Accuracy of discrete analogue of the spectral problem with mixed boundary value condition for operator of linear elasticity theory (Q2703335)

In the region \(\bar\Omega=\Omega\cup\Gamma=\{x=(x_1,x_2,x_3): 0\leq x_{i}\leq l_{i},i=1,2,3\}\) the authors consider the problem \(-\sum\limits_{i,k=1}^{3}(\partial \tau_{ki}/\partial x_{i})e_{k}= \nu u , x\in\Omega\); \(\sum\limits_{i,k=1}^{3}\tau_{ki}\cos(n,e_{i}) e_{k}=0, x\in \Gamma_1\); \(u=0, x\in \Gamma\setminus \Gamma_1\), where \(u=(u^{(1)},u^{(2)},u^{(3)})\) is a vector of elastic displacement; \(\tau_{ik}\) are the components of the stress tensor, \(i,k=1,2,3;\) \(\tau_{ik}=\tau_{ki}=\mu\left({\partial u^{(i)}\over\partial x_{k}}+ {\partial u^{(k)}\over\partial x_{i}}\right), i\neq k\), \(\tau_{ii}= \lambda\;\text{div}\, u+2\mu{\partial u^{(i)}\over\partial x_{k}}\); \(\lambda,\mu>0\) are the Lame coefficients; \(e_{k}\) is the unit vector of axis \(0,X_{k}, k=1,2,3\); \(\Gamma_1=\{x=(0,x_2,x_3)\in\Gamma\}\); \(n\) is the external normal to \(\Gamma\). The discrete analogue for considered problem is constructed and the principal term in the expansion in powers of the step of the cubic grid for the error of the eigenvalues is obtained.