A finite-difference scheme of second-order accuracy for elliptic equations with discontinuous coefficients (Q5942185)

The author considers the two-dimensional Dirichlet problem for a second-order elliptic equation in a rectangle \(G\): \[ \operatorname {div} (D \operatorname {grad} u)=-f(x_1, x_2),(x_1, x_2) \in G, \tag{1} \] \[ u=q(x_1, x_2), (x_1, x_2) \in \partial G. \tag{2} \] Here \(D=D(x_1, x_2)\) is the piecewise constant diffusion coefficient with the lines of the discontinuous coefficients parallel to the coordinate axe. In addition the function \(u(x_1, x_2)\) and the normal component of the flux are continuous on the discontinuity surface: \[ [u]=0,\quad [D\delta u/\delta n]=0,\quad (x_1, x_2) \in \Gamma.\tag{3} \] The author approximates problem (1)--(3) by the integro-interpolational method on a uniform grid shifted by the half-increment and constructs a new five-points scheme of seventh-order accuracy in \(W^{1} _{2}\).