Document Zbl 1373.82101

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Comput. Phys.</a> <a href="/?q=in%3A371445" title="Articles in this Issue">327, 140-167 (2016)</a>. </div> <div class="abstract">Summary: This work concerns the numerical solution of a coupled system of self-consistent reaction-drift-diffusion-Poisson equations that describes the macroscopic dynamics of charge transport in photoelectrochemical (PEC) solar cells with reactive semiconductor and electrolyte interfaces. We present three numerical algorithms, mainly based on a mixed finite element and a local discontinuous Galerkin method for spatial discretization, with carefully chosen numerical fluxes, and implicit-explicit time stepping techniques, for solving the time-dependent nonlinear systems of partial differential equations. We perform computational simulations under various model parameters to demonstrate the performance of the proposed numerical algorithms as well as the impact of these parameters on the solution to the model.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6798174">Cited in <strong>3</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A82D37" title="MSC2020">82D37</a> </td> <td class="space"> Statistical mechanics of semiconductors </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M60" title="MSC2020">65M60</a> </td> <td class="space"> Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aphotoelectrochemical+cell+modeling">photoelectrochemical cell modeling</a>; <a href="/?q=ut%3Asolar+cell+simulation">solar cell simulation</a>; <a href="/?q=ut%3Adrift-diffusion-Poisson+systems">drift-diffusion-Poisson systems</a>; <a href="/?q=ut%3Areactive+interfaces">reactive interfaces</a>; <a href="/?q=ut%3Ainterfacial+charge+transfer">interfacial charge transfer</a>; <a href="/?q=ut%3Amixed+finite+element+method">mixed finite element method</a>; <a href="/?q=ut%3Alocal+discontinuous+Galerkin+method">local discontinuous Galerkin method</a>; <a href="/?q=ut%3Adomain+decomposition">domain decomposition</a>; <a href="/?q=ut%3Aimplicit-explicit+time+stepping">implicit-explicit time stepping</a></div> <div class="software"> <h3>Software:</h3><a href="/software/989">UMFPACK</a>; <a href="/software/930">SuperLU</a>; <a href="/software/3516">deal.ii</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1373.82101" data-ciurl="/ci/06798174" data-biburl="/bibtex/06798174.bib" data-amsurl="/amsrefs/06798174.bib" data-xmlurl="/xml/06798174.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06798174.pdf" title="Zbl 1373.82101 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jcp.2016.08.026" aria-label="DOI for “Numerical algorithms based on Galerkin methods for the modeling of reactive interfaces in photoelectrochemical (PEC) solar cells”" title="10.1016/j.jcp.2016.08.026">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1604.07855"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Anile, A. M.; Allegretto, W.; Ringhofer, C., Mathematical Problems in Semiconductor Physics, Lecture Notes in Mathematics (2003), Springer-Verlag: Springer-Verlag Berlin · <a href="/1024.00023" class="nowrap">Zbl 1024.00023</a></td> </tr><tr> <td>[2]</td> <td class="space">Arbogast, T.; Juntunen, M.; Pool, J.; Wheeler, M. F., A discontinuous Galerkin method for two-phase flow in a porous medium enforcing h(div) velocity and continuous capillary pressure, Comput. Geosci., 17, 1055-1078 (2013) · <a href="/1393.76059" class="nowrap">Zbl 1393.76059</a></td> </tr><tr> <td>[3]</td> <td class="space">Aziz, J.; Genov, R.; Bardakjian, B.; Derchansky, M.; Carlen, P., Brain-silicon interface for high-resolution