A combined mixed finite element ADI scheme for solving Richards' equation with mixed derivatives on irregular grids

We propose and analyze an efficient numerical method for solving semilinear parabolic problems with mixed derivative terms on non-rectangular domains. The spatial semidiscretization process is based on an expanded mixed finite element scheme which, combined with suitable quadrature rules, is converted into a cell-centered finite difference scheme. This choice preserves the asymptotic accuracy and local conservation of mass of the method, while substantially reducing the computational cost of the totally discrete scheme. To obtain it, an alternating direction implicit scheme is used for the integration in time. The resulting numerical algorithm involves sets of uncoupled tridiagonal systems which can be solved in parallel. We set out some theoretical results of unconditional convergence (of second order in space and first order in time) for our method. Finally, a numerical experiment is shown in order to illustrate the theoretical results.