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The difference method for non-linear elliptic differential equations with mixed derivatives. (English) Zbl 0534.65051
Consider the following Dirichlet problem \(F(x,u,u_ x,u_{xx})=0\), \(x\in \Omega \subset {\mathbb{R}}^ n\), \(u(x)=\bar u(x)\), \(x\in \partial \Omega\) where F satisfies the following conditions: \(F_{w_{ij}}(x,z,q,w)=F_{w_{ji}}(x,z,q,w)\), \(F_ z(x,z,q,w)\leq - L\), \(L>0\), there exists some bounded function \(G^{ij}\), such that: \(| F_{w_{ij}}(x,z,q,w)| \leq G^{ij}(x)\) and \(H/2| F_{qi}(x,z,q,w)| \leq F_{w_{ii}}(x,z,q,w)-1/\rho \sum_{i\neq j}G^{ij}(x), H>0\). A rectangular net is considered; the corresponding difference system is of the form: \(F(x_ m,v_ m,(v_ m)_ I,(v_ m)_{II}+(v_ m)_{\Delta}=0\) in the interior net-points, \(v_ m=\bar u(x_ m)\), \(x_ m\in \partial \Omega\) resp. \(v_ m=\bar u(x_ n)\) if \(x_ n\in \Omega\) is not an interior point. The following theorem is established: Let \(u\in C^ 2({\bar \Omega})\) be a solution of the Dirichlet problem and v a solution of the difference equation. If F fulfils the above assumptions and \(F_{w_{ii}}\) and \(F_{qi}\) are bounded, then \(\lim_{H\to 0}\| u-v\|_ H=0\) and \(\| u- v\|_ H\leq \max \{\epsilon(H),\frac{\eta(H)}{L}\},\) where \(\| y\|_ H=\max \{y(m):m\in M_{{\hat \Sigma}}\}\) and \(M_{{\hat \Sigma}}\) is the set of interior net-points.
Reviewer: C.Kalik
MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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