in vitro neural recording, IEEE Trans. Biomed. Circuits Syst., 1, 56-62 (2007)</td> </tr><tr> <td>[4]</td> <td class="space">Babuska, I., The finite element method with Lagrange multipliers, Numer. Math., 20, 179-192 (1973) · <a href="/0258.65108" class="nowrap">Zbl 0258.65108</a></td> </tr><tr> <td>[5]</td> <td class="space">Bangerth, W.; Hartmann, R.; Kanschat, G., deal.II - a general purpose object oriented finite element library, ACM Trans. Math. Softw., 33, 24/1-24/27 (2007) · <a href="/1365.65248" class="nowrap">Zbl 1365.65248</a></td> </tr><tr> <td>[7]</td> <td class="space">Bank, R. E.; Rose, D. J.; Fichtner, W., Numerical methods for semiconductor device simulation, SIAM J. Sci. Stat. Comput., 4, 416-435 (1983) · <a href="/0521.65086" class="nowrap">Zbl 0521.65086</a></td> </tr><tr> <td>[8]</td> <td class="space">Bard, A. J.; Faulkner, L. R., Electrochemical Methods: Fundamentals and Applications (2000), Wiley: Wiley New York</td> </tr><tr> <td>[9]</td> <td class="space">Baronas, R.; Ivanauskas, F.; Kulis, I.; Kulys, J., Mathematical Modeling of Biosensors: An Introduction for Chemists and Mathematicians (2009), Springer</td> </tr><tr> <td>[10]</td> <td class="space">Baumgartner, S.; Heitzinger, C., A one-level FETI method for the drift-diffusion-Poisson system with discontinuities at an interface, J. Comput. Phys., 243, 74-86 (2013) · <a href="/1349.65610" class="nowrap">Zbl 1349.65610</a></td> </tr><tr> <td>[11]</td> <td class="space">Bazant, M. Z.; Thornton, K.; Ajdari, A., Diffuse-charge dynamics in electrochemical systems, Phys. Rev. E, 70, Article 021506 pp. (2004)</td> </tr><tr> <td>[12]</td> <td class="space">Bell, J.; Farrell, T.; Penny, M.; Will, G., A mathematical model of the semiconductor-electrolyte interface in dye sensitised solar cells, (May, R.; Blyth, W. F., Proceedings of EMAC 2003 (2003), Australian Mathematical Society), 193-198</td> </tr><tr> <td>[13]</td> <td class="space">Ben Abdallah, N.; Cáceres, M. J.; Carrillo, J. A.; Vecil, F., A deterministic solver for a hybrid quantum-classical transport model in nanoMOSFETs, J. Comput. Phys., 228, 6553-6571 (2009) · <a href="/1175.82072" class="nowrap">Zbl 1175.82072</a></td> </tr><tr> <td>[14]</td> <td class="space">Biesheuvel, P. M.; Fu, Y.; Bazant, M. Z., Electrochemistry and capacitive charging of porous electrodes in asymmetric multicomponent electrolytes, Russian J. Electrochem., 48, 580-592 (2012)</td> </tr><tr> <td>[15]</td> <td class="space">Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications (2013), Springer · <a href="/1277.65092" class="nowrap">Zbl 1277.65092</a></td> </tr><tr> <td>[16]</td> <td class="space">Bosisio, F.; Micheletti, S.; Sacco, R., A discretization scheme for an extended drift-diffusion model including trap-assisted phenomena, J. Comput. Phys., 159, 197-212 (2000) · <a href="/1073.82624" class="nowrap">Zbl 1073.82624</a></td> </tr><tr> <td>[17]</td> <td class="space">Brezzi, F., On the existence uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, ESAIM: Mathematical Modelling and Numerical Analysis, 8, 129-151 (1974) · <a href="/0338.90047" class="nowrap">Zbl 0338.90047</a></td> </tr><tr> <td>[18]</td> <td class="space">Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer · <a href="/0788.73002" class="nowrap">Zbl 0788.73002</a></td> </tr><tr> <td>[19]</td> <td class="space">Brezzi, F.; Marini, L. D.; Pietra, P., Two-dimensional exponential fitting and applications to drift-diffusion models, SIAM J. Numer. Anal., 26, 1342-1355 (1989) · <a href="/0686.65088" class="nowrap">Zbl 0686.65088</a></td> </tr><tr> <td>[20]</td> <td