{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T23:40:22Z","timestamp":1680306022842},"reference-count":14,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The results of this contribution are derived in the framework of functional type\na posteriori error estimates. The error is measured in a combined norm which takes into account both\nthe primal and dual variables denoted by <jats:italic>x<\/jats:italic> and <jats:italic>y<\/jats:italic>, respectively.\nOur first main result is an error equality for all equations of the class\n<jats:inline-formula id=\"j_cmam-2016-0016_ineq_9999_w2aab3b7b1b1b6b1aab1c13b1b5Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:msup>\n                                       <m:mi>A<\/m:mi>\n                                       <m:mo>*<\/m:mo>\n                                    <\/m:msup>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>A<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>x<\/m:mi>\n                                 <\/m:mrow>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mi>x<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>f<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathrm{A}^{*}\\mathrm{A}x+x=f}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>\nor in mixed formulation\n<jats:inline-formula id=\"j_cmam-2016-0016_ineq_9998_w2aab3b7b1b1b6b1aab1c13b1b7Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:msup>\n                                       <m:mi>A<\/m:mi>\n                                       <m:mo>*<\/m:mo>\n                                    <\/m:msup>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>y<\/m:mi>\n                                 <\/m:mrow>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mi>x<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>f<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathrm{A}^{*}y+x=f}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, <jats:inline-formula id=\"j_cmam-2016-0016_ineq_9997_w2aab3b7b1b1b6b1aab1c13b1b9Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mi>A<\/m:mi>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mi>x<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>y<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathrm{A}x=y}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>,\nwhere the exact solution <jats:inline-formula id=\"j_cmam-2016-0016_ineq_9996_w2aab3b7b1b1b6b1aab1c13b1c11Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mi>x<\/m:mi>\n                              <m:mo>,<\/m:mo>\n                              <m:mi>y<\/m:mi>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$(x,y)$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is in <jats:inline-formula id=\"j_cmam-2016-0016_ineq_9995_w2aab3b7b1b1b6b1aab1c13b1c13Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:mi>D<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mrow>\n                                       <m:mo>(<\/m:mo>\n                                       <m:mi>A<\/m:mi>\n                                       <m:mo>)<\/m:mo>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                                 <m:mo>\u00d7<\/m:mo>\n                                 <m:mi>D<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:msup>\n                                    <m:mi>A<\/m:mi>\n                                    <m:mo>*<\/m:mo>\n                                 <\/m:msup>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$D(\\mathrm{A})\\times D(\\mathrm{A}^{*})$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.\nHere <jats:inline-formula id=\"j_cmam-2016-0016_ineq_9994_w2aab3b7b1b1b6b1aab1c13b1c15Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>A<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathrm{A}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is a linear, densely defined and closed (usually a differential)\noperator and <jats:inline-formula id=\"j_cmam-2016-0016_ineq_9993_w2aab3b7b1b1b6b1aab1c13b1c17Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>A<\/m:mi>\n                              <m:mo>*<\/m:mo>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathrm{A}^{*}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> its adjoint. In this paper we deal with very conforming\nmixed approximations, i.e., we assume that the approximation\n<jats:inline-formula id=\"j_cmam-2016-0016_ineq_9992_w2aab3b7b1b1b6b1aab1c13b1c19Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mover>\n                                 <m:mi>x<\/m:mi>\n                                 <m:mo>~<\/m:mo>\n                              <\/m:mover>\n                              <m:mo>,<\/m:mo>\n                              <m:mover>\n                                 <m:mi>y<\/m:mi>\n                                 <m:mo>~<\/m:mo>\n                              <\/m:mover>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${(\\tilde{x},\\tilde{y})}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> belongs to <jats:inline-formula id=\"j_cmam-2016-0016_ineq_9991_w2aab3b7b1b1b6b1aab1c13b1c21Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:mi>D<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mrow>\n                                       <m:mo>(<\/m:mo>\n                                       <m:mi>A<\/m:mi>\n                                       <m:mo>)<\/m:mo>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                                 <m:mo>\u00d7<\/m:mo>\n                                 <m:mi>D<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo>(<\/m:mo>\n                                 <m:msup>\n                                    <m:mi>A<\/m:mi>\n                                    <m:mo>*<\/m:mo>\n                                 <\/m:msup>\n                                 <m:mo>)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${D(\\mathrm{A})\\times D(\\mathrm{A}^{*})}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. In order to obtain the exact global error value\nof this approximation one only needs the problem data and\nthe mixed approximation itself, i.e.,\nwe have the <jats:italic>equality<\/jats:italic>\n               <\/jats:p>\n               <jats:p>\n                  <jats:disp-formula id=\"j_cmam-2016-0016_eq_9999_w2aab3b7b1b1b6b1aab1c13b2aAa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:msup>\n                                       <m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>x<\/m:mi>\n                                             <m:mo>-<\/m:mo>\n                                             <m:mover>\n                                                <m:mi>x<\/m:mi>\n                                                <m:mo>~<\/m:mo>\n                                             <\/m:mover>\n                                          <\/m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                       <\/m:mrow>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:msup>\n                                    <m:mo>+<\/m:mo>\n                                    <m:msup>\n                                       <m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>A<\/m:mi>\n                                             <m:mo>\u2062<\/m:mo>\n                                             <m:mrow>\n                                                <m:mo>(<\/m:mo>\n                                                <m:mrow>\n                                                   <m:mi>x<\/m:mi>\n                                                   <m:mo>-<\/m:mo>\n                                                   <m:mover>\n                                                      <m:mi>x<\/m:mi>\n                                                      <m:mo>~<\/m:mo>\n                                                   <\/m:mover>\n                                                <\/m:mrow>\n                                                <m:mo>)<\/m:mo>\n                                             <\/m:mrow>\n                                          <\/m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                       <\/m:mrow>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:msup>\n                                    <m:mo>+<\/m:mo>\n                                    <m:msup>\n                                       <m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>y<\/m:mi>\n                                             <m:mo>-<\/m:mo>\n                                             <m:mover>\n                                                <m:mi>y<\/m:mi>\n                                                <m:mo>~<\/m:mo>\n                                             <\/m:mover>\n                                          <\/m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                       <\/m:mrow>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:msup>\n                                    <m:mo>+<\/m:mo>\n                                    <m:msup>\n                                       <m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                          <m:mrow>\n                                             <m:msup>\n                                                <m:mi>A<\/m:mi>\n                                                <m:mo>*<\/m:mo>\n                                             <\/m:msup>\n                                             <m:mo>\u2062<\/m:mo>\n                                             <m:mrow>\n                                                <m:mo>(<\/m:mo>\n                                                <m:mrow>\n                                                   <m:mi>y<\/m:mi>\n                                                   <m:mo>-<\/m:mo>\n                                                   <m:mover>\n                                                      <m:mi>y<\/m:mi>\n                                                      <m:mo>~<\/m:mo>\n                                                   <\/m:mover>\n                                                <\/m:mrow>\n                                                <m:mo>)<\/m:mo>\n                                             <\/m:mrow>\n                                          <\/m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                       <\/m:mrow>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:msup>\n                                 <\/m:mrow>\n                                 <m:mo>=<\/m:mo>\n                                 <m:mrow>\n                                    <m:mi>\u2133<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mrow>\n                                       <m:mo>(<\/m:mo>\n                                       <m:mover>\n                                          <m:mi>x<\/m:mi>\n                                          <m:mo>~<\/m:mo>\n                                       <\/m:mover>\n                                       <m:mo>,<\/m:mo>\n                                       <m:mover>\n                                          <m:mi>y<\/m:mi>\n                                          <m:mo>~<\/m:mo>\n                                       <\/m:mover>\n                                       <m:mo>)<\/m:mo>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>,<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$\\lvert x-\\tilde{x}\\rvert^{2}+\\lvert\\mathrm{A}(x-\\tilde{x})\\rvert^{2}+\\lvert y-%\n\\tilde{y}\\rvert^{2}+\\lvert\\mathrm{A}^{*}(y-\\tilde{y})\\rvert^{2}=\\mathcal{M}(%\n\\tilde{x},\\tilde{y}),$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:disp-formula>\n               <\/jats:p>\n               <jats:p>where\n<jats:inline-formula id=\"j_cmam-2016-0016_ineq_9990_w2aab3b7b1b1b6b1aab1c13b3b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mi>\u2133<\/m:mi>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mrow>\n                                    <m:mo>(<\/m:mo>\n                                    <m:mover>\n                                       <m:mi>x<\/m:mi>\n                                       <m:mo>~<\/m:mo>\n                                    <\/m:mover>\n                                    <m:mo>,<\/m:mo>\n                                    <m:mover>\n                                       <m:mi>y<\/m:mi>\n                                       <m:mo>~<\/m:mo>\n                                    <\/m:mover>\n                                    <m:mo>)<\/m:mo>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>:=<\/m:mo>\n                              <m:mrow>\n                                 <m:msup>\n                                    <m:mrow>\n                                       <m:mo>|<\/m:mo>\n                                       <m:mrow>\n                                          <m:mi>f<\/m:mi>\n                                          <m:mo>-<\/m:mo>\n                                          <m:mover>\n                                             <m:mi>x<\/m:mi>\n                                             <m:mo>~<\/m:mo>\n                                          <\/m:mover>\n                                          <m:mo>-<\/m:mo>\n                                          <m:mrow>\n                                             <m:msup>\n                                                <m:mi>A<\/m:mi>\n                                                <m:mo>*<\/m:mo>\n                                             <\/m:msup>\n                                             <m:mo>\u2062<\/m:mo>\n                                             <m:mover>\n                                                <m:mi>y<\/m:mi>\n                                                <m:mo>~<\/m:mo>\n                                             <\/m:mover>\n                                          <\/m:mrow>\n                                       <\/m:mrow>\n                                       <m:mo>|<\/m:mo>\n                                    <\/m:mrow>\n                                    <m:mn>2<\/m:mn>\n                                 <\/m:msup>\n                                 <m:mo>+<\/m:mo>\n                                 <m:msup>\n                                    <m:mrow>\n                                       <m:mo>|<\/m:mo>\n                                       <m:mrow>\n                                          <m:mover>\n                                             <m:mi>y<\/m:mi>\n                                             <m:mo>~<\/m:mo>\n                                          <\/m:mover>\n                                          <m:mo>-<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>A<\/m:mi>\n                                             <m:mo>\u2062<\/m:mo>\n                                             <m:mover>\n                                                <m:mi>x<\/m:mi>\n                                                <m:mo>~<\/m:mo>\n                                             <\/m:mover>\n                                          <\/m:mrow>\n                                       <\/m:mrow>\n                                       <m:mo>|<\/m:mo>\n                                    <\/m:mrow>\n                                    <m:mn>2<\/m:mn>\n                                 <\/m:msup>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathcal{M}(\\tilde{x},\\tilde{y}):=\\lvert f-\\tilde{x}-\\mathrm{A}^{*}\\tilde{y}%\n\\rvert^{2}+\\lvert\\tilde{y}-\\mathrm{A}\\tilde{x}\\rvert^{2}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>\ncontains only known data.