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  • 1
    Electronic Resource
    Electronic Resource
    Conjugate gradient methods and ILU preconditioning of non-symmetric matrix systems with arbitrary sparsity patterns (1989)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 9 (1989), S. 213-233 
    Details
    ISSN: 0271-2091
    Keywords: Preconditioning ; Conjugate gradients ; Non-symmetric matrices ; Finite elements ; Convective transport ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Preconditioning techniques based on incomplete Gaussian elimination for large, sparse, non-symmetric matrix systems are described. A certain level of fill-in may be specified in the incomplete factorizations. All methods considered may be applied to matrices with arbitrary sparsity patterns, for instance those associated with the general preprocessor algorithms or adaptive mesh techniques. The preconditioners have been combined with five conjugate gradient-like methods and tested on finite element discretized scalar convection-diffusion equations in 2D and 3D. It is found from numerical experiments that an amount of fill-in corresponding to about 50% of the number of original non-zero matrix entries is the optimal choice for this class of preconditioners. The preconditioners show almost no sensitivity to grid distortion. In problems with significantly variable coefficients or anisotropy the preconditioners stabilize the basic iterative schemes in addition to reducing the computational work substantially, mostly by more than 90%. The modified preconditioning technique, where fill-in is added on the main diagonal, performs in general better than the standard incomplete LU factorization, but is inferior to the latter in 3D problems and for matrix systems with complicated sparsity patterns.
    Additional Material: 16 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650090207
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  • 2
    Electronic Resource
    Electronic Resource
    Implicit finite element methods for two-phase flow in oil reservoirs (1990)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 10 (1990), S. 651-681 
    Details
    ISSN: 0271-2091
    Keywords: Porous media ; Two-phase flow ; Oil recovery ; Finite elements ; Preconditioning ; Conjugate gradients ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The equations governing immiscible, incompressible, two-phase, porous media flow are discretized by generalized streamline diffusion Petrov-Galerkin methods in space and by implicit differences in time. Systems of non-linear algebraic equations are solved by Newton-Raphson iteration employing ILU-preconditioned conjugate-gradient-like methods to the non-symmetric matrix system in each iteration. The resulting solution methods are robust, enable complex grids with irregular nodal orderings and allow capillary effects.Several numerical formulations are tested and compared for one-, two- and three-dimensional flow cases, with emphasis on problems involving saturation shocks, heterogeneous media and curved boundaries. For reservoirs consisting of multiple rock types with differing capillary pressure properties, it is shown that traditional Bubnov-Galerkin methods give poor results and the new Petrov-Galerkin formulations are required. Investigations regarding the behaviour of several preconditioned conjugate-gradient-like methods in these type of problems are also reported.
    Additional Material: 14 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650100605
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  • 3
    Electronic Resource
    Electronic Resource
    A unified mesh refinement method with applications to porous media flow (1998)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 28 (1998), S. 679-702 
    Details
    ISSN: 0271-2091
    Keywords: finite elements ; h-refinement ; porous media flow ; object-oriented programming ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: A unified algorithm is presented for the refinement of finite element meshes consisting of tensor product Lagrange elements in any number of space dimensions. The method leads to repeatedly refined n-irregular grids with associated constraint equations. Through an object-oriented implementation existing solvers can be extended to handle mesh refinements without modifying the implementation of the finite element equations. Various versions of the refinement procedure are investigated in a porous media flow problem involving singularities around wells. A domain decomposition-type finite element method is also proposed based on the refinement technique. This method is applied to flow in heterogeneous porous media. © 1998 John Wiley & Sons, Ltd.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    A numerical comparison of conjugate gradient-like methods (1988)
    New York, NY [u.a.] : Wiley-Blackwell
    Communications in Applied Numerical Methods 4 (1988), S. 793-798 
    Details
    ISSN: 0748-8025
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Three second-order, nonselfadjoint partial differential equations in two dimensions are discretized by the finite element method, leading to non-symmetric systems of linear equations. The systems are solved by several conjugate gradient-like methods, and compared for their computational cost. All methods are tested with and without preconditioning.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1630040613
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  • 5
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    Solving PDEs in Python: The FEniCS Tutorial I (2021)
    Springer Nature
    Details
    Publication Date: 2024-04-14
    Description: computer science; algorithms; visualization; software; programming
    Keywords: computer science ; algorithms ; visualization ; software ; programming ; thema EDItEUR::U Computing and Information Technology::UY Computer science ; thema EDItEUR::U Computing and Information Technology::UY Computer science
    Language: English
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  • 6
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    Programming for Computations - MATLAB/Octave (2021)
    Springer Nature
    Details
    Publication Date: 2024-04-14
    Description: Computational Science and Engineering; Numerical Analysis;
    Keywords: Computational Science and Engineering ; Numerical Analysis ; thema EDItEUR::U Computing and Information Technology::UY Computer science
    Language: English
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  • 7
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    Scaling of Differential Equations (2021)
    Springer Nature | Springer International Publishing
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    Publication Date: 2024-04-04
    Description: The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models. Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics. The text is easily accessible and example-driven. The first part on ODEs fits even a lower undergraduate level, while the most advanced multiphysics fluid mechanics examples target the graduate level. The scientific literature is full of scaled models, but in most of the cases, the scales are just stated without thorough mathematical reasoning. This book explains how the scales are found mathematically. This book will be a valuable read for anyone doing numerical simulations based on ordinary or partial differential equations.