class="space">Brinkman, D.; Fellner, K.; Markowich, P. A.; Wolfram, M. T., A drift-diffusion reaction model for excitonic photovoltaic bilayers: asymptotic analysis and a 2D HDG finite element scheme, Math. Models Methods Appl. Sci., 23, 839-872 (2013) · <a href="/1264.35118" class="nowrap">Zbl 1264.35118</a></td> </tr><tr> <td>[21]</td> <td class="space">Burger, M.; Pinnau, R., A globally convergent Gummel map for optimal dopant profiling, Math. Models Methods Appl. Sci., 19, 769-786 (2009) · <a href="/1173.35706" class="nowrap">Zbl 1173.35706</a></td> </tr><tr> <td>[22]</td> <td class="space">Cassano, G.; de Falco, C.; Giulianetti, C.; Sacco, R., Numerical simulation of tunneling effects in nanoscale semiconductor devices using quantum corrected drift-diffusion models, Comput. Methods Appl. Mech. Eng., 195, 2193-2208 (2006) · <a href="/1214.82123" class="nowrap">Zbl 1214.82123</a></td> </tr><tr> <td>[23]</td> <td class="space">Cauley, S.; Balakrishnan, V.; Klimeck, G.; Koh, C.-K., A two-dimensional domain decomposition technique for the simulation of quantum-scale devices, J. Comput. Phys., 231, 1293-1313 (2012) · <a href="/1242.65263" class="nowrap">Zbl 1242.65263</a></td> </tr><tr> <td>[24]</td> <td class="space">Chen, S.; E, W.; Liu, Y.; Shu, C.-W., A discontinuous Galerkin implementation of a domain decomposition method for kinetic-hydrodynamic coupling multiscale problems in gas dynamics and device simulations, J. Comput. Phys., 225, 1314-1330 (2007) · <a href="/1343.76041" class="nowrap">Zbl 1343.76041</a></td> </tr><tr> <td>[25]</td> <td class="space">Chen, Z., Finite element analysis of the one-dimensional full drift-diffusion semiconductor model, SIAM J. Numer. Anal., 32, 455-483 (1995) · <a href="/0822.65115" class="nowrap">Zbl 0822.65115</a></td> </tr><tr> <td>[26]</td> <td class="space">Chen, Z.; Cockburn, B., Analysis of a finite element method for the drift-diffusion semiconductor device equations: the multidimensional case, Numer. Math., 71, 1-28 (1995) · <a href="/0830.65116" class="nowrap">Zbl 0830.65116</a></td> </tr><tr> <td>[27]</td> <td class="space">Chidyagwai, P.; Rivière, B., On the solution of the coupled Navier-Stokes and Darcy equations, Comput. Methods Appl. Mech. Eng., 198, 3806-3820 (2009) · <a href="/1230.76023" class="nowrap">Zbl 1230.76023</a></td> </tr><tr> <td>[28]</td> <td class="space">Coats, K. H., IMPES stability: selection of stable timesteps, SPE J., 8, 181-187 (2003)</td> </tr><tr> <td>[29]</td> <td class="space">Cockburn, B.; Dawson, C., Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions, (Whiteman, J. R., Proceedings of the Conference on the Mathematics of Finite Elements and Applications: MAFELAP X (2000), Elsevier: Elsevier New York), 225-238 · <a href="/0960.65107" class="nowrap">Zbl 0960.65107</a></td> </tr><tr> <td>[30]</td> <td class="space">Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin methods for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · <a href="/0927.65118" class="nowrap">Zbl 0927.65118</a></td> </tr><tr> <td>[31]</td> <td class="space">Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173-261 (2002) · <a href="/1065.76135" class="nowrap">Zbl 1065.76135</a></td> </tr><tr> <td>[32]</td> <td class="space">Davis, T. A., A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw., 30, 165-195 (2004) · <a href="/1072.65036" class="nowrap">Zbl 1072.65036</a></td> </tr><tr> <td>[33]</td> <td class="space">Davis, T. A., Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw., 30, 196-199 (2004) · <a href="/1072.65037" class="nowrap">Zbl 1072.65037</a></td> </tr><tr> <td>[34]</td> <td