# Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations

Molecular Based Mathematical Biology (2013)

- Volume: 1, page 26-41
- ISSN: 2299-3266

## Access Full Article

top## Abstract

top## How to cite

topM. R. Swager, and Y. C. Zhou. "Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations." Molecular Based Mathematical Biology 1 (2013): 26-41. <http://eudml.org/doc/267279>.

@article{M2013,

abstract = {A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-to-one correspondence between the continuous piecewise polynomial space of degree k + 1 and the divergencefree vector space of degree k, one can construct high-order two-dimensional exponentially fitted basis functions that are strictly interpolative at a selected node set but are discontinuous on edges in general, spanning nonconforming finite element spaces. First order convergence was proved for the methods constructed from divergence-free Raviart-Thomas space RT 00 at two different node sets.},

author = {M. R. Swager, Y. C. Zhou},

journal = {Molecular Based Mathematical Biology},

keywords = {Drift-diffusion equations; Exponential fitting; Multidimensional; Divergence-free basis functions; High order methods; drift-diffusion equations; exponential fitting; multi-dimensional drift-diffusion equations; divergence-free basis functions; high order methods},

language = {eng},

pages = {26-41},

title = {Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations},

url = {http://eudml.org/doc/267279},

volume = {1},

year = {2013},

}

TY - JOUR

AU - M. R. Swager

AU - Y. C. Zhou

TI - Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations

JO - Molecular Based Mathematical Biology

PY - 2013

VL - 1

SP - 26

EP - 41

AB - A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-to-one correspondence between the continuous piecewise polynomial space of degree k + 1 and the divergencefree vector space of degree k, one can construct high-order two-dimensional exponentially fitted basis functions that are strictly interpolative at a selected node set but are discontinuous on edges in general, spanning nonconforming finite element spaces. First order convergence was proved for the methods constructed from divergence-free Raviart-Thomas space RT 00 at two different node sets.

LA - eng

KW - Drift-diffusion equations; Exponential fitting; Multidimensional; Divergence-free basis functions; High order methods; drift-diffusion equations; exponential fitting; multi-dimensional drift-diffusion equations; divergence-free basis functions; high order methods

