A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems
In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form − u (4) = f ( x , u ). We combine the advantages of the local discontinuous Galerkin (LDG) method and th...
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| Veröffentlicht in: | Numerical algorithms 2023-04, Vol.92 (4), p.1983-2023 |
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| container_title | Numerical algorithms |
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| creator | Baccouch, Mahboub |
| description | In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form −
u
(4)
=
f
(
x
,
u
). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the
L
2
-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order
p
+ 1 in the
L
2
-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most
p
and mesh size
h
are used. We then show that the UWLDG solutions are superconvergent with order
p
+ 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using
P
p
polynomials with
p
≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method. |
| doi_str_mv | 10.1007/s11075-022-01374-z |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2918616333</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2918616333</sourcerecordid><originalsourceid>FETCH-LOGICAL-c319t-2378791ce1757e041b0f205e1dcd4486efae5c32148e102db4a3b62d7a96ae8c3</originalsourceid><addsrcrecordid>eNp9UE1PwzAMjRBIjMEf4BSJcyBO2qY9ThMMJCQucI7S1t26dclImiH26wkMiRu-2Jb9PvQIuQZ-C5yruwDAVc64EIyDVBk7nJAJ5EqwShT5aZo5KAayKs_JRQhrzhNMqAkJMxriDn3j7B79Eu1I4zB6wz7QbOjgGjPQtg_pPPY2uhjowgzoN72lWxxXrqWd89Q6O_QWjU9b9OOKOd-ip7WLtjX-k-3NEJHuvKsH3IZLctaZIeDVb5-St4f71_kje35ZPM1nz6yRUI1MSFWqChoElSvkGdS8EzxHaJs2y8oCO4N5IwVkJQIXbZ0ZWReiVaYqDJaNnJKbI28Sfo8YRr1O7myS1KKCsoBCppoScfxqvAvBY6d3vt8m1xq4_g5XH8PVKVz9E64-JJA8gkJ6tkv0f9T_oL4AdDCAEg</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2918616333</pqid></control><display><type>article</type><title>A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems</title><source>SpringerLink Journals - AutoHoldings</source><creator>Baccouch, Mahboub</creator><creatorcontrib>Baccouch, Mahboub</creatorcontrib><description>In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form −
u
(4)
=
f
(
x
,
u
). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the
L
2
-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order
p
+ 1 in the
L
2
-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most
p
and mesh size
h
are used. We then show that the UWLDG solutions are superconvergent with order
p
+ 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using
P
p
polynomials with
p
≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method.</description><identifier>ISSN: 1017-1398</identifier><identifier>EISSN: 1572-9265</identifier><identifier>DOI: 10.1007/s11075-022-01374-z</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Algorithms ; Approximation ; Boundary conditions ; Boundary value problems ; Computer Science ; Estimates ; Exact solutions ; Finite element analysis ; Finite volume method ; Galerkin method ; Numeric Computing ; Numerical Analysis ; Original Paper ; Partial differential equations ; Polynomials ; Theory of Computation ; Variables</subject><ispartof>Numerical algorithms, 2023-04, Vol.92 (4), p.1983-2023</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c319t-2378791ce1757e041b0f205e1dcd4486efae5c32148e102db4a3b62d7a96ae8c3</citedby><cites>FETCH-LOGICAL-c319t-2378791ce1757e041b0f205e1dcd4486efae5c32148e102db4a3b62d7a96ae8c3</cites><orcidid>0000-0002-6721-309X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11075-022-01374-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11075-022-01374-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,778,782,27907,27908,41471,42540,51302</link.rule.ids></links><search><creatorcontrib>Baccouch, Mahboub</creatorcontrib><title>A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems</title><title>Numerical algorithms</title><addtitle>Numer Algor</addtitle><description>In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form −
u
(4)
=
f
(
x
,
u
). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the
L
2
-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order
p
+ 1 in the
L
2
-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most
p
and mesh size
h
are used. We then show that the UWLDG solutions are superconvergent with order
p
+ 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using
P
p
polynomials with
p
≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method.</description><subject>Algebra</subject><subject>Algorithms</subject><subject>Approximation</subject><subject>Boundary conditions</subject><subject>Boundary value problems</subject><subject>Computer Science</subject><subject>Estimates</subject><subject>Exact solutions</subject><subject>Finite element analysis</subject><subject>Finite volume method</subject><subject>Galerkin method</subject><subject>Numeric Computing</subject><subject>Numerical Analysis</subject><subject>Original Paper</subject><subject>Partial differential equations</subject><subject>Polynomials</subject><subject>Theory of Computation</subject><subject>Variables</subject><issn>1017-1398</issn><issn>1572-9265</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNp9UE1PwzAMjRBIjMEf4BSJcyBO2qY9ThMMJCQucI7S1t26dclImiH26wkMiRu-2Jb9PvQIuQZ-C5yruwDAVc64EIyDVBk7nJAJ5EqwShT5aZo5KAayKs_JRQhrzhNMqAkJMxriDn3j7B79Eu1I4zB6wz7QbOjgGjPQtg_pPPY2uhjowgzoN72lWxxXrqWd89Q6O_QWjU9b9OOKOd-ip7WLtjX-k-3NEJHuvKsH3IZLctaZIeDVb5-St4f71_kje35ZPM1nz6yRUI1MSFWqChoElSvkGdS8EzxHaJs2y8oCO4N5IwVkJQIXbZ0ZWReiVaYqDJaNnJKbI28Sfo8YRr1O7myS1KKCsoBCppoScfxqvAvBY6d3vt8m1xq4_g5XH8PVKVz9E64-JJA8gkJ6tkv0f9T_oL4AdDCAEg</recordid><startdate>20230401</startdate><enddate>20230401</enddate><creator>Baccouch, Mahboub</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>8FE</scope><scope>8FG</scope><scope>ABJCF</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>L6V</scope><scope>M7S</scope><scope>P62</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><orcidid>https://orcid.org/0000-0002-6721-309X</orcidid></search><sort><creationdate>20230401</creationdate><title>A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems</title><author>Baccouch, Mahboub</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-2378791ce1757e041b0f205e1dcd4486efae5c32148e102db4a3b62d7a96ae8c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Algebra</topic><topic>Algorithms</topic><topic>Approximation</topic><topic>Boundary conditions</topic><topic>Boundary value problems</topic><topic>Computer Science</topic><topic>Estimates</topic><topic>Exact solutions</topic><topic>Finite element analysis</topic><topic>Finite volume method</topic><topic>Galerkin method</topic><topic>Numeric Computing</topic><topic>Numerical Analysis</topic><topic>Original Paper</topic><topic>Partial differential equations</topic><topic>Polynomials</topic><topic>Theory of Computation</topic><topic>Variables</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Baccouch, Mahboub</creatorcontrib><collection>CrossRef</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>Materials Science & Engineering 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 (ProQuest)</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>ProQuest Engineering Collection</collection><collection>Engineering 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><collection>Engineering Collection</collection><jtitle>Numerical algorithms</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Baccouch, Mahboub</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems</atitle><jtitle>Numerical algorithms</jtitle><stitle>Numer Algor</stitle><date>2023-04-01</date><risdate>2023</risdate><volume>92</volume><issue>4</issue><spage>1983</spage><epage>2023</epage><pages>1983-2023</pages><issn>1017-1398</issn><eissn>1572-9265</eissn><abstract>In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form −
u
(4)
=
f
(
x
,
u
). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the
L
2
-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order
p
+ 1 in the
L
2
-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most
p
and mesh size
h
are used. We then show that the UWLDG solutions are superconvergent with order
p
+ 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using
P
p
polynomials with
p
≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s11075-022-01374-z</doi><tpages>41</tpages><orcidid>https://orcid.org/0000-0002-6721-309X</orcidid></addata></record> |
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| subjects | Algebra Algorithms Approximation Boundary conditions Boundary value problems Computer Science Estimates Exact solutions Finite element analysis Finite volume method Galerkin method Numeric Computing Numerical Analysis Original Paper Partial differential equations Polynomials Theory of Computation Variables |
| title | A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems |
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