Compensated optimal grids for elliptic boundary-value problems

A method is proposed which allows to efficiently treat elliptic problems on unbounded domains in two and three spatial dimensions in which one is only interested in obtaining accurate solutions at the domain boundary. The method is an extension of the optimal grid approach for elliptic problems, based on optimal rational approximation of the associated Neumann-to-Dirichlet map in Fourier space. It is shown that, using certain types of boundary discretization, one can go from second-order accurate schemes to essentially spectrally accurate schemes in two-dimensional problems, and to fourth-order accurate schemes in three-dimensional problems without any increase in the computational complexity. The main idea of the method is to modify the impedance function being approximated to compensate for the numerical dispersion introduced by a small finite-difference stencil discretizing the differential operator on the boundary. We illustrate how the method can be efficiently applied to nonlinear problems arising in modeling of cell communication.

[1]  Vladimir Druskin,et al.  Optimal finite difference grids and rational approximations of the square root I. Elliptic problems , 2000 .

[2]  S. Tsynkov.  Numerical solution of problems on unbounded domains. a review , 1998 .

[3]  Zydrunas Gimbutas,et al.  Coulomb Interactions on Planar Structures: Inverting the Square Root of the Laplacian , 2000, SIAM J. Sci. Comput..

[4]  R. Peierls.  The size of a dislocation , 1940 .

[5]  Gregory T. Reeves,et al.  Quantitative models of developmental pattern formation. , 2006, Developmental cell.

[6]  Nir Yakoby,et al.  Drosophila eggshell is patterned by sequential action of feedforward and feedback loops , 2007, Development.

[7]  Michael E. Taylor,et al.  Partial Differential Equations II: Qualitative Studies of Linear Equations , 1996 .

[8]  Lung-an Ying.  Infinite Element Methods , 1995 .

[9]  Stanislav Y Shvartsman,et al.  Signal propagation and failure in discrete autocrine relays. , 2004, Physical review letters.

[10]  Silvina Ponce Dawson,et al.  Sheet excitability and nonlinear wave propagation. , 2003, Physical review letters.

[11]  M. Freeman.  Feedback control of intercellular signalling in development , 2000, Nature.

[12]  Dmitri Tseluiko,et al.  Nonlinear Dynamics of Electrified Thin Liquid Films , 2007, SIAM J. Appl. Math..

[13]  J. Toland,et al.  The Peierls–Nabarro and Benjamin–Ono Equations , 1997 .

[14]  Robert V. Kohn,et al.  Magnetic microstructures - a paradigm of multiscale problems , 1999 .

[15]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[16]  Vladimir Druskin,et al.  Gaussian Spectral Rules for the Three-Point Second Differences: I. A Two-Point Positive Definite Problem in a Semi-Infinite Domain , 1999, SIAM J. Numer. Anal..

[17]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[18]  J. Barkley Rosser,et al.  Nine-point difference solutions for Poisson's equation , 1975 .

[19]  Eric H Davidson,et al.  Gene regulation: gene control network in development. , 2007, Annual review of biophysics and biomolecular structure.

[20]  John L. Tassoulas,et al.  CONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION , 2000 .

[21]  John L. Hudson,et al.  Spatiotemporal patterns during electrodissolution of a metal ring: three-dimensional simulations , 2001 .

[22]  Stanislav Y Shvartsman,et al.  Long-range signal transmission in autocrine relays. , 2003, Biophysical journal.

[23]  Vladimir Druskin,et al.  Application of the difference Gaussian rules to solution of hyperbolic problems: global expansion , 2002 .

[24]  François Alouges,et al.  Exponential meshes and three‐dimensional computation of a magnetic field , 2003 .

[25]  S. Shvartsman,et al.  Discrete models of autocrine cell communication in epithelial layers. , 2003, Biophysical journal.

[26]  Katharina Krischer,et al.  TUNING THE RANGE OF SPATIAL COUPLING IN ELECTROCHEMICAL SYSTEMS : FROM LOCAL VIA NONLOCAL TO GLOBAL COUPLING , 1997 .

[27]  A. Pasquinelli,et al.  Regulatory principles of developmental signaling. , 2002, Annual review of cell and developmental biology.

[28]  Vladimir Druskin,et al.  Application of the Difference Gaussian Rules to Solution of Hyperbolic Problems , 2000 .

[29]  Cyrill B. Muratov,et al.  Optimal grid-based methods for thin film micromagnetics simulations , 2006, J. Comput. Phys..