Estimation of the rate of convergence of difference schemes for the system of elasticity theory equations with discontinuous coefficients (Q2761530)

Let \(\Omega_0=\{x=(x_1,x_2):0<x_{\alpha}<l_{\alpha}\), \(\alpha=1,2\}\), let \(\Gamma_0\) be the boundary of \(\Omega_0\) and let the curve \(\Gamma\) divide the domain \(\Omega_0\) into subdomains \(\Omega_1, \Omega_2\), and \(\Gamma\in C^2\), \(\Gamma\cap\Gamma_0=\emptyset\). The author considers the Dirichlet problem for the system of elasticity theory equations with discontinuous coefficients NEWLINE\[NEWLINE-\sum_{\alpha,\beta=1}^{2}{\partial\over\partial x_{\alpha}} \Biggl(K_{\alpha\beta}(x){\partial\vec u\over\partial x_{\beta}}\Biggr)= \vec f(x),\quad x\in\Omega_0;\;\vec u(x)=0, x\in \Gamma_0,NEWLINE\]NEWLINE where \(\vec u(x)\) is the vector of elastic displacements, \(\vec f(x)\) is the vector of external forces, \(\vec f(x)\in L_2(\Omega_0)\); the matrices \(K_{\alpha\beta}(x), \alpha,\beta=1,2\) have some special form depending on the elasticity coefficients. The solution of the considered system satisfies the following conjunction conditions: \([\vec u]_{\Gamma}=0, [\sum_{\alpha,\beta=1}^{2}K_{\alpha\beta}{\partial\vec u\over\partial x_{\beta}}\cos(n,x_{\alpha})]_{\Gamma}=0\), where \(n\) is the external normal to \(\Gamma\); the symbol \([\varphi]_{\Gamma}\) means the jump of the function \(\varphi(x)\) on the curve \(\Gamma\). For the considered problem a difference scheme with accuracy order \(O(\sqrt{h})\) in the mesh norm \(W_2^1(\omega)\) is constructed.