{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T19:12:24Z","timestamp":1760728344082},"reference-count":17,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,6,27]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart\u2013Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as \ud835\udcaa(<jats:italic>h<\/jats:italic>\n                  <jats:sup>\u22121<\/jats:sup>), while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as \ud835\udcaa(<jats:italic>h<\/jats:italic>\n                  <jats:sup>\u2212<jats:italic>k<\/jats:italic>\u22121<\/jats:sup>), <jats:italic>k<\/jats:italic> being the order of the Raviart\u2013Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual <jats:italic>L<\/jats:italic>\n                  <jats:sup>2<\/jats:sup>-norm. However, we are still able to recover the optimal a priori <jats:italic>L<\/jats:italic>\n                  <jats:sup>2<\/jats:sup>-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart\u2013Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited.<\/jats:p>","DOI":"10.1515\/jnma-2021-0042","type":"journal-article","created":{"date-parts":[[2022,1,18]],"date-time":"2022-01-18T00:37:03Z","timestamp":1642466223000},"page":"141-162","source":"Crossref","is-referenced-by-count":2,"title":["Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow"],"prefix":"10.1515","volume":"30","author":[{"given":"Erik","family":"Burman","sequence":"first","affiliation":[{"name":"Chair of Computational Mathematics, University College London , London , UK"}]},{"given":"Riccardo","family":"Puppi","sequence":"additional","affiliation":[{"name":"Chair of Modelling and Numerical Simulation, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne , Lausanne , Switzerland"}]}],"member":"374","published-online":{"date-parts":[[2021,10,18]]},"reference":[{"key":"2022060700372521109_j_jnma-2021-0042_ref_001","unstructured":"R. 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