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Comp."],"abstract":"<p>We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff\u2013Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov\u2013Galerkin type with optimal test functions (DPG). We prove well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme.<\/p>\n                  <p>\n                    The variational formulation and its analysis require tools that control traces and jumps in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (standard Sobolev space of scalar functions) and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis div bold div right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext>div<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">div<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\textrm {div}\\,\\, \\textbf {div}\\!)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (symmetric tensor functions with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">L_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -components whose twice iterated divergence is in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L 2\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">L_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis div bold div right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext>div<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">div<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\textrm {div}\\,\\, \\textbf {div}\\!)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . They are essential to construct basis functions for an approximation of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis div bold div right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext>div<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mtext mathvariant=\"bold\">div<\/mml:mtext>\n                            <\/mml:mrow>\n                            <mml:mspace width=\"negativethinmathspace\"\/>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\textrm {div}\\,\\, \\textbf {div}\\!)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>\n                  <p>To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.<\/p>","DOI":"10.1090\/mcom\/3381","type":"journal-article","created":{"date-parts":[[2018,6,6]],"date-time":"2018-06-06T10:19:23Z","timestamp":1528280363000},"page":"1587-1619","source":"Crossref","is-referenced-by-count":23,"title":["An ultraweak formulation of the Kirchhoff\u2013Love plate bending model and DPG approximation"],"prefix":"10.1090","volume":"88","author":[{"given":"Thomas","family":"F\u00fchrer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Norbert","family":"Heuer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Antti","family":"Niemi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2018,10,5]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"1632","DOI":"10.1137\/S0036142900379680","article-title":"Bending moment mixed method for the Kirchhoff-Love plate model","volume":"40","author":"Amara, Mohamed","year":"2002","journal-title":"SIAM J. 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