class="space">Davis, T. A.; Duff, I. S., An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM J. Matrix Anal. Appl., 1, 140-158 (1997) · <a href="/0884.65021" class="nowrap">Zbl 0884.65021</a></td> </tr><tr> <td>[35]</td> <td class="space">Davis, T. A.; Duff, I. S., A combined unifrontal/multifrontal method for unsymmetric sparse matrices, ACM Trans. Math. Softw., 25, 1-19 (1999) · <a href="/0962.65027" class="nowrap">Zbl 0962.65027</a></td> </tr><tr> <td>[36]</td> <td class="space">Dawson, C.; Sun, S.; Wheeler, M. F., Compatible algorithms for coupled flow and transport, Comput. Methods Appl. Mech. Eng., 193, 2565-2580 (2004) · <a href="/1067.76565" class="nowrap">Zbl 1067.76565</a></td> </tr><tr> <td>[37]</td> <td class="space">de Falco, C.; Jerome, J. W.; Sacco, R., Quantum-corrected drift-diffusion models: solution fixed point map and finite element approximation, J. Comput. Phys., 228, 1770-1789 (2009) · <a href="/1158.82012" class="nowrap">Zbl 1158.82012</a></td> </tr><tr> <td>[38]</td> <td class="space">Degond, P.; El Ayyadi, A., A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations, J. Comput. Phys., 181, 222-259 (2002) · <a href="/1008.82033" class="nowrap">Zbl 1008.82033</a></td> </tr><tr> <td>[39]</td> <td class="space">Degond, P.; Gallego, S.; Méhats, F., An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes, J. Comput. Phys., 221, 226-249 (2007) · <a href="/1107.82401" class="nowrap">Zbl 1107.82401</a></td> </tr><tr> <td>[40]</td> <td class="space">Demmel, J. W.; Eisenstat, S. C.; Gilbert, J. R.; Li, X. S.; Liu, J. W.H., A supernodal approach to sparse partial pivoting, SIAM J. Matrix Anal. Appl., 20, 720-755 (1999) · <a href="/0931.65022" class="nowrap">Zbl 0931.65022</a></td> </tr><tr> <td>[41]</td> <td class="space">Douglas, J.; Gamba, I. M.; Squeff, M. C.J., Simulation of the transient behavior of a one dimensional semiconductor device, Mat. Apl. Comput., 5, 103-122 (1986) · <a href="/0625.65123" class="nowrap">Zbl 0625.65123</a></td> </tr><tr> <td>[42]</td> <td class="space">Douglas, J.; Russell, T. J., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19, 871-885 (1982) · <a href="/0492.65051" class="nowrap">Zbl 0492.65051</a></td> </tr><tr> <td>[43]</td> <td class="space">The Eigen Library (2010)</td> </tr><tr> <td>[44]</td> <td class="space">Eisenberg, B., Ionic channels: natural nanotubes described by the drift diffusion equations, Superlattices Microstruct., 27, 545-549 (2000)</td> </tr><tr> <td>[45]</td> <td class="space">Fajardo, A. M.; Lewis, N. S., Free-energy dependence of electron-transfer rate constants at Si/liquid interfaces, J. Phys. Chem. B, 101, 11136-11151 (1997)</td> </tr><tr> <td>[46]</td> <td class="space">Fawcett, W. R., Liquids, Solutions, and Interfaces: From Classical Macroscopic Descriptions to Modern Microscopic Details (2004), Oxford University Press: Oxford University Press New York</td> </tr><tr> <td>[47]</td> <td class="space">Foley, J. M.; Price, M. J.; Feldblyum, J. I.; Maldonado, S., Analysis of the operation of thin nanowire photoelectrodes for solar energy conversion, Energy Environ. Sci., 5, 5203-5220 (2012)</td> </tr><tr> <td>[48]</td> <td class="space">Galler, M., Multigroup Equations for the Description of the Particle Transport in Semiconductors (2005), World Scientific · <a href="/1099.82023" class="nowrap">Zbl 1099.82023</a></td> </tr><tr> <td>[49]</td> <td class="space">Gao, Y.; Georgievskii, Y.; Marcus, R., On the theory of electron transfer reactions at semiconductor electrode/liquid interfaces, J. Phys. Chem., 112, 3358-3369 (2000)</td> </tr><tr> <td>[50]</td> <td class="space">Gao, Y.; Marcus, R., On the theory of electron