Auxiliary space preconditioning for mixed finite element discretizations of Richards’ equation

We propose an auxiliary space method for the solution of the indefinite problem arising from mixed method finite element discretizations of scalar elliptic problems. The proposed technique uses conforming elements as an auxiliary space and utilizes special interpolation operators for the transfer of residuals and corrections between the spaces. We show that the corresponding method provides optimal solver for the indefinite problem by only solving symmetric and positive definite auxiliary problems. We apply this preconditioner to the mixed form discretization of Richards’ equation linearized with the L-scheme. We provide numerical tests validating the theoretical estimates.