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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
##### Keywords:
convergence; error estimate; Dirichlet problem
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