
)
=
a
1
(
x
)
λ
min
(
Q
)
+
a
2
(
x
)
λ
max
(
Q
)
. Linearization of the operator leads to a non-divergence form homogenization problem, which can be solved by averaging against the invariant measure. We estimate the error obtained by linearization based on semi-concavity estimates on the nonlinear operator. These estimates show that away from high curvature regions, the linearization can be accurate. Numerical results show that for many values of
Q
, the linearization is highly accurate, and that even near corners, the error can be small (a few percent) even for relatively wide ranges of the coefficients. |
| doi_str_mv | 10.1007/s10915-018-0730-x |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2918313800</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2918313800</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-412e3c44d99d88bf804cda14c775288a6678e591f58fb4974dd8a13b43ae02e33</originalsourceid><addsrcrecordid>eNp1kEFLwzAYhoMoOKc_wFvAczVp0ibxNmbnBJlD5jlkaToysqZLWtj89Xar4MnTd3mf9-N9ALjH6BEjxJ4iRgJnCcI8QYyg5HABRjhjJGG5wJdghDjPEkYZvQY3MW4RQoKLdAS6SdMEf7A71Ro49zu_MbX9Vq31NfQVnHXOHeHC187WRgVYOGeb1mq4fCniMyxieyYjVHUJF93OBKuVg58mdq6NsPIBLjutLVwdGwOLfXdujrfgqlIumrvfOwZfs2I1nSfvH69v08l7ognO24Ti1BBNaSlEyfm64ojqUmGqGctSzlWeM24ygauMV2sqGC1LrjBZU6IM6lEyBg9Db79x35nYyq3vQt2_lKnAnGDCEepTeEjp4GMMppJN6GeFo8RInuzKwa7s7cqTXXnomXRgYp-tNyb8Nf8P_QB8sX5o</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2918313800</pqid></control><display><type>article</type><title>Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations</title><source>Springer Nature - Complete Springer Journals</source><source>ProQuest Central UK/Ireland</source><source>ProQuest Central</source><creator>Finlay, Chris ; Oberman, Adam M.</creator><creatorcontrib>Finlay, Chris ; Oberman, Adam M.</creatorcontrib><description>We are interested in the shape of the homogenized operator
F
¯
(
Q
)
for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is
H
a
1
,
a
2
(
Q
,
x
)
=
a
1
(
x
)
λ
min
(
Q
)
+
a
2
(
x
)
λ
max
(
Q
)
. Linearization of the operator leads to a non-divergence form homogenization problem, which can be solved by averaging against the invariant measure. We estimate the error obtained by linearization based on semi-concavity estimates on the nonlinear operator. These estimates show that away from high curvature regions, the linearization can be accurate. Numerical results show that for many values of
Q
, the linearization is highly accurate, and that even near corners, the error can be small (a few percent) even for relatively wide ranges of the coefficients.</description><identifier>ISSN: 0885-7474</identifier><identifier>EISSN: 1573-7691</identifier><identifier>DOI: 10.1007/s10915-018-0730-x</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algorithms ; Approximation ; Computational Mathematics and Numerical Analysis ; Concavity ; Divergence ; Eigenvalues ; Elliptic differential equations ; Error analysis ; Estimates ; Homogenization ; Linearization ; Mathematical and Computational Engineering ; Mathematical and Computational Physics ; Mathematics ; Mathematics and Statistics ; Partial differential equations ; Theoretical</subject><ispartof>Journal of scientific computing, 2018-11, Vol.77 (2), p.936-949</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2018</rights><rights>Springer Science+Business Media, LLC, part of Springer Nature 2018.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-412e3c44d99d88bf804cda14c775288a6678e591f58fb4974dd8a13b43ae02e33</citedby><cites>FETCH-LOGICAL-c316t-412e3c44d99d88bf804cda14c775288a6678e591f58fb4974dd8a13b43ae02e33</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10915-018-0730-x$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://www.proquest.com/docview/2918313800?pq-origsite=primo$$EHTML$$P50$$Gproquest$$H</linktohtml><link.rule.ids>314,777,781,21369,27905,27906,33725,41469,42538,43786,51300,64364,64368,72218</link.rule.ids></links><search><creatorcontrib>Finlay, Chris</creatorcontrib><creatorcontrib>Oberman, Adam M.</creatorcontrib><title>Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations</title><title>Journal of scientific computing</title><addtitle>J Sci Comput</addtitle><description>We are interested in the shape of the homogenized operator
F
¯
(
Q
)
for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is
H
a
1
,
a
2
(
Q
,
x
)
=
a
1
(
x
)
λ
min
(
Q
)
+
a
2
(
x
)
λ
max
(
Q
)
. Linearization of the operator leads to a non-divergence form homogenization problem, which can be solved by averaging against the invariant measure. We estimate the error obtained by linearization based on semi-concavity estimates on the nonlinear operator. These estimates show that away from high curvature regions, the linearization can be accurate. Numerical results show that for many values of