UR - http://eudml.org/doc/267279

ER -

## References

top- D. N. Allen and R. V. Southwell. Relaxation method applied to determine the motion, in two dimensions, of viscous fluid past a fixed cylinder. Q. J. Mech. Appl. Math., 8(2):129-145, 1955.[Crossref]
- C. Amatore, O. V. Klymenko, A. I. Oleinick, and I. Svir. Diffusion with moving boundary on spherical surfaces. Chemphyschem., 10:1593-1602, 2009.[PubMed][WoS][Crossref]
- C. Amatore, A. I. Oleinick, O. V. Klymenko, and I. Svir. Theory of long-range diffusion of proteins on a spherical biological membrane: Application to protein cluster formation and actin-comet tail growth. Chemphyschem., 10:1586-1592, 2009.[WoS][PubMed][Crossref]
- Lutz Angermann and Song Wang. On convergence of the exponentially fitted finite volume method with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation. Computational Methods in Applied Mathematics, 3:493-512, 2003. Zbl1038.65112
- Lutz Angermann and Song Wang. Three-dimensional exponentially fitted conforming tetrahedral finite elements for the semiconductor continuity equations. Applied Numerical Mathematics, 46(1):19 - 43, 2003. Zbl1028.82024
- Douglas N. Arnold, Richard S. Falk, and Ragnar Winther. Multigrid in h(div) and h(curl). Numer. Math., 85:197-218, 2000. Zbl0974.65113
- S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer, 2008. Zbl1135.65042
- F. Brezzi, J. Douglas, and L. Marini. Two families of mixed finite elements for second order elliptic problem. Math. Comp., 47:217-235, 1985. Zbl0599.65072
- F. Brezzi and M. Fortin. Mixed and Hybrid Finite Elements. Springer-Verlag, 1991. Zbl0788.73002
- F. Brezzi, L. D. Marini, and P. Pietra. Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal., 26:1342-1355, 1989.[Crossref] Zbl0686.65088
- Y. J. Peng C. Chainais-Hillairet. Finite volume approximation for degenerate drift-diffusion system in several space dimensions. Mathematical Models and Methods in Applied Sciences, 14(03):461-481, 2004.
- Jehanzeb Hameed Chaudhry, Jeffrey Comer, Aleksei Aksimentiev, and Luke N. Olson. A finite element method for modified Poisson-Nernst-Planck equations to determine ion flow though a nanopore. Preprint.
- P.G. Ciarlet and J.L. Lions, editors. Handbook of Numerical Analysis: Finite Element Methods (Part 1). North- Holland, 1991. Zbl0712.65091
- M. Crouzeix and P.A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numer., 7:33-76, 1973. Zbl0302.65087
- Carlo de Falco, Joseph W. Jerome, and Riccardo Sacco. Quantum-corrected drift-diffusion models: Solution fixed point map and finite element approximation. J. Comput. Phys., 228(5):1770 - 1789, 2009.[WoS] Zbl1158.82012
- E. Gatti, S. Micheletti, and R. Sacco. A new Galerkin framework for the drift-diffusion equation in semiconductors. East-West J. Numer. Math., 6:101-135, 1998. Zbl0915.65128
- Alan F. Hegarty, Eugene O’Riordan, and Martin Stynes. A comparison of uniformly convergent difference schemes for two-dimensional convectionâATdiffusion problems. J. Comput. Phys., 105(1):24 - 32, 1993. Zbl0769.65071
- J. W. Jerome. Analysis of Charge Transport: A Mathematical Study of Semiconductor Devices. Springer, 1996.
- Vladimir Yu. Kiselev, Marcin Leda, Alexey I. Lobanov, Davide Marenduzzo, and Andrew B. Goryachev. Lateral dynamics of charged lipids and peripheral proteins in spatially heterogeneous membranes: Comparison of continuous and monte carlo approaches. J. Chem. Phys., 135:155103, 2011.[WoS]
- R. D. Lazarov, Ilya D. Mishev, P. S. Vassilevski, and L. Finite volume methods for convection-diffusion problems. SIAM J. Numer. Anal., 33:31-55, 1996.[Crossref] Zbl0847.65075
- B. Z. Lu, Y. C. Zhou, Gary A. Huber, Steve D. Bond, Michael J. Holst, and J. A. McCammon. Electrodiffusion: A continuum modeling framework for biomolecular systems with realistic spatiotemporal resolution. J. Chem. Phys., 127:135102, 2007.[WoS][Crossref]
- Benzhuo Lu, M. J. Holst, J. A. McCammon, and Y. C. Zhou. Poisson-Nernst-Planck Equations for simulating biomolecular diffusion-reaction processes I: Finite element solutions. J. Comput. Phys., 229:6979-6994, 2010. Zbl1195.92004
- Benzhuo Lu and Y. C. Zhou. 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.[WoS]
- J. J. H. Miller and S. Wang. A new non-conforming petrov-galerkin finite-element method with triangular elements for a singularly perturbed advection-diffusion problem. IMA J. Numer. Anal., 14(2):257-276, 1994. Zbl0806.65111
- Eugene O’Riordan and Martin Stynes. A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions. Math. Comp., 57:47-62, 1991. Zbl0733.65063
- René Pinnau. Uniform convergence of an exponentially fitted scheme for the quantum drift diffusion model. SIAMJ. Numer. Anal., 42:1648-1668, 2004.[Crossref] Zbl1080.65107
- P.A. Raviart and J.M. Thomas. A mixed finite element method for second order elliptic problems. In I. Galligani and E. Magenes, editors, Lecture notes in Mathematics, Vol. 606. Springer-Verlag, 1977. Zbl0362.65089
- R. Sacco, E. Gatti, and L. Gotusso. The patch test as a validation of a new finite element for the solution of convection-diffusion equations. Comput. Methods Appl. Mech. Eng., 124:113-124, 1995.[Crossref] Zbl0948.78013
- R. Sacco and F. Saleri. Stabilized mixed finite volume methods for convection-diffusion problems. East West J. Numer. Math., 5:291-311, 1997. Zbl0895.65050
- R. Sacco and M. Stynes. Finite element methods for convection-diffusion problems using exponential splines on triangles. Computers Math. Applic., 35(3):35 - 45, 1998. Zbl0907.65110
- Riccardo Sacco, Emilio Gatti, and Laura Gotusso. A nonconforming exponentially fitted finite element method for two-dimensional drift-diffusion models in semiconductors. Numerical Methods for Partial Differential Equations, 15(2):133-150, 1999. Zbl0926.65119
- D. L. Scharfetter and H. K. Gummel. Large-signal analysis of a silicon read diode oscillator. IEEE T. Electron Dev., 16:64-77, 1969.[Crossref]
- R. Scheichl. Decoupling three-dimensional mixed problems using divergence-free finite elements. SIAM J. Sci. Comput., 23(5):1752-1776, 2002. Zbl1038.65125
- J. Wang, Y. Wang, and X. Ye. A robust numerical method for stokes equations based on divergence-free h(div) finite element methods. SIAM J. Sci. Comput., 31(4):2784-2802, 2009. Zbl05770810
- J. Wang and X. Ye. New finite element methods in computational fluid dynamics by H(div) elements. SIAM J. Numer. Anal., 45:1269-1286, 2007.[WoS][Crossref] Zbl1138.76049
- Y. C. Zhou. Electrodiffusion of lipids on membrane surfaces. J. Chem. Phys., 136:205103, 2012.[Crossref][WoS]