Compatible spatial discretizations
Douglas N. Arnold,Pavel B. Bochev,Richard B. Lehoucq,Roy A. Nicolaides,Mikhail Shashkov +4 more
- 01 Jan 2006
179
TL;DR: The de Rham Complex and Elasticity Complex of the MPFA O-Method for General Quadrilateral Grids in Two and Three Dimensions are discussed in this article, as well as a Cell-Centered Finite Difference Method on Quadrilaterals and the development and application of compatible Discretizations of Maxwell's Equations.
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Abstract: Numerical Convergence of the MPFA O-Method for General Quadrilateral Grids in Two and Three Dimensions.- Differential Complexes and Stability of Finite Element Methods I. The de Rham Complex.- Defferential Complexes and Stability of Finite Element Methods II: The Elasticity Complex.- On the Role of Involutions in the Discontinuous Galerkin Discretization of Maxwell and Magnetohydrodynamic Systems.- Principles of Mimetic Discretizations of Differential Operators.- Compatible Discretizations for Eigenvalue Problems.- Conjugated Bubnov-Galerkin Infinite Element for Maxwell Equations.- Covolume Discretization of Differential Forms.- Mimetic Reconstruction of Vectors.- A Cell-Centered Finite Difference Method on Quadrilaterals.- Development and Application of Compatible Discretizations of Maxwell's Equations.
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Citations
Finite element exterior calculus: From hodge theory to numerical stability
TL;DR: In this article, the authors consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem.
Finite element exterior calculus: from Hodge theory to numerical stability
TL;DR: In this article, the authors developed an abstract Hilbert space framework for analyzing stability and convergence of finite element approximations of the Hodge Laplacian in the continuous problem.
Finite element approximation of eigenvalue problems
TL;DR: The final part tries to introduce the reader to the fascinating setting of differential forms and homological techniques with the description of the Hodge–Laplace eigenvalue problem and its mixed equivalent formulations.
Mimetic finite difference method
TL;DR: Flexibility and extensibility of the mimetic methodology are shown by deriving higher-order approximations, enforcing discrete maximum principles for diffusion problems, and ensuring the numerical stability for saddle-point systems.
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An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User Guide for the MATLAB Reservoir Simulation Toolbox (MRST)
Knut-Andreas Lie
- 21 Jul 2019
TL;DR: This book provides a self-contained introduction to the simulation of flow and transport in porous media, written by a developer of numerical methods, and will prove invaluable for researchers, professionals and advanced students using reservoir simulation methods.
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References
A Cell-Centered Finite Difference Method on Quadrilaterals
Mary F. Wheeler,Ivan Yotov +1 more
- 01 Jan 2006
TL;DR: In this paper, a cell-centered finite difference method for elliptic problems on curvilinear quadrilateral grids was proposed, which is based on the lowest order Brezzi-Douglas-Marini (BDM) mixed finite element method.
Mimetic Reconstruction of Vectors
J. Blair Perot,Dragan Vidovic,P. Wesseling +2 more
- 01 Jan 2006
TL;DR: It is demonstrated how explicit reconstruction can be used to define discrete Hodge star interpolation operators, and how some reconstruction approaches can lead to local conservation statements for vector derived quantities such as momentum and kinetic energy.
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Development and Application of Compatible Discretizations of Maxwell’s Equations
Daniel A. White,Joseph Koning,R. Rieben +2 more
- 01 Jan 2006
TL;DR: An extensible C++ framework that supports a variety of specific instantiations of these such as standard interpolatory bases, spectral bases, hierarchical bases, and semi-orthogonal bases is designed and Virtually any electromagnetics problem that can be cast in the language of differential forms can be solved using this framework.