Q
, the linearization is highly accurate, and that even near corners, the error can be small (a few percent) even for relatively wide ranges of the coefficients.</description><subject>Algorithms</subject><subject>Approximation</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Concavity</subject><subject>Divergence</subject><subject>Eigenvalues</subject><subject>Elliptic differential equations</subject><subject>Error analysis</subject><subject>Estimates</subject><subject>Homogenization</subject><subject>Linearization</subject><subject>Mathematical and Computational Engineering</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Partial differential equations</subject><subject>Theoretical</subject><issn>0885-7474</issn><issn>1573-7691</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNp1kEFLwzAYhoMoOKc_wFvAczVp0ibxNmbnBJlD5jlkaToysqZLWtj89Xar4MnTd3mf9-N9ALjH6BEjxJ4iRgJnCcI8QYyg5HABRjhjJGG5wJdghDjPEkYZvQY3MW4RQoKLdAS6SdMEf7A71Ro49zu_MbX9Vq31NfQVnHXOHeHC187WRgVYOGeb1mq4fCniMyxieyYjVHUJF93OBKuVg58mdq6NsPIBLjutLVwdGwOLfXdujrfgqlIumrvfOwZfs2I1nSfvH69v08l7ognO24Ti1BBNaSlEyfm64ojqUmGqGctSzlWeM24ygauMV2sqGC1LrjBZU6IM6lEyBg9Db79x35nYyq3vQt2_lKnAnGDCEepTeEjp4GMMppJN6GeFo8RInuzKwa7s7cqTXXnomXRgYp-tNyb8Nf8P_QB8sX5o</recordid><startdate>20181101</startdate><enddate>20181101</enddate><creator>Finlay, Chris</creator><creator>Oberman, Adam M.</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>8FE</scope><scope>8FG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K7-</scope><scope>P5Z</scope><scope>P62</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope></search><sort><creationdate>20181101</creationdate><title>Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations</title><author>Finlay, Chris ; Oberman, Adam M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-412e3c44d99d88bf804cda14c775288a6678e591f58fb4974dd8a13b43ae02e33</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Algorithms</topic><topic>Approximation</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Concavity</topic><topic>Divergence</topic><topic>Eigenvalues</topic><topic>Elliptic differential equations</topic><topic>Error analysis</topic><topic>Estimates</topic><topic>Homogenization</topic><topic>Linearization</topic><topic>Mathematical and Computational Engineering</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Partial differential equations</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Finlay, Chris</creatorcontrib><creatorcontrib>Oberman, Adam M.</creatorcontrib><collection>CrossRef</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>Computer Science Database</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><jtitle>Journal of scientific computing</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Finlay, Chris</au><au>Oberman, Adam M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations</atitle><jtitle>Journal of scientific computing</jtitle><stitle>J Sci Comput</stitle><date>2018-11-01</date><risdate>2018</risdate><volume>77</volume><issue>2</issue><spage>936</spage><epage>949</epage><pages>936-949</pages><issn>0885-7474</issn><eissn>1573-7691</eissn><abstract>We are interested in the shape of the homogenized operator
F
¯
(
Q
)
for PDEs which have the structure of a nonlinear Pucci operator. A typical operator is
H
a
1
,
a
2
(
Q
,
x
)
=
a
1
(
x
)
λ
min
(
Q
)
+
a
2
(
x
)
λ
max
(
Q
)
. Linearization of the operator leads to a non-divergence form homogenization problem, which can be solved by averaging against the invariant measure. We estimate the error obtained by linearization based on semi-concavity estimates on the nonlinear operator. These estimates show that away from high curvature regions, the linearization can be accurate. Numerical results show that for many values of
Q
, the linearization is highly accurate, and that even near corners, the error can be small (a few percent) even for relatively wide ranges of the coefficients.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10915-018-0730-x</doi><tpages>14</tpages></addata></record> |
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| ispartof | Journal of scientific computing, 2018-11, Vol.77 (2), p.936-949 |
| issn | 0885-7474 1573-7691 |
| language | eng |
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| source | Springer Nature - Complete Springer Journals; ProQuest Central UK/Ireland; ProQuest Central |
| subjects | Algorithms Approximation Computational Mathematics and Numerical Analysis Concavity Divergence Eigenvalues Elliptic differential equations Error analysis Estimates Homogenization Linearization Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Partial differential equations Theoretical |
| title | Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations |
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