Finite Element Method Aar´ on Romo Hern´ andez Introduction Finite Element Method General Approach Background Distributions Sobolev Spaces Variational Problem Lax-Milgram Theorem The Finite Element Method Discrete Hilbert Space The String Problem FEM for problems in 2D and 3D Conclusion Finite Element Method Partial Differential Equations Final Project Aar´on Romo Hern´ andez Centro de Investigaci´on en Matem´ aticas Jalisco S/N, Col. Valenciana. Guanajuato, Gto. December-2012
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Finite Element Method
Aaron Romo Hernandez
Introduction
Finite Element Method
General Approach
Background
Distributions
Sobolev Spaces
Variational Problem
Lax-Milgram Theorem
The Finite Element Method
Discrete Hilbert Space
The String Problem
FEM for problems in 2D and3D
Conclusion
Finite Element MethodPartial Differential Equations Final Project
Aaron Romo Hernandez
Centro de Investigacion en MatematicasJalisco S/N, Col. Valenciana. Guanajuato, Gto.
The Finite Element MethodDiscrete Hilbert SpaceThe String ProblemFEM for problems in 2D and 3D
Conclusion
Finite Element Method
Aaron Romo Hernandez
Introduction
Finite Element Method
General Approach
Background
Distributions
Sobolev Spaces
Variational Problem
Lax-Milgram Theorem
The Finite Element Method
Discrete Hilbert Space
The String Problem
FEM for problems in 2D and3D
Conclusion
Background: Distributions I
Definition (Distribution)Let Ω ⊂ Rn be an open non-empty subset. Let D(Ω) the space of functionsϕ : Ω→ R with compact support contained in Ω, such that every ϕ ∈ D(Ω)is also in C∞(Ω). A distribution u ∈ D′(Ω) is a lineal, continuous functional
u : D(Ω) → Rϕ 7→ (u, ϕ)
Definition (Regular Distribution)Let f be a locally integrable function in Ω. Then, f defines a distributionin Ω
(f , φ) =
∫Ω
f (x)φ(x)dx .
A distribution generated from a locally integrable function is called aregular distribution. If a distribution is not regular, it is said to be singular.
Example (Dirac Distribution)The Dirac distribution,
(δx0 , φ) = φ(x0),
is a singular distribution.
Finite Element Method
Aaron Romo Hernandez
Introduction
Finite Element Method
General Approach
Background
Distributions
Sobolev Spaces
Variational Problem
Lax-Milgram Theorem
The Finite Element Method
Discrete Hilbert Space
The String Problem
FEM for problems in 2D and3D
Conclusion
Background: Distributions II
Definition (Partial Differential Operator)Let Ω ∈ Rn be an open non-empty subset. Let ϕ ∈ C k (Ω), and α ∈ Nn,|α| ≤ k. The general partial differential operator is defined as
∂αϕ =∂|α|ϕ
∂xα11 · · · ∂xαn
n, |α| = α1 + . . .+ αn
Definition (Distributional Derivative)Let α ∈ N be a multi-index given. Let u ∈ D′(Ω) be a distribution. Wedefine de distributional derivative of u as a distribution
∂αu : D(Ω) → Rϕ 7→ (∂αu, ϕ)
where (∂αu, ϕ) = (−1)|α|(u, ∂αϕ)
Definition (Distributional Product)Let f ∈ C∞(Ω) and u ∈ D′(Ω). The distribution fu is defined by
The Finite Element MethodDiscrete Hilbert SpaceThe String ProblemFEM for problems in 2D and 3D
Conclusion
Finite Element Method
Aaron Romo Hernandez
Introduction
Finite Element Method
General Approach
Background
Distributions
Sobolev Spaces
Variational Problem
Lax-Milgram Theorem
The Finite Element Method
Discrete Hilbert Space
The String Problem
FEM for problems in 2D and3D
Conclusion
FEM for higher dimensional problems I
To solve de boundary problem
−∇ · (c∇u(x)) + λu(x) = f (x) x ∈ Ω(c∇u) · n + qu(x) = g(x) x ∈ ∂Ω
we take the product between the first equation with the test function v andintegrate over the problem domain
−∫
Ω∇ · (c∇u(x))v(x) + λu(x)v(x)dx =
∫Ω
f (x)v(x)dx ,
integrating by parts,∫Ω
(c∇u(x)) · ∇v(x) + λu(x)v(x)dx −∫∂Ω
(c∇u) · nds =
∫Ω
f (x)v(x)dx ,
considering the boundary conditions∫Ω
(c∇u(x))·∇v(x)+λu(x)v(x)dx−∫∂Ω
(−qu(x)+g(x))ds =
∫Ω
f (x)v(x)dx ,
finally, we define the variational problem∫Ω
c∇u · ∇v + λuvdx −∫∂Ω−qu + gds =
∫Ω
fvdx , ∀v ∈ H1(Ω)
Finite Element Method
Aaron Romo Hernandez
Introduction
Finite Element Method
General Approach
Background
Distributions
Sobolev Spaces
Variational Problem
Lax-Milgram Theorem
The Finite Element Method
Discrete Hilbert Space
The String Problem
FEM for problems in 2D and3D
Conclusion
FEM for higher dimensional problems II
∫Ω
c∇uk · ∇vk + λuk vk dx −∫∂Ω−quk + gds =
∫Ω
fvk dx , ∀vk ∈ H1k (Ω)
to solve the variational problemin higher dimensions we pro-ceed in a similar way as we dowith the string problem and ap-proximate the infinite dimen-sional domain with a finite di-mensional one and transformthe variational problem in asystem of algebraic equations.
Finite Element Method
Aaron Romo Hernandez
Introduction
Finite Element Method
General Approach
Background
Distributions
Sobolev Spaces
Variational Problem
Lax-Milgram Theorem
The Finite Element Method
Discrete Hilbert Space
The String Problem
FEM for problems in 2D and3D
Conclusion
Conclusion
The Finite Element Method provides a useful tool for solving boundary valueproblems. Using FEM we are able to transform a hard PDE problem into asystem of linear equations, or a system of ODEs, which is solved relativelyeasy. However, we have to be careful defining the variational problem andwe must care about when the weak solution converges to the solution ofthe original problem.