{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,4]],"date-time":"2026-05-04T06:06:50Z","timestamp":1777874810705,"version":"3.51.4"},"reference-count":61,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100002920","name":"University Grants Committee Research Grants Council","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004377","name":"Hong Kong Polytechnic University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004377","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Computers & Mathematics with Applications"],"published-print":{"date-parts":[[2026,5]]},"DOI":"10.1016\/j.camwa.2026.02.019","type":"journal-article","created":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T14:51:00Z","timestamp":1772549460000},"page":"43-59","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Optimal L2 error estimates for a fully discrete method of the Cahn-Hilliard-MHD model"],"prefix":"10.1016","volume":"210","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-6260-1015","authenticated-orcid":false,"given":"Dongmei","family":"Duan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3982-2381","authenticated-orcid":false,"given":"Fuzheng","family":"Gao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1270-4972","authenticated-orcid":false,"given":"Xiaoming","family":"He","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2461-7166","authenticated-orcid":false,"given":"Yanping","family":"Lin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"key":"10.1016\/j.camwa.2026.02.019_bib0001","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1007\/BF01396363","article-title":"A second order splitting method for the Cahn-Hilliard equation","volume":"54","author":"Elliott","year":"1989","journal-title":"Numerische Mathematik"},{"issue":"3\u20134","key":"10.1016\/j.camwa.2026.02.019_bib0002","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0167-2789(03)00030-7","article-title":"A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method","volume":"179","author":"Liu","year":"2003","journal-title":"Phys. D"},{"key":"10.1016\/j.camwa.2026.02.019_bib0003","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1017\/S0022112004000370","article-title":"A diffuse-interface method for simulating two-phase flows of complex fluids","volume":"515","author":"Yue","year":"2004","journal-title":"J. Fluid Mech."},{"key":"10.1016\/j.camwa.2026.02.019_bib0004","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1017\/S0022112006001935","article-title":"A variational approach to the moving contact line hydrodynamics","volume":"564","author":"Qian","year":"2006","journal-title":"J. Fluid Mech."},{"issue":"258","key":"10.1016\/j.camwa.2026.02.019_bib0005","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1090\/S0025-5718-06-01915-6","article-title":"Analysis of finite element approximations of a phase field model for two-phase fluids","volume":"76","author":"Feng","year":"2007","journal-title":"Math. Comput."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0006","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1137\/090752675","article-title":"An energy stable and convergent finite-difference scheme for the modified phase field crystal equation","volume":"49","author":"Wang","year":"2011","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0007","doi-asserted-by":"crossref","first-page":"1320","DOI":"10.1137\/110827119","article-title":"Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation","volume":"50","author":"Feng","year":"2012","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0008","doi-asserted-by":"crossref","first-page":"1167","DOI":"10.1093\/imanum\/dru035","article-title":"Adaptive, second-order in time, primitive-variable discontinuous Galerkin schemes for a Cahn\u2013Hilliard equation with a mass source","volume":"35","author":"Aristotelous","year":"2015","journal-title":"IMA J. Numer. Analy."},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0009","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1137\/140971154","article-title":"Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows","volume":"53","author":"Shen","year":"2015","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.camwa.2026.02.019_bib0010","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/j.jcp.2015.09.044","article-title":"A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids","volume":"305","author":"Zhao","year":"2016","journal-title":"J. Comput. Phys."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0011","doi-asserted-by":"crossref","first-page":"1899","DOI":"10.1137\/15M1039857","article-title":"Asymptotically compatible Fourier spectral approximations of nonlocal Allen-Cahn equations","volume":"54","author":"Du","year":"2016","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.camwa.2026.02.019_bib0012","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.ijmultiphaseflow.2017.04.008","article-title":"Three dimensional phase-field investigation of droplet formation in microfluidic flow focusing devices with experimental validation","volume":"93","author":"Bai","year":"2017","journal-title":"Int. J. Multiph. Flow"},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0013","doi-asserted-by":"crossref","first-page":"B110","DOI":"10.1137\/16M1100885","article-title":"Decoupled, linear, and energy stable finite element method for the Cahn-Hilliard-Navier-Stokes-Darcy phase field model","volume":"40","author":"Gao","year":"2018","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.02.019_bib0014","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.jcp.2017.10.021","article-title":"The scalar auxiliary variable (SAV) approach for gradient flows","volume":"353","author":"Shen","year":"2018","journal-title":"J. Comput. Phys."