The residual velocity method applied to a steady free boundary-value problem of vector Laplacian type

We consider a free boundary-value problem based on a simplifiedmodel of two-phase flow in porous media. The model has two independentvariables on each side of the free interface. At the interfaceat steady state, five mixed Dirichlet and Neumann conditionsare given. The movement of the interface in time-dependent situationscan be reduced to a normal motion proportional to the residualin one of the steady-state interface conditions (the ellipticinterior problems and the other interface conditions are satisfiedat each time). Following previous work, we consider the useof other residuals for the normal velocity that have superiornumerical properties. The well-posedness criteria for this vectorexample are particularly clear. The advantages of the correctlychosen, non-physical residual velocities are demonstrated innumerical computations. Although the finite-difference implementationin this work is not applicable to general problems, it has superiorperformance to previous implementations.