Fundamental Solutions of 9-point Discrete Laplacians; Derivation and Tables
Robert E. Lynch · Purdue e-Pubs (Purdue University System) · 1992
The only value of a which yields a higher order of accuracy for the Laplace or the Poisson equation is a = 1{3; this gives the optimal 9-point discrete Laplacian:,1: + (for application to the Poisson equation, see Birkhoff~Lynch [84, p. 92J).For a = 2{3, La is Palya's blinear finite element approximation of the Laplacian (see Birkhoff-Lynch [84, pp.190-191J).Values 0/ G(O) and G(I{3).For a = 0, the solution of (la-b) is well•known: see McCrea-Whipple [40] (are Stohr [50, III], Sobolev [52], Duffin [56}, Duffin-Shelly [58), and van der Pol [59]).These authors (as do we) first obtain values of G, by evaluating integrals with (j, k) at mesh points along a straight line, and then employ the difference equation and symmetry to obtain values at other mesh points in the plane.Duffin [59] used the fact that discrete harmonic functions satisfy discrete Cauchy~Ricmann equations to extend values from a line to the plane; we do not know if the concept of 'discrete harmonic function' can be generalized to apply to solutions of 9-point discrete Laplacians and accomplish a similar extension of values to the plane.Some values ncar the origin are given in Table 1; Table 2 lists them accurate to 5 digits.We are unaware of published solutions C(a) with a different from zero.Tables 3 and 4 give results for a = 1{3 from our general analysis. Analysis.As can be verified by direct substitution, a solution of (la) can be written as 2 .. 2".I ff (eijr+iJ:Y_l)dxdy Gj ,J: (ct) = ._2-;;2:-~---';--'-="--':""=:-:--:"':=:T';-::-=c:c:-:-,, " ... 4+2{(1 a)[cosx+ cosy] +acosxcosy} , , where the denominator in the integrand is equal to (L.. eijr+ikY)/e'jz+iky.It is a consequence of the asymptotic result (3) that (4) satisfies the boundary conditions (lb).Following Duffin [59, p. 348].