Finite difference approximations to the Dirichlet problem for elliptic systems
The authors consider systems of difference equations \[ (1)\quad \Delta^ hu+f(x,u)=0,\quad in\quad \Omega_ h\subset {\mathbb{R}}^ m,\quad u\in {\mathbb{R}}^ n,\quad u=0\quad on\quad \Gamma_ h \] where \(\Delta^ h\) is usual difference approximation of the Laplacian and \(\partial f_ i/\partial u_ j\geq 0\), \(i\neq j\), \(\partial f_ i/\partial u_ i\geq - L\). Based on inverse isotonity of \(I-\lambda \Delta^ h\), \(\lambda\geq 0\), monotone iterative schemes for solving (1) are constructed. These schemes are motivated by Crank-Nicholson and alternating direction implicit discretization of the corresponding parabolic problem. Unlike the stability problem the monotonity condition obtained in the article for the Crank-Nicholson scheme only distinguishes the purely implicit case, when no restriction on the time step (iteration parameter) arises. The necessity of this condition is supported by a numerical experiment. The authors use their schemes to obtain upper and lower solutions converging to minimal and maximal solutions, respectively, and to check uniqueness of the numerical solutions. Numerical results are given for a system from mathematical biology \((m=1\), \(n=2)\).
alternating direction implicit discretization
scientific article; zbMATH DE number 3972119