Finite Definition and 1000 Threads

The finite element method (FEM) is a widely used method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points.
The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain.
The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.
Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).

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  1. B

    Linear Algebra: A Basis for a Finite Dim VS

    Why is it enough to prove that a set of vectors is a BASIS to a FINITE DIMENSIONAL Vector Space, it is enough to show that it is Linearly Independent. No Need to prove that it spans the whole vector space?
  2. P

    Can Iterative Methods Optimize Non-Linear Finite Differences?

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  3. R

    Can the intersection over a finite set be written as a sum?

    I know the union can be, but how about the intersection? I am trying to prove that: Suppose (X,T) is a finite topological space, n is a positive integer and U_i\in T for 1<= i <= n. Use mathematical induction to prove \bigcap U_i \in T, where the intersection goes from i=1 to n.
  4. B

    Is a Set Linearly Independent if Every Finite Subset Is?

    Homework Statement S is linearly independent iff every finite subset of S is linearly independent. Homework Equations The Attempt at a Solution letting S be linearly independent is pretty easy. i am slightly worried about my logic for the other way though. it goes like this...
  5. M

    Is coutnable unions of finite sets an infinite set?

    Hiya. :) While doing an assignment I ran into this little problem. We are working in the set of natural numbers \mathbb{N}. If i collect each natural number in a set S_1 = \{1\}, S_2 = \{2\},\ldots, S_n = \{n\},\ldots What happens when I take the countable union of all these? S =...
  6. A

    Material for learning finite difference solution to Hamilton-Jacobi equation

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  7. L

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  8. J

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  9. R

    Probability- finite n-th moment

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  10. F

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  11. C

    Proving the Existence of F from a Family of Finite Subsets of Natural Numbers

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  12. J

    Prove that an improper fraction with a finite binary expansion

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  13. A

    Calculating the electric field due to a wire of finite length

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  14. A

    Solving Finite Series with Real Y

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  15. 1

    MATLAB Finite difference method with matlab- square grid, cavity inside

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  16. L

    Finite fields and products of polynomials

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  17. M

    Jellium Model: Finite Confinement & Coulomb Interactions

    Is the Jellium model only suitable for an electron gas of infinite volume? If I confined a gas to a finite volume using an infinite potential well, is there still a way to cancel out the infinities in the coulomb interactions between electrons?
  18. M

    Potential of Finite Quadrupole and Zonal Harmonics

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  19. B

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  20. V

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  21. S

    Population Growth with Finite Resources:

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  22. K

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  23. D

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  24. V

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  25. C

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  26. A

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  27. G

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  28. Shackleford

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  29. P

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  30. radou

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  31. N

    Can Elements and Their Inverses Occupy Distinct Cosets in Nonabelian Groups?

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  32. Z

    If the universe is flat and finite?

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  33. R

    Solving the diffusion equation finite difference technique

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  34. P

    Business Calculus or Finite Mathematics

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  35. M

    How Do Subfields of Finite Fields Relate to Their Elements?

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  36. D

    How to Apply Boundary Conditions in Finite Element Analysis Using C++?

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  37. T

    Eternal universe vs finite life of stars

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  38. D

    Electric field at a point on the perpendicular bisector of a finite line charge

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  39. radou

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  40. M

    2D Finite Element Transient Heat Problem

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  41. I

    Are Cyclic Groups with x^n = 1 the Only Finite Groups?

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  42. T

    Finite Intersection Property Question

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  43. B

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  44. N

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  45. F

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  46. H

    Exploring the Age of the Universe: Is It Finite or Beyond Our Horizon?

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  47. M

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  48. Z

    Heat Capacity in Adiabatic Vessels: Corrections for Errors

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  49. K

    How Do Irreducible Representations of Finite Groups Work?

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  50. J

    Is the Universe Infinite or Finite?

    So this has bugged me for a long time. I'm a physicist in training, and have very little knowledge of the cosmos, but: I've heard for a long time the notion that the universe is infinite. For a long time, this troubled me, because I really couldn't conceptualize how something could be...
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