Finite element exterior calculus (FEEC) is a mathematical framework that formulates finite element methods using chain complexes. Its main application has been a comprehensive theory for finite element methods in , computational solid and fluid mechanics. FEEC was developed in the early 2000s by Douglas N. Arnold, and Ragnar Winther, among others. Finite element exterior calculus is sometimes called as an example of a compatible discretization technique, and bears similarities with discrete exterior calculus, although they are distinct theories. One starts with the recognition that the used differential operators are often part of complexes: successive application results in zero. Then, the phrasing of the differential operators of relevant differential equations and relevant boundary conditions as a Hodge Laplacian. The Hodge Laplacian terms are split using the Hodge decomposition. A related variational saddle-point formulation for mixed quantities is then generated. Discretization to a mesh-related subcomplex is done requiring a collection of projection operators which commute with the differential operators. One can then prove uniqueness and optimal convergence as function of mesh density. FEEC is of immediate relevancy for diffusion, elasticity, electromagnetism, Stokes flow. For the important de Rham complex, pertaining to the grad, curl and div operators, suitable family of elements have been generated not only for tetrahedrons, but also for other shaped elements such as bricks. Moreover, also conforming with them, prism and pyramid shaped elements have been generated. For the latter, uniquely, the shape functions are not polynomial. The quantities are 0-forms (scalars), 1-forms (gradients), 2-forms (fluxes), and 3-forms (densities). Diffusion, electromagnetism, and elasticity, Stokes flow, general relatively, and actually all known complexes, can all be phrased in terms the de Rham complex. For Navier-Stokes, there may be possibilities too. (Wikipedia).
Infinite Limits With Equal Exponents (Calculus)
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From playlist Calculus
Applications of double integrals: examples
Free ebook http://tinyurl.com/EngMathYT Example of how to apply double integrals to compute mass and moments of thin plates.
From playlist Engineering Mathematics
Epsilon delta limit (Example 3): Infinite limit at a point
This is the continuation of the epsilon-delta series! You can find Examples 1 and 2 on blackpenredpen's channel. Here I use an epsilon-delta argument to calculate an infinite limit, and at the same time I'm showing you how to calculate a right-hand-side limit. Enjoy!
From playlist Calculus
FEM@LLNL | Unifying the Analysis of Geometric Decomposition in FEEC
Sponsored by the MFEM project, the FEM@LLNL Seminar Series focuses on finite element research and applications talks of interest to the MFEM community. On March 22, 2022, Tobin Isaac of Georgia Tech presented "Unifying the Analysis of Geometric Decomposition in FEEC." Two operations take
From playlist FEM@LLNL Seminar Series
An introduction to the double integral. Whereas the single integral determines the area under a curve, the double integral of a two variable function determines the volume under a surface as marked out by a region on the XY plane.
From playlist Advanced Calculus / Multivariable Calculus
DSI | Diagrammatic Differential Equations in Physics Modeling and Simulation
Abstract: I’ll discuss some results from a recent paper on applying categories of diagrams for specifying multiphysics models for PDE-based simulations. We developed a graphical formalism inspired by the graphical approach to physics pioneered by the late Enzo Tonti. We will discuss the gr
From playlist DSI Virtual Seminar Series
Download the free PDF http://tinyurl.com/EngMathYT A tutorial on the basics of setting up and evaluating double integrals. We show how to sketch regions of integration, their description, and how to reverse the order of integration.
From playlist Several Variable Calculus / Vector Calculus
Lecture 9: Discrete Exterior Calculus (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Ari Stern: Hybrid finite element methods preserving local symmetries and conservation laws
Abstract: Many PDEs arising in physical systems have symmetries and conservation laws that are local in space. However, classical finite element methods are described in terms of spaces of global functions, so it is difficult even to make sense of such local properties. In this talk, I wil
From playlist Numerical Analysis and Scientific Computing
Lecture 3: Exterior Algebra (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Lecture 8: Discrete Differential Forms (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Branimir Cacic, Classical gauge theory on quantum principalbundles
Noncommutative Geometry Seminar (Europe), 20 October 2021
From playlist Global Noncommutative Geometry Seminar (Europe)
Lecture 7: Integration (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Masoud Khalkhali: Curvature of the determinant line bundle for noncommutative tori
I shall first survey recent progress in understanding differential and conformal geometry of curved noncommutative tori. This is based on work of Connes-Tretkoff, Connes-Moscovici, and Fathizadeh and myself. Among other results I shall recall the computation of spectral invariants, includi
From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"
Calculus: Absolute Maximum and Minimum Values
In this video, we discuss how to find the absolute maximum and minimum values of a function on a closed interval.
From playlist Calculus
Definition of an Injective Function and Sample Proof
We define what it means for a function to be injective and do a simple proof where we show a specific function is injective. Injective functions are also called one-to-one functions. Useful Math Supplies https://amzn.to/3Y5TGcv My Recording Gear https://amzn.to/3BFvcxp (these are my affil
From playlist Injective, Surjective, and Bijective Functions
Quantum Finite Elements: Lattice Field Theory on Curved Manifolds by Richard Brower
PROGRAM NONPERTURBATIVE AND NUMERICAL APPROACHES TO QUANTUM GRAVITY, STRING THEORY AND HOLOGRAPHY (HYBRID) ORGANIZERS: David Berenstein (University of California, Santa Barbara, USA), Simon Catterall (Syracuse University, USA), Masanori Hanada (University of Surrey, UK), Anosh Joseph (II
From playlist NUMSTRING 2022
14_9 The Volume between Two Functions
Calculating the volume of a shape using the double integral. In this example problem a part of the volume is below the XY plane.
From playlist Advanced Calculus / Multivariable Calculus