transfer reactions at semiconductor electrode/liquid interfaces. II. A free electron model, J. Phys. Chem., 112, 6351-6360 (2000)</td> </tr><tr> <td>[51]</td> <td class="space">Gamba, I. M.; Squeff, M. C.J., Simulation of the transient behavior of a one dimensional semiconductor device II, SIAM J. Numer. Anal., 2, 539-552 (1989) · <a href="/0669.65086" class="nowrap">Zbl 0669.65086</a></td> </tr><tr> <td>[52]</td> <td class="space">Gatica, G. N., A Simple Introduction to the Mixed Finite Element Method (2014), Springer · <a href="/1293.65152" class="nowrap">Zbl 1293.65152</a></td> </tr><tr> <td>[53]</td> <td class="space">Girault, V.; Sun, S.; Wheeler, M. F.; Yotov, I., Coupling discontinuous Galerkin and mixed finite element discretizations using mortar finite elements, SIAM J. Numer. Anal., 46, 949-979 (2008) · <a href="/1165.65075" class="nowrap">Zbl 1165.65075</a></td> </tr><tr> <td>[54]</td> <td class="space">Glowinski, R.; Wheeler, M. F., Domain decomposition and mixed finite element methods for elliptic problems, (Glowinski, R.; Golub, G. H.; Meurant, G. A.; Périaux, J., Domain Decomposition Methods for Partial Differential Equations (1988), SIAM: SIAM Philadelphia), 144-172 · <a href="/0661.65105" class="nowrap">Zbl 0661.65105</a></td> </tr><tr> <td>[55]</td> <td class="space">(Grasser, T., Advanced Device Modeling and Simulation (2003), World Scientific)</td> </tr><tr> <td>[56]</td> <td class="space">Grätzel, M., Photoelectrochemical cells, Nature, 414, 338-344 (2001)</td> </tr><tr> <td>[57]</td> <td class="space">Grätzel, M., Dye-sensitized solar cells, J. Photochem. Photobiol. C, 4, 145-153 (2003)</td> </tr><tr> <td>[58]</td> <td class="space">Green, M. A.; Keevers, M. J., Optical properties of intrinsic silicon at 300 k, Prog. Photovolt., 3, 189-192 (1995)</td> </tr><tr> <td>[59]</td> <td class="space">The GRVY Library (2008)</td> </tr><tr> <td>[60]</td> <td class="space">He, Y.; Gamba, I. M.; Lee, H. C.; Ren, K., On the modeling and simulation of reaction-transfer dynamics in semiconductor-electrolyte solar cells, SIAM J. Appl. Math., 75, 2515-2539 (2015) · <a href="/1351.82113" class="nowrap">Zbl 1351.82113</a></td> </tr><tr> <td>[61]</td> <td class="space">Horng, T. L.; Lin, T. C.; Liu, C.; Eisenberg, B., PNP equations with steric effects: a model of ion flow through channels, J. Phys. Chem. B, 116, 11422-11441 (2012)</td> </tr><tr> <td>[62]</td> <td class="space">Jerome, J. W., Analysis of Charge Transport: A Mathematical Study of Semiconductor Devices (1996), Springer-Verlag: Springer-Verlag Berlin</td> </tr><tr> <td>[63]</td> <td class="space">Jerome, J. W.; Kerkhoven, T., A finite element approximation theory for the drift-diffusion semiconductor model, SIAM J. Numer. Anal., 28, 403-422 (1991) · <a href="/0725.65120" class="nowrap">Zbl 0725.65120</a></td> </tr><tr> <td>[64]</td> <td class="space">Jin, S.; Pareschi, L., Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes, J. Comput. Phys., 161, 312-330 (2000) · <a href="/1156.82408" class="nowrap">Zbl 1156.82408</a></td> </tr><tr> <td>[65]</td> <td class="space">Jüngel, A., Transport Equations for Semiconductors (2009), Springer: Springer Berlin</td> </tr><tr> <td>[66]</td> <td class="space">Kamat, P. V.; Tvrdy, K.; Baker, D. R.; Radich, J. G., Beyond photovoltaics: semiconductor nanoarchitectures for liquid-junction solar cells, Chem. Rev., 110, 6664-6688 (2010)</td> </tr><tr> <td>[67]</td> <td class="space">Kayes, B. M.; Atwater, H. A.; Lewis, N. S., Comparison of the device physics principles of planar and radial p-n junction nanorod solar cells, J. Appl. Phys., 97, Article 114302 