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The main tool in the a priori error analysis is the connection between the nonconforming virtual element space and the Sobolev space <jats:inline-formula><jats:alternatives><jats:tex-math>$$H^1_0(\\Omega )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mn>0<\/mml:mn>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msubsup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>\u03a9<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> by a right-inverse <jats:italic>J<\/jats:italic> of the interpolation operator <jats:inline-formula><jats:alternatives><jats:tex-math>$$I_h$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>I<\/mml:mi>\n                    <mml:mi>h<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The stability of the discrete solution allows for the proof of existence of a unique discrete solution, of a discrete inf-sup estimate and, consequently, for optimal error estimates in the <jats:inline-formula><jats:alternatives><jats:tex-math>$$H^1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> norms. The explicit residual-based a posteriori error estimate for the NCVEM is reliable and efficient up to the oscillation terms. Numerical experiments on different types of polygonal meshes illustrate the robustness of an error estimator and support the improved convergence rate of an adaptive mesh-refinement in comparison to the uniform mesh-refinement.<\/jats:p>","DOI":"10.1007\/s00211-022-01296-x","type":"journal-article","created":{"date-parts":[[2022,6,20]],"date-time":"2022-06-20T08:05:57Z","timestamp":1655712357000},"page":"551-600","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["A priori and a posteriori error analysis of the lowest-order NCVEM for second-order linear indefinite elliptic problems"],"prefix":"10.1007","volume":"151","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rekha","family":"Khot","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amiya K.","family":"Pani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,6,20]]},"reference":[{"issue":"3","key":"1296_CR1","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1016\/j.camwa.2013.05.015","volume":"66","author":"B Ahmad","year":"2013","unstructured":"Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. 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