A method of obtaining (numerically) approximate solutions to (usually) differential equations. It consists of a method of discretization splitting the domain into disjoint subdomains over each of which the problem has a simpler (approximate) solution, and a method of reassembling those pieces to obtain a solution over the whole domain. It is closely tied to the calculus of variations.
Questions tagged [finite-element-method]
637 questions
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1 answer
Stokes problem fem
I'm looking for a book or pdf to study the Stokes problem with finite elements method
$\Delta u+\nabla p=f$ in $\Omega$
$\nabla\cdot u=0$ in $\Omega$
$+$ boundary conditions (example: $u=0$ on $\partial\Omega$.
I'm interested in study the existence…
yemino
- 814
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3 answers
10 basis functions in the reference tetrahedro (fem)
I'm writting a Finite Element code and I need the 10 basis functions in $\mathbb{P}^2(\hat K)$ (polynomial of degree less than or equal to 2 defined over the reference tetrahedro $\hat K$) in the reference tetrahedro with…
yemino
- 814
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1 answer
Determine Functions of a Reference Triangle for Finite Element
I am given a problem to define the functions $\phi_1(x, y)$, $\phi_2(x, y)$ and $\phi_3(x, y)$ for a single triangular element as a reference to do element assembly. However I'm not sure why the answer is:
φ1(x, y) := 1 − (1/2 x) − (1/2 y)
φ2(x, y)…
essy.lin
- 11
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Dimension of a meshing space for finite-element methods
In my Finite-Element Method class, we defined a meshing $\tau_{h}$ for an interval $I = [0,10]$ with $h$ being the size of a cell (unit in the partition of $I$). Then, internal approximation requires a finite dimensional subspace of the bigger…
friend-math
- 45
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Mass matrix of a Tetra10 element
How or where can I find the mass matrix of a 10 nodes tetraedrhal element for FEM computation? Every nodes has 3 dof (x,y,z) so that the element has 30 dof. The Stiffness Matrix is a 30x30 matrix and the Mass matrix should have the same dimensions
aSpagno
- 73
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1 answer
Solving Poisson equation with variable coefficient with Finite element method
I need to solve the poisson equation with varibale coefficient $\nabla\cdot(k\nabla u)=f$ with the finite element method (FEM). I really don't know how to obtain the weak formulation of the problem.
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Finite elements in $\mathbb R^d$ with obstable in $\mathbb R^{d-1}$
Is it possible to use some version of finite element method in a fluid problem (for example Stokes or Navier--Stokes problem) in a bounded domain $\Omega\subset\mathbb{R}^d$ with an obstacle (velocity on the obstacle equal to zero) in…
yemino
- 814
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1 answer
What is the semilinear form for this poisson equation?
I am using finite element software to solve ODEs and PDEs. It requires the semilinear form. E.g When solving the poisson equation
$$\nabla\cdot((1+u^2)\nabla u) = -f$$
the semilinear form
$$\int_\Omega (1+u^2)\nabla u\cdot\nabla v - fvdx$$
is…
DJames
- 155
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1 answer
Second order shape funtions on first order element
In a Finite Element context, say I have a 1D finite element with 2 nodes, $x=0$ and $x=1$. The typically used shape funcions are
$N_1(x) = 1 - x$
$N_2(x) = x$
Would it make sense to use second order polynomials on that same element (without…
mfnx
- 197
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1 answer
Basis funcions Discontinuous Galerkin
I am learning about Disconinuous Galerkin methods. I fail to understand how he basis funcions are constructed. I understand that typically Legendre polynomial are used, but I can't see how they relate to the nodes.
In Continuous Galerkin methods,…
mfnx
- 197
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1 answer
Finite Element Code for Stokes Problem using P2P1 Quad Elements
I am trying to write some MATLAB code to solve fluid flow problems using the finite element method. I am starting with the relatively-simple steady Stokes problem. I am using the quadrilateral P2P1 element, and following Reddy and Gartling 2010. As…
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1 answer
Deriving shape function for 2d finite element analysis
I have recently learned about finite element analysis. I mainly focus on structural mechanics.
A common element for 2d is the triangle with 3 nodes. I will first make an example of how I would compute the shape functions for a 3-node triangular…
Finn Eggers
- 932
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1 answer
What is this "1" in this stiffness matrix?
What is this "1" in this stiffness matrix?
http://www.annualreviews.org/article/suppl/10.1146/annurev.earth.35.031306.140104?file=ea.35.rayfield.pdf
$$K_e= \frac{E^e A^e 1}{L^e} \begin{bmatrix}1&-1\\1&-1\end{bmatrix}$$
eq. 6, p. 2.
Typo or some kind…
mavavilj
- 7,270
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1 answer
$L_2$-projection estimates.
Let $\Omega$ be some domain. Let $V $ be a function space over this domain. Let the $L^2(\Omega)$ projection $Pf$ of an $f \in L^2(\Omega)$ be given by the function such that
$$
(f-Pf,g)=0,
$$
for every $g \in V$.
Then, my question!
I can't…
edo
- 87
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1 answer
Finite Element for PDE having both spatial and temporal Derivatives
Can anyone guide or suggest any examples for implementing the finite element method to 1D PDEs that have both temporal and spatial partial derivatives? Im having trouble understanding the FEM so a simple example that shows how to make a state space…
amima
- 1