pp. (2005)</td> </tr><tr> <td>[68]</td> <td class="space">Kilic, M. S.; Bazant, M. Z.; Ajdari, A., Steric effects in the dynamics of electrolytes at large applied voltages: II. Modified Nernst-Planck equations, Phys. Rev. E, 75, Article 021503 pp. (2007)</td> </tr><tr> <td>[69]</td> <td class="space">Kircher, R.; Bergner, W., Three-Dimensional Simulation of Semiconductor Devices (1991), Birkhäuser Verlag: Birkhäuser Verlag Basel</td> </tr><tr> <td>[70]</td> <td class="space">Kulikovsky, A. A., A more accurate Scharfetter-Gummel algorithm of electron transport for semiconductor and gas discharge simulation, J. Comput. Phys., 119, 149-155 (1995) · <a href="/0828.76055" class="nowrap">Zbl 0828.76055</a></td> </tr><tr> <td>[71]</td> <td class="space">Laser, D.; Bard, A. J., Semiconductor electrodes: VII. Digital simulation of charge injection and the establishment of the space charge region in the absence of surface states, J. Electrochem. Soc., 123, 1828-1832 (1976)</td> </tr><tr> <td>[72]</td> <td class="space">Laser, D.; Bard, A. J., Semiconductor electrodes: VIII. Digital simulation of open-circuit photopotentials, J. Electrochem. Soc., 123, 1833-1837 (1976)</td> </tr><tr> <td>[73]</td> <td class="space">Lee, P.; Griffith, B. E.; Peskin, C. S., The immersed boundary method for advection-electrodiffusion with implicit timestepping and local mesh refinement, J. Comput. Phys., 229, 5208-5227 (2010) · <a href="/1346.76109" class="nowrap">Zbl 1346.76109</a></td> </tr><tr> <td>[74]</td> <td class="space">Lewis, N. S., Mechanistic studies of light-induced charge separation at semiconductor/liquid interfaces, Acc. Chem. Res., 23, 176-183 (1990)</td> </tr><tr> <td>[75]</td> <td class="space">Lewis, N. S., Progress in understanding electron-transfer reactions at semiconductor/liquid interfaces, J. Phys. Chem. B, 102, 4843-4855 (1998)</td> </tr><tr> <td>[76]</td> <td class="space">Lions, P.-L., On the Schwarz alternating method. I, (Domain Decomposition Methods (1988), SIAM: SIAM Philadelphia), 1-42 · <a href="/0658.65090" class="nowrap">Zbl 0658.65090</a></td> </tr><tr> <td>[77]</td> <td class="space">Lions, P.-L., On the Schwarz alternating method. II. Stochastic interpretation and order properties, (Domain Decomposition Methods (1989), SIAM: SIAM Philadelphia), 47-70 · <a href="/0681.65072" class="nowrap">Zbl 0681.65072</a></td> </tr><tr> <td>[78]</td> <td class="space">Lions, P.-L., On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, (Domain Decomposition Methods (1990), SIAM: SIAM Philadelphia), 202-223 · <a href="/0704.65090" class="nowrap">Zbl 0704.65090</a></td> </tr><tr> <td>[79]</td> <td class="space">Liu, W., One-dimensional steady-state Poisson-Nernst-Planck systems for ion channels with multiple ion species, J. Differ. Equ., 246, 428-451 (2009) · <a href="/1165.34032" class="nowrap">Zbl 1165.34032</a></td> </tr><tr> <td>[80]</td> <td class="space">Lu, B.; Holst, M. J.; McCammon, J. A.; Zhou, Y. C., Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes I: finite element solutions, J. Comput. Phys., 229, 6979-6994 (2010) · <a href="/1195.92004" class="nowrap">Zbl 1195.92004</a></td> </tr><tr> <td>[81]</td> <td class="space">Lu, B.; Zhou, Y. C., Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates, Biophys. J., 100, 2475-2485 (2011)</td> </tr><tr> <td>[82]</td> <td class="space">Mafé, S.; Pellicer, J.; Aguilella, V. M., A numerical approach to ionic transport through charged membranes, J. Comput. Phys., 75, 1-14 (1988) · <a href="/0637.65129" class="nowrap">Zbl 0637.65129</a></td> </tr><tr> <td>[83]</td> <td class="space">Markowich, P. A.; Ringhofer, C. A.; Schmeiser, C., Semiconductor