{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T06:14:09Z","timestamp":1775888049759,"version":"3.50.1"},"reference-count":48,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T00:00:00Z","timestamp":1663632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Swedish Research Council","award":["VR 2018-03661"],"award-info":[{"award-number":["VR 2018-03661"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Stability and convergence analyses for the domain decomposition finite element\/finite difference (FE\/FD) method are presented. The analyses are developed for a semi-discrete finite element scheme for time-dependent Maxwell\u2019s equations. The explicit finite element schemes in different settings of the spatial domain are constructed and a domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies.<\/jats:p>","DOI":"10.3390\/a15100337","type":"journal-article","created":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T21:12:53Z","timestamp":1663708373000},"page":"337","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Stability and Convergence Analysis of a Domain Decomposition FE\/FD Method for Maxwell\u2019s Equations in the Time Domain"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3687-1866","authenticated-orcid":false,"given":"Mohammad","family":"Asadzadeh","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Chalmers University of Technology, 41296 Gothenburg, Sweden"}]},{"given":"Larisa","family":"Beilina","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Gothenburg, 41296 Gothenburg, Sweden"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,20]]},"reference":[{"key":"ref_1","unstructured":"Iserles, A. (1994). Domain decomposition algorithms. Acta Numerica, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Toselli, A., and Widlund, B. (2005). Domain Decomposition Methods, Springer.","DOI":"10.1007\/b137868"},{"key":"ref_3","first-page":"176","article-title":"Adaptive Hybrid Finite Element\/Difference method for Maxwell\u2019s equations","volume":"1","author":"Beilina","year":"2010","journal-title":"TWMS J. Pure Appl. Math."},{"key":"ref_4","first-page":"702","article-title":"Energy estimates and numerical verification of the stabilized Domain Decomposition Finite Element\/Finite Difference approach for time-dependent Maxwell\u2019s system","volume":"11","author":"Beilina","year":"2013","journal-title":"Cent. Eur. J. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/S0010-4655(99)00463-4","article-title":"Stable FEM-FDTD hybrid method for Maxwell\u2019s equations","volume":"125","author":"Rylander","year":"2000","journal-title":"J. Comput. Phys. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1006\/jcph.2002.7063","article-title":"Stability of Explicit-Implicit Hybrid Time-Stepping Schemes for Maxwell\u2019s Equations","volume":"179","author":"Rylander","year":"2002","journal-title":"J. Comput. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1023\/A:1007625629485","article-title":"Explicit hybrid solver for the Maxwell equations in 3D","volume":"15","author":"Edelvik","year":"2000","journal-title":"J. Sci. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1109\/TAP.1966.1138693","article-title":"Numerical solution of initial boundary value problems involving Maxwell\u2019s equations in isotropic media","volume":"14","author":"Yee","year":"1966","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18576\/amis\/120101","article-title":"An Adaptive Finite Element Method in Quantitative Reconstruction of Small Inclusions from Limited Observations","volume":"12","author":"Malmberg","year":"2018","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1122","DOI":"10.1016\/j.jcp.2007.05.029","article-title":"Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries","volume":"226","author":"Ciarlet","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1016\/0022-247X(91)90104-8","article-title":"A coercive bilinear form for Maxwell\u2019s equations","volume":"157","author":"Costabel","year":"1991","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s002050050197","article-title":"Singularities of Electromagnetic Fieldsn in Polyhedral Domains","volume":"151","author":"Costabel","year":"2000","journal-title":"Arch. Rational Mech. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s002110100388","article-title":"Weighted Regularization of Maxwell Equations in Polyhedral Domains. A rehabilitation of nodal finite elements","volume":"93","author":"Dauge","year":"2002","journal-title":"Numer. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1051\/m2an\/2016066","article-title":"Finite element quasi-interpolation and best approximation","volume":"51","author":"Ern","year":"2017","journal-title":"ESAIM Math. Mod. Numer. Anal."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1016\/j.camwa.2017.10.017","article-title":"Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions","volume":"75","author":"Ern","year":"2018","journal-title":"Comput. Math. Appl."},{"key":"ref_16","unstructured":"Aram, M.G. (2021). Antenna Design, Radiobiological Modelling, and Non-Invasive Monitoring for Microwave Hyperthermia. [Licentiate Thesis, Chalmers University of Technology]. Available online: https:\/\/research.chalmers.se\/en\/publication\/?id=528711."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"998","DOI":"10.1137\/20M1315798","article-title":"Carleman-based reconstruction algorithm for the waves","volume":"59","author":"Baudouin","year":"2020","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1109\/TBME.2014.2307072","article-title":"Real-time Microwave Imaging of Differential Temperature for Thermal Therapy Monitoring","volume":"61","author":"Haynes","year":"2014","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1080\/17415977.2020.1802447","article-title":"An inverse problem of a simultaneous reconstruction of the dielectric constant and conductivity from experimental backscattering data","volume":"29","author":"Khoa","year":"2021","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ito, K., and Jin, B. (2015). Inverse Problems: Tikhonov Theory and Algorithms, World Scientific.","DOI":"10.1142\/9120"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Groetsch, C.W. (1993). Inverse Problems in the Mathematical Sciences, Friedr. Vieweg & Sohn Verlagsgesellschaft.","DOI":"10.1007\/978-3-322-99202-4"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., and Yagola, A.G. (1995). Numerical Methods for the Solution of Ill-Posed Problems, Kluwer.","DOI":"10.1007\/978-94-015-8480-7"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2637","DOI":"10.1088\/0031-9155\/52\/10\/001","article-title":"A large-scale study of the ultrawideband microwave dielectric properties of normal breast tissue obtained from reduction surgeries","volume":"52","author":"Lazebnik","year":"2007","journal-title":"Phys. Med. Biol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6093","DOI":"10.1088\/0031-9155\/52\/20\/002","article-title":"A large-scale study of the ultrawideband microwave dielectric properties of normal, benign, and malignant breast tissues obtained from cancer surgeries","volume":"52","author":"Lazebnik","year":"2007","journal-title":"Phys. Med. Biol."