Compensated optimal grids for elliptic boundary-value problems

Abstract

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.

Fig. 1

The schematics of cell-to-cell signaling in an epithelial layer: the geometry of the epithelial layer (a) and the summary of the physical processes at the cell surface (b). In (a), red and blue circles show signaling molecules that are secreted by the epithelial cells, orange ovals represent the molecules of an imposed morphogen gradient. Both the signaling molecules and the morphogen bind to their specific cell-surface receptors, initiating responses by the intracellular machinery, represented by various symbols within cells. Details are taken from the signaling circuitry involved in the Drosophila egg development [21].

Fig. 2

The relative error in approximating F_c in (20) with F_n from (22)

Fig. 3

Comparison of the performance of different optimal grids for solving the boundary-value problem in (23) with m fixed. In both (a) and (b), the relative error of the solution is shown for all admissible values of q; n = 8 in (a) while n = 14 in (b). Red, green, and blue lines show the results of using geometric, Zolotarev, and compensated optimal grids, respectively. In all cases m = 1000.

Fig. 4

Convergence study for the solution of (23) with q = 4. Results for the compensated grids with n = 10 and n = 14, as well as geometric and Zolotarev grids with n = 14 are shown.

Fig. 5

(a) The form of the solutions of (27) for several values of α. (b) Results of the convergence studies of the numerical solution using a compensated grid with n = 14 obtained in Sec. 3.2

Fig. 6

(a) The discretization of the rectangular domain using a hexagonal grid. (b) The 9-point anisotropy-adjusted stencil for Δ_⊥ on a square grid. In (a), the solid circles show the discretization nodes, while the empty circles correspond to the ghost nodes of the reflecting boundary. Similarly, solid lines in (a) show the connections between the discretization nodes, while dashed lines show the connections to the ghost nodes. In (b), the fractions give, apart from the factor of $h_{\perp }^{-2}$, the weights of different nodes in the stencil.

Fig. 7

The L^∞ norm of the relative error of the numerical solution of (32) and (33) obtained, using hexagonal grids in the xy-plane and compensated optimal grids with n = 8 in (a) and n = 14 in (b) (see text for complete details). In (b), the straight line indicates the O(|q|⁴) dependence.

Fig. 8

The L^∞ norm of the relative error of the numerical solution of (32) and (33) obtained, using anisotropy-adjusted square grids in the xy-plane and compensated optimal grids with n = 8 in (a) and n = 14 in (b) (see text for complete details). In (b), the straight line indicates the O(|q|⁴) dependence.

Fig. 9

(a) The profile of the solution of (6) and (49) obtained using the n = 14 compensated optimal grid of Sec. 4 on a hexagonal lattice with $h_{\perp }=0.25,L_x=60h_{\perp },L_y=68h_{\perp }\sqrt{3}/2.$ (b) The relative error of ū(0, 0) obtained using the n = 14 compensated optimal grids of Sec. 4 for hexagonal and anisotropy-adjusted square lattices. In (b), the straight line indicates the $O\left(h_{\perp }^4\right)$ dependence.