Equations (1990), Springer: Springer New York · <a href="/0765.35001" class="nowrap">Zbl 0765.35001</a></td> </tr><tr> <td>[84]</td> <td class="space">Mathur, S. R.; Murthy, J. Y., A multigrid method for the Poisson-Nernst-Planck equations, Int. J. Heat Mass Transf., 52, 4031-4039 (2009) · <a href="/1167.76343" class="nowrap">Zbl 1167.76343</a></td> </tr><tr> <td>[85]</td> <td class="space">Mauro, A. J.; Sigurdsson, J. K.; Shrake, J.; Atzberger, P. J.; Isaacson, S. A., A first-passage kinetic Monte Carlo method for reaction-drift-diffusion processes, J. Comput. Phys., 259, 536-567 (2014) · <a href="/1349.82047" class="nowrap">Zbl 1349.82047</a></td> </tr><tr> <td>[86]</td> <td class="space">Memming, R., Semiconductor Electrochemistry (2001), Wiley-VCH: Wiley-VCH New York</td> </tr><tr> <td>[87]</td> <td class="space">Micheletti, S.; Quarteroni, A.; Sacco, R., Current-voltage characteristics simulation of semiconductor devices using domain decomposition, J. Comput. Phys., 119, 46-61 (1995) · <a href="/0822.65116" class="nowrap">Zbl 0822.65116</a></td> </tr><tr> <td>[88]</td> <td class="space">Moyles, I.; Wetton, B., A numerical framework for singular limits of a class of reaction diffusion problems, J. Comput. Phys., 300, 308-326 (2015) · <a href="/1349.65325" class="nowrap">Zbl 1349.65325</a></td> </tr><tr> <td>[89]</td> <td class="space">Nelson, J., The Physics of Solar Cells (2003), Imperial College Press: Imperial College Press London</td> </tr><tr> <td>[90]</td> <td class="space">Newman, J.; Thomas-Alyea, K. E., Electrochemical Systems (2004), Wiley-Interscience: Wiley-Interscience New York</td> </tr><tr> <td>[91]</td> <td class="space">Nozik, A. J.; Memming, R., Physical chemistry of semiconductor-liquid interfaces, J. Phys. Chem., 100, 13061-13078 (1996)</td> </tr><tr> <td>[92]</td> <td class="space">Park, J. H.; Jerome, J. W., Qualitative properties of steady-state Poisson-Nernst-Planck systems: mathematical study, SIAM J. Appl. Math., 57, 609-630 (1997) · <a href="/0874.34017" class="nowrap">Zbl 0874.34017</a></td> </tr><tr> <td>[93]</td> <td class="space">Peitz, I.; Fromherz, P., Electrical interfacing of neurotransmitter receptor and field effect transistor, Eur. Phys. J. E, 30, 223-231 (2009)</td> </tr><tr> <td>[94]</td> <td class="space">Penny, M.; Farrell, T.; Please, C., A mathematical model for interfacial charge transfer at the semiconductor-dye-electrolyte interface of a dye-sensitised solar cell, Sol. Energy Mater. Sol. Cells, 92, 11-23 (2008)</td> </tr><tr> <td>[95]</td> <td class="space">Penny, M. A.; Farrell, T. W.; Will, G. D.; Bell, J. M., Modelling interfacial charge transfer in dye-sensitised solar cells, J. Photochem. Photobiol. A, 164, 41-46 (2004)</td> </tr><tr> <td>[96]</td> <td class="space">Pomykal, K. E.; Lewis, N. S., Measurement of interfacial charge-transfer rate constants at n-type InP/CHOH junctions, J. Phys. Chem. B, 101, 2476-2484 (1997)</td> </tr><tr> <td>[97]</td> <td class="space">Quarteroni, A.; Valli, A., Domain Decomposition Methods for Partial Differential Equations (1999), Oxford Science Publications: Oxford Science Publications Oxford · <a href="/0931.65118" class="nowrap">Zbl 0931.65118</a></td> </tr><tr> <td>[98]</td> <td class="space">Raviart, P.; Thomas, J.