<p>A block-structured approach for solving 2-dimensional finite element approximations of the Poisson equation on graphics processing units(GPUs) is developed. Linear triangular elements are used, and a matrix-free version of the conjugate gradient method is utilized for solving test problems with over 30 million elements. A speedup of 24 is achieved on a NVIDIA Tesla C1060 GPU when compared to a serial CPU version of the same solution approach, and a comparison is made with previous GPU implementations of the same problem.</p>

A block-structured approach for solving 2-dimensional finite element approximations of the Poisson equation on graphics processing units(GPUs) is developed. Linear triangular elements are used, and a matrix-free version of the conjugate gradient method is utilized for solving test problems with over 30 million elements. A speedup of 24 is achieved on a NVIDIA Tesla C1060 GPU when compared to a serial CPU version of the same solution approach, and a comparison is made with previous GPU implementations of the same problem.

In this thesis the mimetic finite difference method for grids with curved faces is presented, implemented and tested with an emphasis on applications in reservoir simulation. The thesis gives a brief introduction to reservoir modeling and introduce the mimetic method for flat and for curved faces. Then the continuity condition for the curved mimetic method is discussed. It is shown that the suggested continuity condition is not valid for cases with a difference in permeability between two cells separated by a curved face. An alternative continuity condition is discussed and implemented. Numerical examples confirm that the original continuity condition is incorrect for general examples with heterogeneous permeability. Numerical examples for the alternative continuity condition shows that it is correct for simple cases, and that it gives no gain in accuracy compared to the mimetic method. In conclusion the curved mimetic method is primarily of academic interest.

In this thesis we look at how it is possible to construct models in quantum mechanics by using p-adic numbers. First we look closely at different quantum mechanical models using the real numbers, as it is necessary to understand them well before moving on to p-adic numbers. The most promising model, where Weyl systems are used, is studied in detail. Here time translation is not generated by the Hamiltonian, but constructed directly as an operator possessing some fundamental structure in relation to the classical dynamics. Then we develop the relevant theory of the field of p-adic numbers Qp , with a focus on the properties of Qp as a locally compact abelian group. Here we present alternative proofs to those found in the literature. In particular, we give an independent proof of the selfduality of Qp. In the last chapters we look at some models using Qp . We generalize the idea of Weyl systems from real to p-adic numbers, and we discuss the physical implications. When using Weyl systems, time is p-adic. We also produce MatLab algorithms for numerical computations in connection with approximations of p-adic models by finite models.

We study Gabriel-Zisman localization, localization by a multiplicative system and by a null system. We define the triangulated category and the derived category. Finally we describe a scheme for localization from a triangulated category to a module category.

We start by giving a brief introduction to reservoirs and reservoir modelling at different scales. We introduce a mathematical model for the two-phase flow, before we look at numerical discretizations. In particular we look at the Multiscale Mixed Finite-Element (MsMFE) Method from the Matlab Reservoir Simulation Toolbox (MRST), developed by SINTEF. Next we introduce a mimetic method, (with the inverse ip_simple inner product, wich is used for solving the local flow problems required to construct the basis functions used in the MsMFE method. After we have given a short introduction to parallel computing, and some common terms, we introduce a parallel MsMFE method. The method makes use of the Matlab Parallel Computing Toolbox, and it lets us calculate the inner products, as well as construct the basis functions of the MsMFE Method, in parallel. The new method makes use of a structure for storing the inner products that proves to be facillitate faster construction of the required basis functions than the regular structure used in MRST. We conclude that the new functions performs quite well, and consequently that the MsMFE method is well-suited for parallelization; additinally we conclude that the Parallel Computing Toolbox works well for this task. We note that, for larger problems, the parallel MsMFE method displays a near linear speedup for up to twelve Matlab workers. The new parallel functions are released as a module for MRST under the GNU General Public License (GPL). They can be downloaded from http://master.andershoff.net.

This report features the isogeometric finite element method applied to the elastodynamic problem in a brittle medium with a potential for cracking. Griffith's theory for fracturing is used. The development of the model is outlined, complete with the Euler-Lagrange equations. The cracking is described with a phase field supplemented with a history field, contrary to the usual way of building the crack directly into the geometry by modification of the basis, facilitating the use of isogeometric analysis even with simplistic basis functions such as Non-Uniform Rational B-Splines (NURBS). The introduction of the crack-phase field results in non-linearity in the coupled problem. The problem is semi-discretized, upon which the spatial sub-problem is treated with isogeometric analysis. The numerical time-stepping solution routine is built around the Newton-Raphson method, but includes both pre-conditioning and correctors and is known as the predictor/multi-corrector time integration scheme. The Jacobian of the semi-discretized system (needed for the Newton-Raphson iteration) is developed analytically. In addition to the numerical tests with NURBS as our basis, we will also test the method with Locally Refined B-splines (LR B-Splines), ensuring better resolution along the crack path. The LR B-spline represents an alternative to the more commonly used T-Spline.

The topic of this thesis is GPU accelerated sparse linear algebra for subsurface reservoir modeling. Numerical techniques for reservoir sim- ulations are described and we present the open source reservoir simula- tion software toolbox MRST. We discuss some of the challenges related to linear algebra and reservoir simulation. Furthermore, we discuss the possibility GPU-acceleraing the linear algebra for reservoir simulation, and implement a GPU based CG solver preconditioned with AMG for MRST, using the open source linear algebra library CUSP.

An artin algebra $l$ over a commutative, local, artinian ring $R$ was fixed, and with this foundation some topics from representation theory were discussed. A series of functors of module categories were defined, and almost split sequences were introduced with some basic results. An isomorphism $omega_{delta,X} : D delta^* rightarrow delta_*(DTr(X))$ of $Gamma$-modules for an artin $R$-algebra $Gamma$ was constructed. The isomorphism $omega_{delta,X}$ was applied to a special case, yielding a deterministic algorithm for computing almost split sequences in the case that $R$ is a field.

We study the Cauchy problem for the 1-D fractal Burgers equation, which is a non-linear and non-local scalar conservation law used to for instance model overdriven detonation in gases. Properties of classical solutions of this problem are studied using techniques mainly developed for the study of entropy solutions. With this approach we prove several a-priori estimates, using techniques such as Kruzkow doubling of variables. The main theoretical result of this study is a L1-type contraction estimate, where we show the contraction in time of the positive part of solutions of the fractal Burgers equation. This result is used to show several other a priori estimates, as well as the uniqueness and regularity in time of solutions. We also solve our Cauchy problem numerically, by proposing, analyzing and implementing one explicit and one implicit-explicit method, both based on finite volume methods. The methods are proved to be monotone, consistent and conservative under suitable CFL conditions. Subsequently, several a priori estimates for the numerical solutions are established. A discussion on how the numerical methods may be implemented efficiently, as well as discussions of some of the numerical results obtained conclude this study.