\nOur second main result is an error estimate for all\nequations of the class\n<jats:inline-formula id=\"j_cmam-2016-0016_ineq_9989_w2aab3b7b1b1b6b1aab1c13b3b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:msup>\n                                       <m:mi>A<\/m:mi>\n                                       <m:mo>*<\/m:mo>\n                                    <\/m:msup>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>A<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>x<\/m:mi>\n                                 <\/m:mrow>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mrow>\n                                    <m:mi>i<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>x<\/m:mi>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>f<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathrm{A}^{*}\\mathrm{A}x+ix=f}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>\nor in mixed formulation\n<jats:inline-formula id=\"j_cmam-2016-0016_ineq_9988_w2aab3b7b1b1b6b1aab1c13b3b5Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:msup>\n                                       <m:mi>A<\/m:mi>\n                                       <m:mo>*<\/m:mo>\n                                    <\/m:msup>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>y<\/m:mi>\n                                 <\/m:mrow>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mrow>\n                                    <m:mi>i<\/m:mi>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mi>x<\/m:mi>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>f<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathrm{A}^{*}y+ix=f}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, <jats:inline-formula id=\"j_cmam-2016-0016_ineq_9987_w2aab3b7b1b1b6b1aab1c13b3b7Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mi>A<\/m:mi>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mi>x<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>=<\/m:mo>\n                              <m:mi>y<\/m:mi>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathrm{A}x=y}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>,\nwhere <jats:italic>i<\/jats:italic> is the imaginary unit. For this problem we have the <jats:italic>two-sided estimate<\/jats:italic>\n               <\/jats:p>\n               <jats:p>\n                  <jats:disp-formula id=\"j_cmam-2016-0016_eq_9998_w2aab3b7b1b1b6b1aab1c13b4aAa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mrow>\n                                    <m:mfrac>\n                                       <m:msqrt>\n                                          <m:mn>2<\/m:mn>\n                                       <\/m:msqrt>\n                                       <m:mrow>\n                                          <m:msqrt>\n                                             <m:mn>2<\/m:mn>\n                                          <\/m:msqrt>\n                                          <m:mo>+<\/m:mo>\n                                          <m:mn>1<\/m:mn>\n                                       <\/m:mrow>\n                                    <\/m:mfrac>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:msub>\n                                       <m:mi>\u2133<\/m:mi>\n                                       <m:mi>i<\/m:mi>\n                                    <\/m:msub>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mrow>\n                                       <m:mo>(<\/m:mo>\n                                       <m:mover>\n                                          <m:mi>x<\/m:mi>\n                                          <m:mo>~<\/m:mo>\n                                       <\/m:mover>\n                                       <m:mo>,<\/m:mo>\n                                       <m:mover>\n                                          <m:mi>y<\/m:mi>\n                                          <m:mo>~<\/m:mo>\n                                       <\/m:mover>\n                                       <m:mo>)<\/m:mo>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                                 <m:mo>\u2264<\/m:mo>\n                                 <m:mrow>\n                                    <m:msup>\n                                       <m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>x<\/m:mi>\n                                             <m:mo>-<\/m:mo>\n                                             <m:mover>\n                                                <m:mi>x<\/m:mi>\n                                                <m:mo>~<\/m:mo>\n                                             <\/m:mover>\n                                          <\/m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                       <\/m:mrow>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:msup>\n                                    <m:mo>+<\/m:mo>\n                                    <m:msup>\n                                       <m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>A<\/m:mi>\n                                             <m:mo>\u2062<\/m:mo>\n                                             <m:mrow>\n                                                <m:mo>(<\/m:mo>\n                                                <m:mrow>\n                                                   <m:mi>x<\/m:mi>\n                                                   <m:mo>-<\/m:mo>\n                                                   <m:mover>\n                                                      <m:mi>x<\/m:mi>\n                                                      <m:mo>~<\/m:mo>\n                                                   <\/m:mover>\n                                                <\/m:mrow>\n                                                <m:mo>)<\/m:mo>\n                                             <\/m:mrow>\n                                          <\/m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                       <\/m:mrow>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:msup>\n                                    <m:mo>+<\/m:mo>\n                                    <m:msup>\n                                       <m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>y<\/m:mi>\n                                             <m:mo>-<\/m:mo>\n                                             <m:mover>\n                                                <m:mi>y<\/m:mi>\n                                                <m:mo>~<\/m:mo>\n                                             <\/m:mover>\n                                          <\/m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                       <\/m:mrow>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:msup>\n                                    <m:mo>+<\/m:mo>\n                                    <m:msup>\n                                       <m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                          <m:mrow>\n                                             <m:msup>\n                                                <m:mi>A<\/m:mi>\n                                                <m:mo>*<\/m:mo>\n                                             <\/m:msup>\n                                             <m:mo>\u2062<\/m:mo>\n                                             <m:mrow>\n                                                <m:mo>(<\/m:mo>\n                                                <m:mrow>\n                                                   <m:mi>y<\/m:mi>\n                                                   <m:mo>-<\/m:mo>\n                                                   <m:mover>\n                                                      <m:mi>y<\/m:mi>\n                                                      <m:mo>~<\/m:mo>\n                                                   <\/m:mover>\n                                                <\/m:mrow>\n                                                <m:mo>)<\/m:mo>\n                                             <\/m:mrow>\n                                          <\/m:mrow>\n                                          <m:mo>|<\/m:mo>\n                                       <\/m:mrow>\n                                       <m:mn>2<\/m:mn>\n                                    <\/m:msup>\n                                 <\/m:mrow>\n                                 <m:mo>\u2264<\/m:mo>\n                                 <m:mrow>\n                                    <m:mfrac>\n                                       <m:msqrt>\n                                          <m:mn>2<\/m:mn>\n                                       <\/m:msqrt>\n                                       <m:mrow>\n                                          <m:msqrt>\n                                             <m:mn>2<\/m:mn>\n                                          <\/m:msqrt>\n                                          <m:mo>-<\/m:mo>\n                                          <m:mn>1<\/m:mn>\n                                       <\/m:mrow>\n                                    <\/m:mfrac>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:msub>\n                                       <m:mi>\u2133<\/m:mi>\n                                       <m:mi>i<\/m:mi>\n                                    <\/m:msub>\n                                    <m:mo>\u2062<\/m:mo>\n                                    <m:mrow>\n                                       <m:mo>(<\/m:mo>\n                                       <m:mover>\n                                          <m:mi>x<\/m:mi>\n                                          <m:mo>~<\/m:mo>\n                                       <\/m:mover>\n                                       <m:mo>,<\/m:mo>\n                                       <m:mover>\n                                          <m:mi>y<\/m:mi>\n                                          <m:mo>~<\/m:mo>\n                                       <\/m:mover>\n                                       <m:mo>)<\/m:mo>\n                                    <\/m:mrow>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>,<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$\\frac{\\sqrt{2}}{\\sqrt{2}+1}\\mathcal{M}_{i}(\\tilde{x},\\tilde{y})\\leq\\lvert x-%\n\\tilde{x}\\rvert^{2}+\\lvert\\mathrm{A}(x-\\tilde{x})\\rvert^{2}+\\lvert