},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0015","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.cam.2018.04.027","article-title":"Linear, second order and unconditionally energy stable schemes for the viscous Cahn\u2013Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method","volume":"343","author":"Yang","year":"2018","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0016","doi-asserted-by":"crossref","first-page":"2535","DOI":"10.1137\/18M1223459","article-title":"Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system","volume":"51","author":"Giorgini","year":"2019","journal-title":"SIAM J. Math. Anal."},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0017","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1137\/18M1206084","article-title":"A second order BDF numerical scheme with variable steps for the Cahn\u2013Hilliard equation","volume":"57","author":"Chen","year":"2019","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"10.1016\/j.camwa.2026.02.019_bib0018","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1137\/18M118236X","article-title":"Maximum principle preserving exponential time differencing schemes for the nonlocal Allen-Cahn equation","volume":"57","author":"Du","year":"2019","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"10.1016\/j.camwa.2026.02.019_bib0019","doi-asserted-by":"crossref","first-page":"2294","DOI":"10.1137\/19M1289157","article-title":"On energy stable, maximum-principle preserving, second-order BDF scheme with variable steps for the Allen-Cahn equation","volume":"58","author":"Liao","year":"2020","journal-title":"IAM J. Numer. Analy."},{"issue":"2","key":"10.1016\/j.camwa.2026.02.019_bib0020","doi-asserted-by":"crossref","first-page":"B479","DOI":"10.1137\/20M1336734","article-title":"On a novel fully decoupled, second-order accurate energy stable numerical scheme for a binary fluid-surfactant phase-field model","volume":"43","author":"Yang","year":"2021","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0021","doi-asserted-by":"crossref","first-page":"B167","DOI":"10.1137\/19M1288280","article-title":"Decoupled, linear, and unconditionally energy stable fully-discrete finite element numerical scheme for a two-phase ferrohydrodynamics model","volume":"43","author":"Zhang","year":"2021","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"10.1016\/j.camwa.2026.02.019_bib0022","doi-asserted-by":"crossref","first-page":"1905","DOI":"10.1137\/21M1446496","article-title":"Generalized SAV-exponential integrator schemes for Allen-Cahn type gradient flows","volume":"60","author":"Ju","year":"2022","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0023","doi-asserted-by":"crossref","first-page":"2621","DOI":"10.1093\/imanum\/drab046","article-title":"Error estimate of a decoupled numerical scheme for the Cahn-Hilliard-Stokes-Darcy system","volume":"42","author":"Chen","year":"2022","journal-title":"IMA J. Numer. Analy."},{"key":"10.1016\/j.camwa.2026.02.019_bib0024","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2023.111997","article-title":"An energy-stable smoothed particle hydrodynamics discretization of the Navier-Stokes-Cahn-Hilliard model for incompressible two-phase flows","volume":"479","author":"Feng","year":"2023","journal-title":"J. Comput. Phys."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0025","doi-asserted-by":"crossref","first-page":"B253","DOI":"10.1137\/22M1499376","article-title":"Reformulated weak formulation and efficient fully-discrete finite element method for a two-phase ferrohydrodynamics Shliomis model","volume":"45","author":"Zhang","year":"2023","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.02.019_bib0026","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2023.115584","article-title":"A SAV finite element method for the Cahn\u2013Hilliard equation with dynamic boundary conditions","volume":"438","author":"Li","year":"2024","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"10.1016\/j.camwa.2026.02.019_bib0027","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1016\/j.jcp.2003.07.035","article-title":"Conservative multigrid methods for Cahn-Hilliard fluids","volume":"193","author":"Kim","year":"2004","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2026.02.019_bib0028","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.jcp.2017.04.039","article-title":"Multiphase Allen-Cahn and Cahn-Hilliard models and their discretizations with the effect of pairwise surface tensions","volume":"343","author":"Wu","year":"2017","journal-title":"J. Comput. Phys."},{"issue":"2","key":"10.1016\/j.camwa.2026.02.019_bib0029","doi-asserted-by":"crossref","first-page":"1234","DOI":"10.1007\/s10915-018-0753-3","article-title":"A robust solver for a mixed finite element method for the Cahn-Hilliard equation","volume":"77","author":"Brenner","year":"2018","journal-title":"J. Sci. Comput."