Compensated optimal grids for elliptic boundary-value problems

• Published in: Journal of computational physics Vol. 227; no. 19; pp. 8622 - 8635
• Main Authors: Posta, F., Shvartsman, S.Y., Muratov, C.B.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01-10-2008; Elsevier
• Subjects: Cell communication; Computational techniques; Dirichlet-to-Neumann map; Exact sciences and technology; Finite-differences; Higher-order schemes; Mathematical methods in physics; Physics; Rational approximation; Finite-differences; Rational approximation; Cell communication; Dirichlet-to-Neumann map; Higher-order schemes; Second order; Domain walls; Nonlinear problems; Complexity; Calculation methods; Discretization; Boundary-value problems; Optimal approximation; Modelling; Differential operator; Calculation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2008.06.026

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.