y-\\tilde{y}%\n\\rvert^{2}+\\lvert\\mathrm{A}^{*}(y-\\tilde{y})\\rvert^{2}\\leq\\frac{\\sqrt{2}}{%\n\\sqrt{2}-1}\\mathcal{M}_{i}(\\tilde{x},\\tilde{y}),$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:disp-formula>\n               <\/jats:p>\n               <jats:p>where <jats:inline-formula id=\"j_cmam-2016-0016_ineq_9986_w2aab3b7b1b1b6b1aab1c13b5b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:msub>\n                                    <m:mi>\u2133<\/m:mi>\n                                    <m:mi>i<\/m:mi>\n                                 <\/m:msub>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mrow>\n                                    <m:mo>(<\/m:mo>\n                                    <m:mover>\n                                       <m:mi>x<\/m:mi>\n                                       <m:mo>~<\/m:mo>\n                                    <\/m:mover>\n                                    <m:mo>,<\/m:mo>\n                                    <m:mover>\n                                       <m:mi>y<\/m:mi>\n                                       <m:mo>~<\/m:mo>\n                                    <\/m:mover>\n                                    <m:mo>)<\/m:mo>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                              <m:mo>:=<\/m:mo>\n                              <m:mrow>\n                                 <m:msup>\n                                    <m:mrow>\n                                       <m:mo>|<\/m:mo>\n                                       <m:mrow>\n                                          <m:mi>f<\/m:mi>\n                                          <m:mo>-<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>i<\/m:mi>\n                                             <m:mo>\u2062<\/m:mo>\n                                             <m:mover>\n                                                <m:mi>x<\/m:mi>\n                                                <m:mo>~<\/m:mo>\n                                             <\/m:mover>\n                                          <\/m:mrow>\n                                          <m:mo>-<\/m:mo>\n                                          <m:mrow>\n                                             <m:msup>\n                                                <m:mi>A<\/m:mi>\n                                                <m:mo>*<\/m:mo>\n                                             <\/m:msup>\n                                             <m:mo>\u2062<\/m:mo>\n                                             <m:mover>\n                                                <m:mi>y<\/m:mi>\n                                                <m:mo>~<\/m:mo>\n                                             <\/m:mover>\n                                          <\/m:mrow>\n                                       <\/m:mrow>\n                                       <m:mo>|<\/m:mo>\n                                    <\/m:mrow>\n                                    <m:mn>2<\/m:mn>\n                                 <\/m:msup>\n                                 <m:mo>+<\/m:mo>\n                                 <m:msup>\n                                    <m:mrow>\n                                       <m:mo>|<\/m:mo>\n                                       <m:mrow>\n                                          <m:mover>\n                                             <m:mi>y<\/m:mi>\n                                             <m:mo>~<\/m:mo>\n                                          <\/m:mover>\n                                          <m:mo>-<\/m:mo>\n                                          <m:mrow>\n                                             <m:mi>A<\/m:mi>\n                                             <m:mo>\u2062<\/m:mo>\n                                             <m:mover>\n                                                <m:mi>x<\/m:mi>\n                                                <m:mo>~<\/m:mo>\n                                             <\/m:mover>\n                                          <\/m:mrow>\n                                       <\/m:mrow>\n                                       <m:mo>|<\/m:mo>\n                                    <\/m:mrow>\n                                    <m:mn>2<\/m:mn>\n                                 <\/m:msup>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\mathcal{M}_{i}(\\tilde{x},\\tilde{y}):=\\lvert f-i\\tilde{x}-\\mathrm{A}^{*}%\n\\tilde{y}\\rvert^{2}+\\lvert\\tilde{y}-\\mathrm{A}\\tilde{x}\\rvert^{2}}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>\ncontains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.