-M., A mixed finite element method for second-order elliptic problems, (Galligani, I.; Magenes, E., Mathematical Aspects of Finite Element Methods (1977), Springer: Springer Berlin), 292-315 · <a href="/0362.65089" class="nowrap">Zbl 0362.65089</a></td> </tr><tr> <td>[99]</td> <td class="space">Schroeder, D., Modelling of Interface Carrier Transport for Device Simulation (1994), Springer-Verlag: Springer-Verlag Vienna · <a href="/0810.68036" class="nowrap">Zbl 0810.68036</a></td> </tr><tr> <td>[100]</td> <td class="space">Schuss, Z.; Nadler, B.; Eisenberg, R. S., Derivation of PNP equations in bath and channel from a molecular model, Phys. Rev. E, 64, Article 036116 pp. (2001)</td> </tr><tr> <td>[101]</td> <td class="space">Selberherr, S., Analysis and Simulation of Semiconductor Devices (2001), Springer: Springer Vienna</td> </tr><tr> <td>[102]</td> <td class="space">Sheldon, J. W.; Cardwell, W. T., One-dimensional, incompressible, noncapillary, two-phase fluid flow in a porous medium, Trans. SPE AIME, 216, 290-296 (1959)</td> </tr><tr> <td>[103]</td> <td class="space">Shockley, W.; Read, W. T., Statistics of recombination of holes and electrons, Phys. Rev., 87, 835-842 (1952) · <a href="/0046.45106" class="nowrap">Zbl 0046.45106</a></td> </tr><tr> <td>[104]</td> <td class="space">Singer, A.; Norbury, J., A Poisson-Nernst-Planck model for biological ion channels—an asymptotic analysis in a three-dimensional narrow fun, SIAM J. Appl. Math., 70, 949-968 (2009) · <a href="/1197.34080" class="nowrap">Zbl 1197.34080</a></td> </tr><tr> <td>[105]</td> <td class="space">Singh, D.; Guo, X.; Murthy, J. Y.; Alexeenko, A.; Fisher, T., Modeling of subcontinuum thermal transport across semiconductor-gas interfaces, J. Appl. Phys., 106, Article 024314 pp. (2009)</td> </tr><tr> <td>[106]</td> <td class="space">Spill, F.; Guerrero, P.; Alarcon, T.; Maini, P. K.; Byrne, H., Hybrid approaches for multiple-species stochastic reaction-diffusion models, J. Comput. Phys., 299, 429-445 (2015) · <a href="/1352.65019" class="nowrap">Zbl 1352.65019</a></td> </tr><tr> <td>[107]</td> <td class="space">Stett, A.; Muller, B.; Fromherz, P., Two-way silicon-neuron interface by electrical induction, Phys. Rev. E, 55, 1779-1782 (1997)</td> </tr><tr> <td>[108]</td> <td class="space">Stone, H.; Gardner, A. O., Analysis of gas-cap or dissolved-gas reservoirs, Trans. SPE AIME, 222, 92-104 (1961)</td> </tr><tr> <td>[109]</td> <td class="space">Stracquadanio, G.; Romano, V.; Nicosia, G., Semiconductor device design using the BiMADS algorithm, J. Comput. Phys., 242, 304-320 (2013) · <a href="/1302.78018" class="nowrap">Zbl 1302.78018</a></td> </tr><tr> <td>[110]</td> <td class="space">Sze, S. M., Physics of Semiconductor Devices (1981), John Wiley & Sons</td> </tr><tr> <td>[111]</td> <td class="space">Thangaraj, D.; Nathan, A., A rotated monotone difference scheme for the two-dimensional anisotropic drift-diffusion equation, J. Comput. Phys., 145, 445-461 (1998) · <a href="/0913.65089" class="nowrap">Zbl 0913.65089</a></td> </tr><tr> <td>[112]</td> <td class="space">Toselli, A.; Widlund, O. B., Domain Decomposition Methods - Algorithms and Theory (2005), Springer-Verlag · <a href="/1069.65138" class="nowrap">Zbl 1069.65138</a></td> </tr><tr> <td>[113]</td> <td class="space">Vassilev, D.; Yotov, I., Coupling Stokes-Darcy flow with transport, SIAM J. Sci. Comput., 31, 3661-3684 (1998) · <a href="/1391.76757" class="nowrap">Zbl 1391.76757</a></td> </tr><tr> <td>[114]</td> <td class="space">Wang, H.; Shu, C.-W.; Zhang, Q., Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems, SIAM J. Numer. Anal., 53, 206-227 (2015) · <a href="/1327.65179" class="nowrap">Zbl 1327.65179</a></td> </tr><tr> <td>[115]</td> <td class="space">Xu, Y.; Shu, C.-W., Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys., 7, 1-46 (2010) · <a href="/1364.65205" class="nowrap">Zbl 1364.65205</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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