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Comp."],"abstract":"<p>\n                    We present an error analysis for an unconditionally energy stable, fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. The scheme, proposed by S.\u00a0M.\u00a0Wise, is based on the idea of convex splitting. In this paper, we rigorously prove first order convergence in time and second order convergence in space. Instead of the (discrete)\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Subscript s Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L Subscript h Superscript 2 Baseline right-parenthesis intersection upper L Subscript s Superscript 2 Baseline left-parenthesis 0 comma upper T semicolon upper H Subscript h Superscript 2 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi>h<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2229\n                              \n                            <\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mi>h<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L_s^\\infty (0,T;L_h^2) \\cap L_s^2 (0,T; H_h^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    error estimate, which would represent the typical approach, we provide a discrete\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Subscript s Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper H Subscript h Superscript 1 Baseline right-parenthesis intersection upper L Subscript s Superscript 2 Baseline left-parenthesis 0 comma upper T semicolon upper H Subscript h Superscript 3 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mi>h<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2229\n                              \n                            <\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mi>h<\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L_s^\\infty (0,T; H_h^1) \\cap L_s^2 (0,T; H_h^3 )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    error estimate for the phase variable, which allows us to treat the nonlinear convection term in a straightforward way. Our convergence is unconditional in the sense that the time step\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n                        <mml:semantics>\n                          <mml:mi>s<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is in no way constrained by the mesh spacing\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This is accomplished with the help of an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Subscript s Superscript 2 Baseline left-parenthesis 0 comma upper T semicolon upper H Subscript h Superscript 3 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mi>h<\/mml:mi>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L_s^2 (0,T;H_h^3)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    bound of the numerical approximation of the phase variable. To facilitate both the stability and convergence analyses, we establish a finite difference analog of a Gagliardo-Nirenberg type inequality.\n                  <\/p>","DOI":"10.1090\/mcom3052","type":"journal-article","created":{"date-parts":[[2015,3,11]],"date-time":"2015-03-11T09:08:39Z","timestamp":1426064919000},"page":"2231-2257","source":"Crossref","is-referenced-by-count":124,"title":["Convergence analysis of a fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation"],"prefix":"10.1090","volume":"85","author":[{"given":"Wenbin","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuan","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cheng","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Steven","family":"Wise","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2015,12,14]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1146\/annurev.fluid.30.1.139","article-title":"Diffuse-interface methods in fluid mechanics","author":"Anderson, D. 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