Schur complement reduction in the mixed-hybrid approximation of Darcy's law: rounding error analysis

AbstractMixed-hybrid finite element approximation of the potential fluid flow problem leads to the solution of a large symmetric indefinite system for the velocity and potential head vector components. Such discretization gives rise to a very accurate approximation of the continuity equation in every element, and for low-order discretizations, the structural properties of the discrete matrix blocks allow cheap block elimination of the positive-definite diagonal block and subsequent reduction to the Schur complement system for the pressure and Lagrangian vector components. This system is then frequently solved by the iterative conjugate gradient-type method. Whereas this approach is well known, considerably less attention has been paid to the numerical stability aspects of such transformation. It was shown in [5] that block LU factorization can be unstable even when the system matrix is symmetric positive definite. In this paper we examine this type of conditional stability for a particular application in the underground water flow modelling. We show that the actual error of the computed approximate solution depends not only on the user-defined tolerance in the conjugate gradient process but also on the spectral properties of the corresponding matrix blocks eliminated during the Schur complement reduction. It is often observed that although the backward error of the approximate solution in the iterative part is reduced to the level of machine accuracy, the total residual norm after the back-substitution process remains at certain accuracy level. We give a bound for this maximal attainable accuracy and illustrate our theoretical results on a model example

Conserving Soil and Water for Society: Sharing Solutions MODELLING OF GROUNDWATER FLOW IN FRACTURED ROCK: THEORETICAL APPROACH AND PRACTICAL APPLICATIONS

The poster gives an overview of the present knowledge achieved by our research group in the field of modelling of the fluid flow in a fractured rock environment. First, we introduce possible approaches. Then we describe a process of generation of a mesh. Our system is based on the discrete stochastic network approach. It means that particular fractures are represented as 2D entities (polygons in our case) placed in 3D space. The resulting mesh has the same statistical characteristic (density of fractures, their orientation, permeability etc.) as real fractured environment in the rock massif. The second part of the poster describes a numerical model of the groundwater flow. This model solves the problem using linear (Darcy) flow. A Mixed-hybrid FEM is used for approximation of the PDE’s. In the third part we show application of our method to a real-world hydrogeological problem-simulation of the injection test and communication between drillholes PTP3 and PTP4a in the Krušné Hory Mountains in the Czech Republic. Additional Keywords: fractured rock, numerical modelling, Darcy law, and finite element method

A single inhalation of vapor from dried toad secretion containing 5-methoxy-N,N-dimethyltryptamine (5-MeO-DMT) in a naturalistic setting is related to sustained enhancement of satisfaction with life, mindfulness-related capacities, and a decrement of psychopathological symptoms

BACKGROUND: 5-methoxy-N,N-dimethyltryptamine (hereinafter referred to as 5-MeO-DMT) is a psychedelic substance found in the secretion from the parotoid glands of the Bufo alvarius toad. Inhalation of vapor from toad secretion containing 5-MeO-DMT has become popular in naturalistic settings as a treatment of mental health problems or as a means for spiritual exploration. However, knowledge of the effects of 5-MeO-DMT in humans is limited. AIMS: The first objective of this study was to assess sub-acute and long-term effects of inhaling vapor from dried toad secretion containing 5-MeO-DMT on affect and cognition. The second objective was to assess whether any changes were associated with the psychedelic experience. METHODS: Assessments at baseline, within 24 h and 4 weeks following intake, were made in 42 individuals who inhaled vapor from dried toad secretion at several European locations. RESULTS: Relative to baseline, ratings of satisfaction with life and convergent thinking significantly increased right after intake and were maintained at follow-up 4 weeks later. Ratings of mindfulness also increased over time and reached statistical significance at 4 weeks. Ratings of depression, anxiety, and stress decreased after the session, and reached significance at 4 weeks. Participants that experienced high levels of ego dissolution or oceanic boundlessness during the session displayed higher ratings of satisfaction with life and lower ratings of depression and stress. CONCLUSION: A single inhalation of vapor from dried toad secretion containing 5-MeO-DMT produces sub-acute and long-term changes in affect and cognition in volunteers. These results warrant exploratory research into therapeutic applications of 5-MeO-DMT

Vývoj metodiky identifikace a matematického modelování proudění a geochemické interakce v rozpukaném prostředí kompaktních hornin:Etapová zpráva Programu státní podpory výzkumu a vývoje MŽP za rok 2003

Cílem projektu je vývoj komplexní metodiky zjišťování a parametrického vyjádření proudění podzemní vody a geochemické interakce v puklinovém prostředí Českého masivu. Metodika zahrnuje metody terénního a laboratorního výzkumu a matematické modelování. Hlavní oblasti plánovaného výzkumu jsou 1. Terénní hydrodynamické a interferenční testy na výzkumných vrtech hlubokých cca 300 m. 2. Stanovení hydraulických, geologických a geochemických vlastností rozpukaných kompaktních hornin a podzemní vody na základě syntézy a analýzy výsledků terénních a laboratorních výzkumů. 3. Matematické modelování proudění a transportu v ucelené hydrogeologické struktuře puklinového prostředí granitového masivu s dvojí pórovitostí, s využitím výsledků terénního a laboratorního výzkumu