The finite element method with piecewise linear functions is considered for a second-order elliptic problem in a specific domain. The domain is the square with 1/8 part of it cut out by the straight line crossing the middles of two adjacent sides. The domain imbedding method reduces the problem to the solution of two subsidiary problems. One of them is the problem for the square and the other is related to the interface boundary (the intersection of the domain boundary with the square) and contains in the left part the capacitance matrix. The authors present numerical results illustrating the effectiveness of different preconditioning techniques for the capacitance matrix method. These techniques are based a) on a hierarchical basis, b) on the square root of the grid analog of the second derivative, c) on a vector probing. The conclusion is made, that b) is the best choice, when the coefficients of the equation are smooth, but it's performance deteriorates, for instance, with increased anisotropy, while c) can be promising.