},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0030","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1137\/18M1206084","article-title":"A second order BDF numerical scheme with variable steps for the Cahn-Hilliard equation","volume":"57","author":"Chen","year":"2019","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"10.1016\/j.camwa.2026.02.019_bib0031","doi-asserted-by":"crossref","first-page":"A3703","DOI":"10.1137\/19M1264412","article-title":"Energy-decaying extrapolated RK-SAV methods for the Allen-Cahn and Cahn-Hilliard equations","volume":"41","author":"Akrivis","year":"2019","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0032","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1137\/19M1280740","article-title":"An efficient and convergent finite element scheme for Cahn-Hilliard equations with dynamic boundary conditions","volume":"59","author":"Metzger","year":"2021","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.camwa.2026.02.019_bib0033","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2021.114186","article-title":"A linear second-order in time unconditionally energy stable finite element scheme for a Cahn-Hilliard phase-field model for two-phase incompressible flow of variable densities","volume":"387","author":"Fu","year":"2021","journal-title":"Comput. Method. Appl. Mech. Eng."},{"issue":"6","key":"10.1016\/j.camwa.2026.02.019_bib0034","first-page":"839","article-title":"A preconditioned steepest descent solver for the Cahn-Hilliard dquation with variable mobility","volume":"19","author":"Chen","year":"2022","journal-title":"Int. J. Numer. Anal. Model."},{"issue":"334","key":"10.1016\/j.camwa.2026.02.019_bib0035","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1090\/mcom\/3704","article-title":"Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation","volume":"91","author":"Li","year":"2022","journal-title":"Math. Comput."},{"issue":"2","key":"10.1016\/j.camwa.2026.02.019_bib0036","doi-asserted-by":"crossref","first-page":"A427","DOI":"10.1137\/21M1459915","article-title":"Optimized Schwarz methods for the Cahn-Hilliard equation","volume":"45","author":"Xu","year":"2023","journal-title":"SIAM J. Sci. Comput."},{"issue":"194","key":"10.1016\/j.camwa.2026.02.019_bib0037","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1090\/S0025-5718-1991-1066834-0","article-title":"On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics","volume":"56","author":"Gunzburger","year":"1991","journal-title":"Math. Comput."},{"issue":"6","key":"10.1016\/j.camwa.2026.02.019_bib0038","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1051\/m2an:2008034","article-title":"Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system","volume":"42","author":"Prohl","year":"2008","journal-title":"ESAIM: Math. Modell. Numer. Analy."},{"key":"10.1016\/j.camwa.2026.02.019_bib0039","series-title":"Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas","author":"Goedbloed","year":"2010"},{"key":"10.1016\/j.camwa.2026.02.019_bib0040","series-title":"Magnetohydrodynamics","volume":"3","author":"Moreau","year":"2013"},{"issue":"4","key":"10.1016\/j.camwa.2026.02.019_bib0041","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1002\/num.21857","article-title":"Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows","volume":"30","author":"Layton","year":"2014","journal-title":"Numer. Method. Part. Differ. Equ."},{"issue":"2","key":"10.1016\/j.camwa.2026.02.019_bib0042","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1093\/imanum\/dru015","article-title":"Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations","volume":"35","author":"He","year":"2015","journal-title":"IMA J. Numer. Analy."},{"issue":"6","key":"10.1016\/j.camwa.2026.02.019_bib0043","doi-asserted-by":"crossref","first-page":"B1009","DOI":"10.1137\/16M1074084","article-title":"Block preconditioners for stable mixed nodal and edge finite element representations of incompressible resistive MHD","volume":"38","author":"Phillips","year":"2016","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0044","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s10915-016-0288-4","article-title":"Decoupled, unconditionally stable, higher order discretizations for MHD flow simulation","volume":"71","author":"Heister","year":"2017","journal-title":"J. Sci. Comput."},{"issue":"04","key":"10.1016\/j.camwa.2026.02.019_bib0045","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1142\/S0218202518500173","article-title":"A fully divergence-free finite element method for magnetohydrodynamic equations","volume":"28","author":"Hiptmair","year":"2018","journal-title":"Math. Model. Method. Appl. Sci."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0046","doi-asserted-by":"crossref","first-page":"1678","DOI":"10.1007\/s10915-019-01059-1","article-title":"A decoupled, linear and unconditionally energy stable scheme with finite element discretizations for magneto-hydrodynamic equations","volume":"81","author":"Zhang","year":"2019","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.02.019_bib0047","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2021.110752","article-title":"A fully decoupled linearized finite element method with second-order temporal accuracy and unconditional energy stability for incompressible MHD equations","volume":"448","author":"Zhang","year":"2022","journal-title":"J. Comput. Phys."},{"issue":"4","key":"10.1016\/j.camwa.2026.02.019_bib0048","doi-asserted-by":"crossref","first-page":"476","DOI":"10.4208\/ijnam2024-1019","article-title":"A voigt-regularization of the thermally coupled inviscid, resistive magnetohydrodynamic","volume":"21","author":"Yang","year":"2024","journal-title":"Int. J. Numer. Analy. Model."