    Keywords: Ordinary Differential Equations ; Partial Differential Equations ; Mathematical Modeling and Industrial Mathematics ; Computational Science and Engineering ; Simulation and Modeling ; Analysis ; Computer Science ; scaling ; non-dimensionalization ; dimensionless numbers ; fluid mechanics ; multiphysics models ; Differential calculus & equations ; Mathematical modelling ; Maths for engineers ; Maths for scientists ; Computer modelling & simulation ; thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKJ Differential calculus and equations ; thema EDItEUR::P Mathematics and Science::PB Mathematics::PBW Applied mathematics::PBWH Mathematical modelling ; thema EDItEUR::P Mathematics and Science::PD Science: general issues::PDE Maths for scientists ; thema EDItEUR::U Computing and Information Technology::UY Computer science::UYM Computer modelling and simulation
    Language: English
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  • 8
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    Solving PDEs in Python (2021)
    Springer Nature | Springer International Publishing
    Details
    Publication Date: 2024-04-04
    Description: This book offers a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. Using a series of examples, including the Poisson equation, the equations of linear elasticity, the incompressible Navier–Stokes equations, and systems of nonlinear advection–diffusion–reaction equations, it guides readers through the essential steps to quickly solving a PDE in FEniCS, such as how to define a finite variational problem, how to set boundary conditions, how to solve linear and nonlinear systems, and how to visualize solutions and structure finite element Python programs. This book is open access under a CC BY license.
    Keywords: Computational Science and Engineering ; Algorithms ; Visualization ; Mathematical Software ; Numerical Analysis ; Software Engineering/Programming and Operating Systems ; Data and Information Visualization ; Software Engineering ; Finite element ; FEniCS ; Partial Differential Equations ; Python ; Simulation ; Open access ; Maths for scientists ; Combinatorics & graph theory ; Mathematical & statistical software ; Operating systems ; thema EDItEUR::P Mathematics and Science::PD Science: general issues::PDE Maths for scientists ; thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis ; thema EDItEUR::P Mathematics and Science::PB Mathematics::PBV Combinatorics and graph theory ; thema EDItEUR::U Computing and Information Technology::UF Business applications::UFM Mathematical and statistical software ; thema EDItEUR::U Computing and Information Technology::UM Computer programming / software engineering::UMZ Software Engineering
    Language: English
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  • 9
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    Programming for Computations - Python (2021)
    Springer Nature
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    Publication Date: 2024-04-04
    Description: Mathematics; Computer mathematics; Numerical analysis; Computer software; Numerical analysis
    Keywords: Mathematics ; Computer mathematics ; Numerical analysis ; Computer software ; Numerical analysis ; thema EDItEUR::P Mathematics and Science::PB Mathematics::PBK Calculus and mathematical analysis::PBKS Numerical analysis ; thema EDItEUR::P Mathematics and Science::PD Science: general issues::PDE Maths for scientists ; thema EDItEUR::U Computing and Information Technology::UF Business applications::UFM Mathematical and statistical software ; thema EDItEUR::U Computing and Information Technology::UY Computer science::UYA Mathematical theory of computation
    Language: English
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  • 10
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    Finite Difference Computing with PDEs: A Modern Software Approach (2021)
    Springer Nature
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    Publication Date: 2024-04-14
    Description: finite difference methods; programming; python; verification; numerical methods; differential equations
    Keywords: finite difference methods ; programming ; python ; verification ; numerical methods ; differential equations ; thema EDItEUR::U Computing and Information Technology::UY Computer science
    Language: English
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