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Difference schemes for elliptic equations with mixed derivatives. (English) Zbl 1070.65110

A strictly elliptic Dirichlet boundary value problem of the following form is considered: \[ -\sum_{\alpha,\beta=1}^n{\partial\over\partial x_\alpha}(k_{\alpha,\beta}(x) {\partial u\over\partial x_\beta})+c(x)u=f(x),\;\;x\in\Omega\subset{\mathbb R}^n, \]
\[ u(x)=\mu(x),\;\;x\in\partial\Omega, \] where \(k_{\alpha,\beta}\in C^1(\overline{\Omega})\) and \(c,f\in C(\overline{\Omega})\). The domain \(\Omega\) is assumed to be rectangular and covered with an uniform grid with step \(h\). The original problem is approximated with the order \(O(h^2)\) by a finite difference equation on the grid \[ -\Lambda y+qy=\phi\;\;\text{ for\;the\;interior\;points}, \]
\[ y=\mu\;\;\text{ on\;the\;boundary\;of\;the\;grid}. \] The finite difference operator \(\Lambda\) is defined with the help of the coefficients \(k_{\alpha,\beta}\) modified in some way, and with the help of the forward and backward finite differences. (I do not cite in extenso its rather complicated form). The authors assume that the coefficients \(k_{\alpha,\beta}\) of the mixed derivatives are non-positive. From this assumption and some other conditions of technical character, it follows that the scheme defined with zero Dirichlet conditions is stable in the discrete norm of \(H^1\) type. This implies the convergence for the original solution \(u\) smooth enough. The stability in the discrete maximum norm is also proved under another condition, also of technical character. Then for the exact solution \(u\) in \(C^4\) the uniform convergence of the second order follows.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35B50 Maximum principles in context of PDEs
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