Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids

In this paper, we consider anisotropic diffusion with decay, and the diffusivity coefficient to be a second-order symmetric and positive definite tensor. It is well-known that this particular equation is a second-order elliptic equation, and satisfies a maximum principle under certain regularity assumptions. However, the finite element implementation of the classical Galerkin formulation for both anisotropic and isotropic diffusion with decay does not respect the maximum principle. We first show that the numerical accuracy of the classical Galerkin formulation deteriorates dramatically with increase in the decay coefficient for isotropic medium and violates the discrete maximum principle. However, in the case of isotropic medium, the extent of violation decreases with mesh refinement. We then show that, in the case of anisotropic medium, the classical Galerkin formulation for anisotropic diffusion with decay violates the discrete maximum principle even at lower values of decay coefficient and does not vanish with mesh refinement. We then present a methodology for enforcing maximum principles under the classical Galerkin formulation for anisotropic diffusion with decay on general computational grids using optimization techniques. Representative numerical results (which take into account anisotropy and heterogeneity) are presented to illustrate the performance of the proposed formulation

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.Comment: 34 page

Python og Lego EV3

Hensikten med oppgaven var å lage et rammeverk i Python for prosjekt som utføres ved hjelp av en programmerbar EV3-robot. Dette rammeverket skal bli brukt til undervisning på UiS i programmering. Det finnes allerede et eksisterende rammeverk som blir brukt på skolen, men dette rammeverket bruker MatLab. Dette rammeverket gir studenter innføring i programmering, og blir brukt til å utføre matematiske prosjekter. Dette rammeverket ble konvertert til en form som er kompatibel med Python. Når vi endret form på dette rammeverket, er det flere problemstillinger som oppsto, og deretter ble løst. Noen av problemene som oppsto og ble løst var: • Hente verdier fra EV3-enheten (MicroPython). • Overføre data til PC (Socket-tilkobling). • Plotte data (MatPlotLib). • Finne en prosjektstruktur som skal være pedagogisk å bruke for nye studenter (Bruk av lister eller dictionary). Etter rammeverket kom på plass, utførte vi ett par prosjekt slik som en ny student må gjøre. Dette ble gjort for å sjekke kompatibiliteten på utførelsen, og for å oppdage problemer som studenter muligens kan møte på.The purpose of the thesis was to create a framework in Python for a programmable lego EV3 robot. The framework is supposed to be used for further teaching at the University of Stavanger. There already exists a framework for this purpose at the school, but this framework uses the language MatLab. The purpose of these frameworks is to give students an introduction to programming, but also give the chance to do mathematical and practical projects. Some of the problems we encountered when we changed language of the framework was: • Get values from EV3 and the sensors (MicroPython) • Transfer data to a PC (Socket-connection) • Plotting the data (MatPlotLib) • Finding a projectstructure that would be easy for a new student to use (Lists or dictionary) After we got the framework done, we did some of the projects that should be expected from a student to do. We did this to test our solution, and to check any problems the students could get

2000. Numerical Modelling of Capillary Transition Zone

Abstract For a large number of reservoirs, a vertical transition zone between water and oil exists. In this zone, both water saturation and capillary pressure vary with height. Traditionally one assumes that there is a relation between capillary pressure and water saturation, given by a Pc-S primary drainage curve prior to any production in the reservoir. The vertical fluid distribution is found by assuming equilibrium between capillary forces and gravity. This paper will focus on numerical modelling of the fluid distribution as production is started in a reservoir. As wells may be open and shut during the (field) lifetime of a reservoir, both imbibition and drainage may occur in different parts of a reservoir. The numerical model will take into account the irreversibility of imbibition and drainage, commonly known as hysteresis, which applies for both capillary pressure and relative permeability. Our test examples will deal with three different production rate regimes; capillary-dominated, capillary-viscous and viscous. We investigate the fluid distribution in the transition zones as the wells are shut down and equilibrium again is reached for the different cases. The tests will show that the fluid distribution differs for different injection-and production rates. For the case where the production in the reservoir is very close to equilibrium, we also show how the fluid distribution can be found analytically

A model-oriented benchmark problem for CO2 storage

Simulation of CO2 storage in geological formations inherently involves decisions concerning relevant physics, upscaling, and numerical modeling. We propose a benchmark study designed to assess the impact of these necessary choices, within the context of storage in a conceptually simple geological formation. The benchmark asks for answers to relevant questions regarding the ultimate fate of the injected CO2 plume