Finite difference approximations to the Dirichlet problem for elliptic systems

Abstract

The Dirichlet problem foru=(u 1,...,u n ) $$\Delta u + f(x,u) = 0in\Omega ,u = 0on\Gamma = \partial \Omega $$ wheref=(f 1,...,f n ), is discretized in the usual way (h mesh size): $$\Delta ^h u + f(x,u) = 0in\Omega _h ,u = 0on\Gamma _h $$ We consider variousmonotone, convergent iterative schemes. Among others, they can be used, together with estimation theorems for upper and lower solutions, to show uniqueness for solutions of (2). Numerical results are given for the system $$\Delta u + u(a - bu - c\upsilon ) = 0,\Delta \upsilon + \upsilon (d - eu - f\upsilon ) = 0$$ from mathematical biology (two competing species). It is shown that there is a unique positive solution for certain values of the positive parametersa,..., f. This result is crucial for the asymptotic behavior of solutions of the corresponding parabolic system ast→∞.