On Discretization of a Two-Dimensional Laplace Operator in a Smooth Two-Dimensional Domain

Currently, the finite element method is the most widespread technique for solving problems of deformable solid mechanics. Its drawbacks are well-known: approximating a displacement with a piecewise-linear function, we obtain that the tensions are discontinuous. At the same time, it is necessary to note that most problems of deformable solid mechanics are described by elliptic-type equations, which have smooth solutions. It seems to be relevant to develop algorithms that would take this smoothness into account. The idea of such algorithms belongs to K.I. Babenko, who stated it in the early seventies of the last century. The author of this study has applied this technique to elliptic eigenproblems for many years and has fully realized its high performance. However, with this technique, the matrix of a finite-dimensional problem turns out to be not symmetric but only close to a symmetrizable one. Below, discretization using the Bubnov-Galerkin method eliminates this disadvantage. Note that the symmetry of matrix of finite-dimensional problem is important in study of stability. Unlike classical difference methods and the finite element method, where the convergence rate is in power dependence on the number of grid nodes, here we have an exponential decrease of the error.