},{"key":"10.1016\/j.camwa.2026.02.019_bib0049","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/j.cma.2019.07.022","article-title":"A diffuse interface model and semi-implicit energy stable finite element method for two-phase magnetohydrodynamic flows","volume":"356","author":"Yang","year":"2019","journal-title":"Comput. Method. Appl. Mech. Eng."},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0050","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s10915-021-01741-3","article-title":"Highly efficient and energy stable schemes for the 2D\/3D diffuse interface model of two-phase magnetohydrodynamics","volume":"90","author":"Su","year":"2022","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.02.019_bib0051","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2023.107477","article-title":"Gauge\u2013Uzawa-based, highly efficient decoupled schemes for the diffuse interface model of two-phase magnetohydrodynamic","volume":"126","author":"Zhang","year":"2023","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.camwa.2026.02.019_bib0052","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/j.matcom.2023.08.039","article-title":"Decoupled, linear, unconditionally energy stable and charge-conservative finite element method for an inductionless magnetohydrodynamic phase-field model","volume":"215","author":"Wang","year":"2024","journal-title":"Math. Comput. Simul."},{"key":"10.1016\/j.camwa.2026.02.019_bib0053","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2023.115409","article-title":"Convergence analysis of a temporally second-order accurate finite element scheme for the Cahn\u2013Hilliard-Magnetohydrodynamics system of equations","volume":"436","author":"Wang","year":"2024","journal-title":"J. Comput. Appl. Math."},{"key":"10.1016\/j.camwa.2026.02.019_bib0054","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2024.108195","article-title":"Error analysis of a fully discrete projection method for Cahn\u2013Hilliard inductionless MHD problems","volume":"138","author":"Ding","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.camwa.2026.02.019_bib0055","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.apnum.2024.09.027","article-title":"Error estimates of time discretizations for a Cahn-Hilliard phase-field model for the two-phase magnetohydrodynamic flows","volume":"207","author":"Shen","year":"2025","journal-title":"Appl. Numer. Math."},{"issue":"2","key":"10.1016\/j.camwa.2026.02.019_bib0056","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10915-024-02773-1","article-title":"Unconditionally optimal convergent zero-energy-contribution scheme for two phase MHD model","volume":"102","author":"Yang","year":"2025","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.02.019_bib0057","article-title":"Unconditional stability and optimal error estimates of first order semi-implicit stabilized finite element method for two phase magnetohydrodynamic diffuse interface model","volume":"429","author":"Chen","year":"2022","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"10.1016\/j.camwa.2026.02.019_bib0058","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s10915-023-02147-z","article-title":"Error analysis of fully discrete scheme for the Cahn-Hilliard-Magneto-Hydrodynamics problem","volume":"95","author":"Qiu","year":"2023","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.camwa.2026.02.019_bib0059","doi-asserted-by":"crossref","DOI":"10.1016\/j.cam.2025.116580","article-title":"Error estimates of semi-implicit numerical scheme for a diffuse interface model of two-phase magnetohydrodynamic flows","volume":"465","author":"Duan","year":"2025","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0060","doi-asserted-by":"crossref","first-page":"1218","DOI":"10.1137\/22M1486844","article-title":"Optimal L2 error estimates of unconditionally stable finite element schemes for the Cahn-Hilliard-Navier-Stokes system","volume":"61","author":"Cai","year":"2023","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"10.1016\/j.camwa.2026.02.019_bib0061","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1051\/m2an\/2022020","article-title":"Optimal error estimates of a Crank\u2013Nicolson finite element projection method for magnetohydrodynamic equations","volume":"56","author":"Wang","year":"2022","journal-title":"ESAIM: Math. Modell. Numer. Analy."}],"container-title":["Computers & Mathematics with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122126000878?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122126000878?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T20:23:14Z","timestamp":1777580594000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0898122126000878"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5]]},"references-count":61,"alternative-id":["S0898122126000878"],"URL":"https:\/\/doi.org\/10.1016\/j.camwa.2026.02.019","relation":{},"ISSN":["0898-1221"],"issn-type":[{"value":"0898-1221","type":"print"}],"subject":[],"published":{"date-parts":[[2026,5]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Optimal L2 error estimates for a fully discrete method of the Cahn-Hilliard-MHD model","name":"articletitle","label":"Article Title"},{"value":"Computers & Mathematics with Applications","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.camwa.2026.02.019","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}]}}