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Comp."],"abstract":"<p>\n                    We present a continuous\/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper R cubed\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {R}^3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . A priori error estimates, taking both the approximation of the surface and the approximation of surface differential operators into account, are proven in a discrete energy norm and in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm. This can be seen as an extension of the formalism and method originally used by Dziuk (1988) for approximating solutions to the Laplace\u2013Beltrami problem, and within this setting this is the first analysis of a surface finite element method formulated using higher order surface differential operators. Using a polygonal approximation\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Gamma Subscript h\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi mathvariant=\"normal\">\n                              \u0393\n                              \n                            <\/mml:mi>\n                            <mml:mi>h<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Gamma _h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of an implicitly defined surface\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Gamma\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u0393\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Gamma<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Gamma\">\n                        <mml:semantics>\n                          <mml:mi mathvariant=\"normal\">\n                            \u0393\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Gamma<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.\n                  <\/p>","DOI":"10.1090\/mcom\/3179","type":"journal-article","created":{"date-parts":[[2016,5,26]],"date-time":"2016-05-26T08:34:50Z","timestamp":1464251690000},"page":"2613-2649","source":"Crossref","is-referenced-by-count":18,"title":["A continuous\/discontinuous Galerkin method and a priori error estimates for the biharmonic problem on surfaces"],"prefix":"10.1090","volume":"86","author":[{"given":"Karl","family":"Larsson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mats","family":"Larson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2017,3,30]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"1145","DOI":"10.1137\/140957172","article-title":"High order discontinuous Galerkin methods for elliptic problems on surfaces","volume":"53","author":"Antonietti, Paola F.","year":"2015","journal-title":"SIAM J. 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