{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,11]],"date-time":"2026-01-11T01:08:27Z","timestamp":1768093707147,"version":"3.49.0"},"reference-count":18,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Reconstructing the pressure from given flow velocities is a task\narising in various applications, and the standard approach uses the\nNavier\u2013Stokes equations to derive a Poisson problem for the pressure <jats:italic>p<\/jats:italic>.\nThat method, however, artificially increases the regularity requirements\non both solution and data. In this context, we propose and analyze two\nalternative techniques to determine <jats:inline-formula id=\"j_cmam-2021-0242_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>p<\/m:mi>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mrow>\n                                 <m:msup>\n                                    <m:mi>L<\/m:mi>\n                                    <m:mn>2<\/m:mn>\n                                 <\/m:msup>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mrow>\n                                    <m:mo stretchy=\"false\">(<\/m:mo>\n                                    <m:mi mathvariant=\"normal\">\u03a9<\/m:mi>\n                                    <m:mo stretchy=\"false\">)<\/m:mo>\n                                 <\/m:mrow>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2021-0242_eq_0228.png\"\/>\n                        <jats:tex-math>{p\\in L^{2}(\\Omega)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. The first is an\nultra-weak variational formulation applying integration by parts to shift\nall derivatives to the test functions. We present conforming finite element discretizations and prove optimal\nconvergence of the resulting Galerkin\u2013Petrov method. The second approach\nis a least-squares method for the original gradient equation, reformulated\nand solved as an artificial Stokes system. To simplify\nthe incorporation of the given velocity within the right-hand side,\nwe assume in the derivations that the\nvelocity field is solenoidal. Yet this assumption is not restrictive,\nas we can use non-divergence-free approximations and even\ncompressible velocities. Numerical experiments confirm\nthe optimal a priori error estimates for both methods considered.<\/jats:p>","DOI":"10.1515\/cmam-2021-0242","type":"journal-article","created":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T10:50:56Z","timestamp":1700477456000},"page":"921-934","source":"Crossref","is-referenced-by-count":3,"title":["Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation"],"prefix":"10.1515","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3494-7118","authenticated-orcid":false,"given":"Douglas R. Q.","family":"Pacheco","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences , Norwegian University of Science and Technology , Trondheim , Norway"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2552-3022","authenticated-orcid":false,"given":"Olaf","family":"Steinbach","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Angewandte Mathematik , TU Graz , Graz , Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,11,21]]},"reference":[{"key":"2024100217254211905_j_cmam-2021-0242_ref_001","doi-asserted-by":"crossref","unstructured":"R.  Araya, C.  Bertoglio, C.  Carcamo, D.  Nolte and S.  Uribe,\nConvergence analysis of pressure reconstruction methods from discrete velocities,\nESAIM Math. Model. Numer. Anal. 57 (2023), no. 3, 1839\u20131861.","DOI":"10.1051\/m2an\/2023021"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_002","doi-asserted-by":"crossref","unstructured":"C.  Bahriawati and C.  Carstensen,\nThree MATLAB implementations of the lowest-order Raviart\u2013Thomas MFEM with a posteriori error control,\nComput. Methods Appl. Math. 5 (2005), no. 4, 333\u2013361.","DOI":"10.2478\/cmam-2005-0016"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_003","doi-asserted-by":"crossref","unstructured":"M.  Berggren,\nApproximations of very weak solutions to boundary-value problems,\nSIAM J. Numer. Anal. 42 (2004), no. 2, 860\u2013877.","DOI":"10.1137\/S0036142903382048"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_004","doi-asserted-by":"crossref","unstructured":"C.  Bertoglio, R.  Nu\u00f1ez, F.  Galarce, D.  Nordsletten and A.  Osses,\nRelative pressure estimation from velocity measurements in blood flows: State-of-the-art and new approaches,\nInt. J. Numer. Methods Biomed. Eng. 34 (2018), no. 2, Article ID e2925.","DOI":"10.1002\/cnm.2925"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_005","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_006","doi-asserted-by":"crossref","unstructured":"J. A.  Burns, T.  Lin and L. G.  