Improved Accuracy for Alternating-Direction Methods for Parabolic Equations Based on Regular and Mixed Finite Elements

An efficient modification by Douglas and Kim of the usual alternating directions method reduces the splitting error from [Formula: see text] to [Formula: see text] in time step k. We prove convergence of this modified alternating directions procedure, for the usual non-mixed Galerkin finite element and finite difference cases, under the restriction that k/h 2 is sufficiently small, where h is the grid spacing. This improves the results of Douglas and Gunn, who require k/h 4 to be sufficiently small, and Douglas and Kim, who require that the locally one-dimensional operators commute. We propose a similar and efficient modification of alternating directions for mixed finite element methods that reduces the splitting error to [Formula: see text], and we prove convergence in the noncommuting case, provided that k/h 2 is sufficiently small. Numerical computations illustrating the mixed finite element results are also presented. They show that our proposed modification can lead to a significant reduction in the alternating direction splitting error.