Аннотация:In this paper, some new direct methods are suggested for solving grid equations appearing in the approximation of the Dirichlet problem for nonwave Helmholtz equation on a structured triangular grid in a rectangle. Difference schemes of the 2nd and 4th order of accuracy are constructed on a seven-point stencil and a new ordering of the variables and a method for their partial elimination that permit reducing the problem to solving a system of three-point vector equations are proposed. Direct methods are elaborated that require O(MN ln N) arithmetic operations to solve the reduced systems, where M and N are proportional to the numbers of grid points in the two corresponding directions.