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 O(k2) to O(k3) 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/h2 is sufficiently small, where h is the grid spacing. This improves the results of Douglas and Gunn, who require k/h4 to be sufficiently small, and Douglas and Kim, who require that the locally onedimensional operators commute. We propose a similar and efficient modification of alternating directions for mixed finite element methods that reduces the splitting error to O(k3), and we prove convergence in the noncommuting case, provided that k/h2 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.