Stanley,\nA Petrov Galerkin finite-element method for interface problems arising in sensitivity computations,\nComput. Math. Appl. 49 (2005), no. 11\u201312, 1889\u20131903.","DOI":"10.1016\/j.camwa.2004.06.037"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_007","doi-asserted-by":"crossref","unstructured":"A.  Ern and J.-L.  Guermond,\nTheory and Practice of Finite Elements,\nAppl. Math. Sci. 159,\nSpringer, New York, 2004.","DOI":"10.1007\/978-1-4757-4355-5"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_008","doi-asserted-by":"crossref","unstructured":"A.  Ern, M.  Vohral\u00edk and M.  Zakerzadeh,\nGuaranteed and robust \n                  \n                     \n                        \n                           L\n                           2\n                        \n                     \n                     \n                     L^{2}\n                  \n               -norm a posteriori error estimates for 1D linear advection problems,\nESAIM Math. Model. Numer. Anal. 55 (2021), S447\u2013S474.","DOI":"10.1051\/m2an\/2020041"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_009","doi-asserted-by":"crossref","unstructured":"M.  Fortin,\nAn analysis of the convergence of mixed finite element methods,\nRAIRO Anal. Num\u00e9r. 11 (1977), no. 4, 341\u2013354.","DOI":"10.1051\/m2an\/1977110403411"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_010","doi-asserted-by":"crossref","unstructured":"J.  Henning, D.  Palitta, V.  Simoncini and K.  Urban,\nAn ultraweak space-time variational formulation for the wave equation: Analysis and efficient numerical solution,\nESAIM Math. Model. Numer. Anal. 56 (2022), no. 4, 1173\u20131198.","DOI":"10.1051\/m2an\/2022035"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_011","doi-asserted-by":"crossref","unstructured":"H.  Johnston and J.-G.  Liu,\nAccurate, stable and efficient Navier\u2013Stokes solvers based on explicit treatment of the pressure term,\nJ. Comput. Phys. 199 (2004), no. 1, 221\u2013259.","DOI":"10.1016\/j.jcp.2004.02.009"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_012","unstructured":"C.  K\u00f6the, R.  L\u00f6scher and O.  Steinbach,\nAdaptive least-squares space-time finite element methods,\npreprint (2023), https:\/\/arxiv.org\/abs\/2309.14300."},{"key":"2024100217254211905_j_cmam-2021-0242_ref_013","unstructured":"R.  L\u00f6scher, D. R. Q.  Pacheco and O.  Steinbach,\nAn adaptive least-squares finite element method for the pressure reconstruction in fluid mechanics,\nin preparation (2023)."},{"key":"2024100217254211905_j_cmam-2021-0242_ref_014","doi-asserted-by":"crossref","unstructured":"D.  Nolte, J.  Urbina, J.  Sotelo, L.  Sok, C.  Montalba, I.  Valverde, A.  Osses, S.  Uribe and C.  Bertoglio,\nValidation of 4D flow based relative pressure maps in aortic flows,\nMed. Image Anal. 74 (2021), Article ID 102195.","DOI":"10.1016\/j.media.2021.102195"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_015","doi-asserted-by":"crossref","unstructured":"D. R. Q.  Pacheco,\nOn the numerical treatment of viscous and convective effects in relative pressure reconstruction methods,\nInt. J. Numer. Meth. Biomed. Eng. 38 (2022), no. 3, Paper No. e3562.","DOI":"10.1002\/cnm.3562"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_016","unstructured":"D. R. Q.  Pacheco,\nStable and Stabilised Finite Element Methods for Incompressible Flows of Generalised Newtonian Fluids,\nComputation Eng. Sci. 42,\nTechnische Universit\u00e4t Graz, Graz, 2021."},{"key":"2024100217254211905_j_cmam-2021-0242_ref_017","doi-asserted-by":"crossref","unstructured":"R. L.  Sani, J.  Shen, O.  Pironneau and P. M.  Gresho,\nPressure boundary condition for the time-dependent incompressible Navier\u2013Stokes equations,\nInternat. J. Numer. Methods Fluids 50 (2006), no. 6, 673\u2013682.","DOI":"10.1002\/fld.1062"},{"key":"2024100217254211905_j_cmam-2021-0242_ref_018","doi-asserted-by":"crossref","unstructured":"O.  Steinbach,\nNumerical Approximation Methods for Elliptic Boundary Value Problems,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-68805-3"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2021-0242\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2021-0242\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,2]],"date-time":"2024-10-02T17:26:56Z","timestamp":1727890016000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2021-0242\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,21]]},"references-count":18,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,1,2]]},"published-print":{"date-parts":[[2024,10,1]]}},"alternative-id":["10.1515\/cmam-2021-0242"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2021-0242","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,21]]}}}