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Comp."],"abstract":"<p>\n                    The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection\u2013reaction\u2013diffusion equation that exhibits both parabolic\u2013hyperbolic and parabolic\u2013elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space\u2013time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1 Baseline left-parenthesis upper H Superscript negative 1 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1(H^{-1})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared left-parenthesis upper L squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2(L^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared left-parenthesis upper H Superscript 1 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2(H^1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space\u2013time efficiency error bounds are then obtained in a standard\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1 Baseline left-parenthesis upper H Superscript negative 1 Baseline right-parenthesis intersection upper L squared left-parenthesis upper H Superscript 1 Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2229\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1(H^{-1})\\cap L^2(H^1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as space discretization, time discretization, quadrature, linearization, and data oscillation are identified and separated. The estimates are also valid in a setting where iterative linearization with inexact solvers is considered. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for a realistic case and a benchmark case. It is shown that the estimators correctly identify the errors up to a factor of the order of unity.\n                  <\/p>","DOI":"10.1090\/mcom\/3932","type":"journal-article","created":{"date-parts":[[2024,2,14]],"date-time":"2024-02-14T13:09:39Z","timestamp":1707916179000},"page":"1053-1096","source":"Crossref","is-referenced-by-count":7,"title":["A posteriori error estimates for the Richards equation"],"prefix":"10.1090","volume":"93","author":[{"given":"K.","family":"Mitra","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Vohral\u00edk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2024,2,14]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics (New York)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032824","volume-title":"A posteriori error estimation in finite element analysis","author":"Ainsworth, Mark","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/047129411X"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/BF01176474","article-title":"Quasilinear elliptic-parabolic differential equations","volume":"183","author":"Alt, Hans Wilhelm","year":"1983","journal-title":"Math. 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