On the Combination of Finite Element and Splitting-Up Methods in the Solution of Parabolic Equations
A scheme combining finite element and splitting-up methods is suggested for the numerical solution of a parabolic equation in two dimensions. Approximation in space variables is implemented by the finite element method on a rectangular grid, triangulated by the diagonals. A finite-difference operator of the problem is split into four positive semidefinite one-dimensional operators acting along coordinate and diagonal directions. For the integration with respect to time, a two-cycle splitting-up scheme of the solution is used. The application of the method to a nonuniform grid topologically equivalent to a rectangular one is studied, and the stability conditions of the splitting-up method in this case are obtained.