Proceeding from the governing equations describing a saturated poroelastic material with intrinsically incompressible solid and fluid constituents, we compare the monolithic and splitting solution of the different multi-field formulations feasible in porous media dynamics. Because of the inherent solid-fluid momentum interactions, one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the momentum equations and the algebraic incompressibility constraint. Here, the resulting set of differential-algebraic equations (DAE) is solved by the finite element method (FEM) following two different strategies: (1) an implicit monolithic approach, where the equations are first discretized in space using stable mixed finite elements and second in time using stiffly accurate implicit time integrators; (2) a semi-explicit-implicit splitting scheme in the sense of a fractional-step method, where the DAE are first discretized in time, split using intermediate variables, and then discretized in space using linear equal-order approximations for all primary unknowns. Finally, a one-and a two-dimensional wave propagation example serve to reveal the pros and cons in regard to accuracy and stability of both solution strategies. Therefore, several test cases differing in the used multi-field formulation, the monolithic time-stepping method, and the approximation order of the individual unknowns are analyzed for varying degrees of coupling controlled by the permeability parameter. In the end, we provide a reliable recommendation which of the presented strategies and formulations is the most suitable for which particular dynamic porous media problem.
MONOLITHIC VS SPLITTING SOLUTIONS IN POROUS MEDIA DYNAMICS1343 PDE, namely the partial solid and fluid momentum balances and the volume balance of the mixture representing a continuity-like constraint for the considered saturated porous medium with incompressible constituents. Necessary modifications and possible variations will be discussed in the framework of the numerical solution procedure, where the FEM is used as a convenient technique for the treatment of coupled problems, cf.[23]. In doing so, the spatial FE discretization of the general three-field (u-v-p or u-w-p) variational problem yields a stiff system of differentialalgebraic equations (DAE) in time. In fact, the semi-discrete volume balance as an algebraic side condition makes the problem ill-conditioned in a computational sense. In the literature, several works exist, however, mainly related to fluid-structure interaction, where coupled equation systems are defined or classified and suitable solution procedures are introduced, see, e.g. [24,25].In this contribution, two solution strategies for the strongly coupled dynamic problem are presented and compared, viz. a monolithic and a splitting solution scheme. In the monolithic approach, the system of equations is solved by one common strategy, where first the spatial FE discretization is carried out as described before yieldi...