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Taflove, A. (1995). Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House.","DOI":"10.1049\/cp:19950258"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/0021-9991(90)90181-Y","article-title":"A mixed finite element formulation for Maxwell\u2019s equations in the time domain","volume":"88","author":"Lee","year":"1990","journal-title":"J. Comput. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Jiang, B. (1998). The Least-Squares Finite Element Method. Theory and Applications in Computational Fluid Dynamics and Electromagnetics, Springer.","DOI":"10.1007\/978-3-662-03740-9"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1006\/jcph.1996.0082","article-title":"The origin of spurious solutions in computational electromagnetics","volume":"125","author":"Jiang","year":"1996","journal-title":"J. Comput. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF01396415","article-title":"Mixed finite elements in R3","volume":"35","year":"1980","journal-title":"Numer. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/BF01389668","article-title":"A new family of mixed finite elements in R3","volume":"50","year":"1986","journal-title":"Numer. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/s10915-004-4152-6","article-title":"High-order RKDG methods for computational electromagnetics","volume":"22","author":"Chen","year":"2005","journal-title":"J. Sci. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1016\/j.jcp.2003.09.007","article-title":"Locally divergence-free discontinuous Galerkin methods for the Maxwell equations","volume":"194","author":"Cockburn","year":"2004","journal-title":"J. Comput. Phys."},{"key":"ref_33","first-page":"411","article-title":"TVB Runge\u2013Kutta local projection discontinuous Galerkin method for conservation laws II: General framework","volume":"52","author":"Cockburn","year":"1989","journal-title":"Math. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.1051\/m2an:2005049","article-title":"Convergence and stability of a discontinuous Galerkin time-domain methods for the 3D heterogeneous Maxwell equations on unstructured meshes","volume":"39","author":"Fezoui","year":"2005","journal-title":"Mod\u00e9l. Math. Anal. Num\u00e9r."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.cam.2006.01.044","article-title":"Interior penalty discontinuous Galerkin method for Maxwell\u2019s equations: Energy norm error estimates","volume":"204","author":"Grote","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Monk, P.B. (2003). Finite Element Methods for Maxwell\u2019s Equations, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198508885.001.0001"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Cohen, G.C. (2002). Higher Order Numerical Methods for Transient Wave Equations, Springer.","DOI":"10.1007\/978-3-662-04823-8"},{"key":"ref_38","first-page":"1287","article-title":"Finite elements and mass lumping for Maxwell\u2019s equations: The 2D case","volume":"11","author":"Elmkies","year":"1997","journal-title":"Comptes Rendus L\u2019Acad. Sci. Ser. I Math."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Joly, P. (2003). Variational Methods for Time-Dependent Wave Propagation Problems, Lecture Notes in Computational Science and Engineering, Springer.","DOI":"10.1007\/978-3-642-55483-4_6"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1137\/0915055","article-title":"A dispersion analysis of finite element methods for Maxwell\u2019s equations","volume":"15","author":"Monk","year":"1994","journal-title":"SIAM J.Sci.Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1109\/22.75280","article-title":"Elimination of vector parasites in Finite Element Maxwell solutions","volume":"39","author":"Paulsen","year":"1991","journal-title":"IEEE Trans. Microw. Theory Technol."},{"key":"ref_42","unstructured":"Jin, J. (1993). The Finite Element Method in Electromagnetics, Wiley."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1006\/jcph.2000.6507","article-title":"Divergence correction techniques for Maxwell Solvers based on a hyperbolic model","volume":"161","author":"Munz","year":"2000","journal-title":"J. Comput. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Beilina, L., and Ruas, V. (2019, January 1\u20135). Convergence of Explicit P1 Finite-Element Solutions to Maxwell\u2019s Equations. Proceedings of the Mathematical and Numerical Approaches for Multi-Wave Inverse Problems, Marseille, France.","DOI":"10.1007\/978-3-030-48634-1_7"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Brenner, S.C., and Scott, L.R. (1994). The Mathematical Theory of Finite Element Methods, Springer.","DOI":"10.1007\/978-1-4757-4338-8"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"105","DOI":"10.3934\/krm.2019005","article-title":"On hp-Streamline Diffusion and Nitsche schemes for the Relativistic Vlasov-Maxwell System","volume":"12","author":"Asadzadeh","year":"2019","journal-title":"Kinet. Relat. Model."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Bergh, J., and L\u00f6fstr\u00f6m, J. (1975). Interpolation Spaces, Springer.","DOI":"10.1007\/978-3-642-66451-9"},{"key":"ref_48","unstructured":"(2022, August 20). Software Package WavES. Available online: http:\/\/www.waves24.com\/."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/337\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:34:55Z","timestamp":1760142895000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/337"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,20]]},"references-count":48,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["a15100337"],"URL":"https:\/\/doi.org\/10.3390\/a15100337","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,20]]}}}