Grid approximation of a singularly perturbed boundary-value problem for a quasi-linear elliptic equation in the completely degenerate case
The author is concerned with the equation \[ \varepsilon^ 2L_ 2u(x) + \varepsilon L_ 1u(x)-g(x,u(x)) =0 \] for \(x = (x_ 1,\dots,x_ n)\), \(0<x_ 1 < d\), \(x_ i \in \mathbb{R}\), \(i = 2,\dots,n\), subjected to a Dirichlet boundary condition. Here \(0 < \varepsilon \leq 1\) denotes a parameter, \(L_ 2\) is a second order elliptic operator, \(L_ 1\) is a first order differential operator, and for the function \(g\) the condition \(\partial/\partial u(g(x)) \geq \text{const} > 0\) is assumed. The coefficients of the operators \(L_ 2\), \(L_ 1\) can depend on \(u\) and on the first order derivatives of \(u\), the parameter \(\varepsilon\) may be involved in the boundary condition. By means of a differential inequality the author obtains a priori estimations for the solution \(u\) and its derivatives. Next he takes a positive integer \(N\) sufficiently large, \(\delta_ 0 = \min\{\varepsilon\ln N,d/2\}\) and constructs a non-uniform mesh on \([0,d]\) with \(h_ 1 = \delta_ 0/N\) on \([0,\delta_ 0]\cup[d-\delta_ 0,d]\) and \(h_ 2 = d/N\) on \((\delta_ 0,d-\delta_ 0)\). For \(0 \leq x_ i \leq 1\) \((i = 2,\dots,n)\) the mesh is uniformly independent of \(\varepsilon\). On this special network the author defines difference schemes in a standard manner. Under some regularity assumptions he proves that the convergence of the solutions of difference schemes is uniform with respect to the parameter \(\varepsilon\). The same is discussed for an iterative difference scheme with an additional parameter \(t\) playing the role of time. The number of iterations is characterized as well.
singularly perturbed boundary value problem
scientific article; zbMATH DE number 56759