<\/jats:p>","DOI":"10.1515\/cmam-2016-0016","type":"journal-article","created":{"date-parts":[[2016,5,9]],"date-time":"2016-05-09T10:01:12Z","timestamp":1462788072000},"page":"609-631","source":"Crossref","is-referenced-by-count":2,"title":["Functional A Posteriori Error Control for Conforming Mixed Approximations\nof Coercive Problems with Lower Order Terms"],"prefix":"10.1515","volume":"16","author":[{"given":"Immanuel","family":"Anjam","sequence":"first","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Duisburg-Essen, Campus Essen, Germany"}]},{"given":"Dirk","family":"Pauly","sequence":"additional","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik, Universit\u00e4t Duisburg-Essen, Campus Essen, Germany"}]}],"member":"374","published-online":{"date-parts":[[2016,5,9]]},"reference":[{"key":"2023033116455895792_j_cmam-2016-0016_ref_001_w2aab3b7b1b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"Anjam I., Mali O., Muzalevskiy A., Neittaanm\u00e4ki P. and Repin S.,\nA posteriori error estimates for a Maxwell type problem,\nRussian J. Numer. Anal. Math. Modelling 24 (2009), no. 5, 395\u2013408.","DOI":"10.1515\/RJNAMM.2009.025"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_002_w2aab3b7b1b1b6b1ab2b1b2Aa","unstructured":"Bauer S., Pauly D. and Schomburg M.,\nThe Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions,\npreprint 2015, http:\/\/arxiv.org\/abs\/1511.06697."},{"key":"2023033116455895792_j_cmam-2016-0016_ref_003_w2aab3b7b1b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"Brezzi F. and Fortin M.,\nMixed and Hybrid Finite Element Methods,\nSpringer, New York, 1991.","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_004_w2aab3b7b1b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"Cai Z. and Zhang S.,\nFlux recovery and a posteriori error estimators: Conforming elements for scalar elliptic equations,\nSIAM J. Numer. Anal. 48 (2010), no. 2, 578\u2013602.","DOI":"10.1137\/080742993"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_005_w2aab3b7b1b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"Jochmann F.,\nA compactness result for vector fields with divergence and curl in Lq\u2062(\u03a9)${{L}^{q}({\\Omega})}$ involving mixed boundary conditions,\nAppl. Anal. 66 (1997), 189\u2013203.","DOI":"10.1080\/00036819708840581"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_006_w2aab3b7b1b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"Mali O., Muzalevskiy A. and Pauly D.,\nConforming and non-conforming functional a posteriori error estimates for elliptic boundary value problems in exterior domains: Theory and numerical tests,\nRussian J. Numer. Anal. Math. Modelling 28 (2013), no. 6, 577\u2013596.","DOI":"10.1515\/rnam-2013-0032"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_007_w2aab3b7b1b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"Mali O., Neittaanm\u00e4ki P. and Repin S.,\nAccuracy Verification Methods, Theory and Algorithms,\nSpringer, Dordrecht, 2014.","DOI":"10.1007\/978-94-007-7581-7"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_008_w2aab3b7b1b1b6b1ab2b1b8Aa","unstructured":"Neittaanm\u00e4ki P. and Repin S.,\nReliable Methods for Computer Simulation, Error Control and a Posteriori Estimates,\nElsevier, New York, 2004."},{"key":"2023033116455895792_j_cmam-2016-0016_ref_009_w2aab3b7b1b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"Pauly D. and Repin S.,\nFunctional a posteriori error estimates for elliptic problems in exterior domains,\nJ. Math. Sci. (N. Y.) 162 (2009), no. 3, 393\u2013406.","DOI":"10.1007\/s10958-009-9643-4"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_010_w2aab3b7b1b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"Pauly D. and Repin S.,\nTwo-sided a posteriori error bounds for electro-magneto static problems,\nJ. Math. Sci. (N. Y.) 166 (2010), no. 1, 53\u201362.","DOI":"10.1007\/s10958-010-9844-x"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_011_w2aab3b7b1b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"Pauly D., Repin S. and Rossi T.,\nEstimates for deviations from exact solutions of Maxwell\u2019s initial boundary value problem,\nAnn. Acad. Sci. Fenn. Math. 36 (2011), no. 2, 661\u2013676.","DOI":"10.5186\/aasfm.2011.3641"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_012_w2aab3b7b1b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"Repin S.,\nA Posteriori Estimates for Partial Differential Equations,\nRadon Ser. Comput. Appl. Math. 4,\nWalter de Gruyter, Berlin, 2008.","DOI":"10.1515\/9783110203042"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_013_w2aab3b7b1b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"Repin S.,\nEstimates of deviations from exact solutions of initial boundary value problems for the wave equation,\nJ. Math. Sci. (N. Y.) 159 (2009), no. 2, 229\u2013240.","DOI":"10.1007\/s10958-009-9435-x"},{"key":"2023033116455895792_j_cmam-2016-0016_ref_014_w2aab3b7b1b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"Repin S., Sauter S. and Smolianski A.,\nTwo-sided a posteriori error estimates for mixed formulations of elliptic problems,\nSIAM J. Numer. Anal. 45 (2007), no. 3, 928\u2013945.","DOI":"10.1137\/050641533"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/16\/4\/article-p609.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0016\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0016\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T23:26:37Z","timestamp":1680305197000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2016-0016\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,9]]},"references-count":14,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,7,6]]},"published-print":{"date-parts":[[2016,10,1]]}},"alternative-id":["10.1515\/cmam-2016-0016"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2016-0016","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2016,5,9]]}}}