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
1
vote
0 answers
Consequence of trace theorem
Trace theorem says that there exists unique continous linear mapping $\gamma : W^{k,p}(\varOmega) \rightarrow L^q(\varOmega)$ that $\gamma(u) = u|_{\partial \varOmega}$.
It is correct to write $||u||_{L_q(\partial \varOmega)} =…
Cygne
- 77
1
vote
1 answer
Prove a finite element space is not conforming
Let $\tau=[-1,1]^2$, consider the finite element $\left(\tau, Q_{1}, \Sigma\right)$, $Q=span\{1,x,y,x^2−y^2 \},$ $\Sigma=\{w(−1,0),w(1,0),w(0,−1),w(0,1)\}$.
Show that the unisolvent element leads to a finite element space,
which is not…
Ariel So
- 359
1
vote
0 answers
Stability analysis of parabolic problem
I am doing a past qualify exam question.
Let
$$W_{h}=\left\{v_{h} \in \mathcal{C}^{0}(\bar{\Omega}) ; \forall \tau \in \mathcal{T}_{h},\left.v_{h}\right|_{\tau} \in \mathcal{P}_{1},\left.v_{h}\right|_{\partial \Omega}=0\right\}$$
Consider the fully…
Ariel So
- 359
1
vote
1 answer
Why does the Jacobian of the Bilinear transform appear in the Piola transform?
I'm doing work with finite element methods and I often do mappings from the reference square $[0,1]^2$ to the physical quadrilateral by a bilinear/trilinear mapping (we'll stick with 2D for now).
Suppose this mapping maps our quadrilateral vertices…
1
vote
0 answers
Solve PDE by FEM
Given PDE: $\nabla^2 u(x,y)=(α^2-β^2 ) u(x,y)$
The domain is the square $[0,1] \times [0,1]$, which the boundary nodes are $(0,0),(1,0),(1,1),(0,1)$.
It is given that $u(0.5,0.5)=1$.
We would like to know how to started with using linear…
Hiven
- 11
- 2
1
vote
0 answers
Right arc less-than bracket, what is the meaning of?
While studying the textbook of finite element method, there's one symbol I don't understand.
It is not listed in Pakin's Comprehensive Latex Symbol List.
I found it via wikipedia, see the following figure:
So it seems to be called "right arc less…
Yanyuan Zhu
- 11
1
vote
0 answers
Finite element method how to implement neumann boundary condition with implicit method
I have implemented a finite element method to solve the 1D heat conduction equation:
$$
\frac{\partial T(x,t)}{\partial t} = k\frac{\partial^2T(x,t)}{\partial x^2}
$$
This is done using an implicit method by solving the set of…
1
vote
0 answers
How to solve simple ODE with FEM
How to solve
$$
\begin{cases}
y'(x)=y(x)\\
y(0)=1
\end{cases}
$$
using finite element method?? I mean Bubnov-Galerkin approximation method along with Ritz minimization. I've try for some times and it looks like i don't understand something.
dann
- 21
1
vote
1 answer
What is the best iteration method used in finite element software?
Finite element analysis software uses iteration to find displacement vector. There are many methods like multigrid and conjugate gradient methods. Which is the best method to iterate the required vector so that the computation time taken is…
1
vote
0 answers
Mixed finite elements type choice
I have the following system of equations on a 2D domain:
\begin{align}
\sum_p \mathbf{a}_l^p \cdot \nabla [\alpha g_p] + \beta f_l &= \mathrm{rhs},
\quad l=1, \ldots, N_f,\\
\sum_l \alpha \mathbf{a}_l^p \cdot \nabla f_l + \beta g_p &= 0,
\quad p=1,…
Evg
- 171
1
vote
0 answers
Why do we eliminate centre of gravity as degree of freedom in FEM
In the finite element lecture we learned, that it is impractical to use centre of gravity of a triangle as a degree of freedom. So we defined a subspace $P_3'$, where $P_2\subseteq P_3'\subseteq P_3$
$$P_3'=\{p\in P_3;\…
Tom83B
- 670
1
vote
0 answers
how to build a solution by FEM in staggered grid of a EDP calculated in ghost points?
Let be the domain $R= \left\{ (x,t) | 0 < x < l, \ t>0 \right\}$ and let be;
\begin{eqnarray}
u_{t}-a(t)u_{xx} &= &f(x,t), \ a>0\\
u(x,0)&=&u_{0}(x), \ x \in (0,l)\\
u(0,t)= 0,& \ u(l,t)=0;
\end{eqnarray}
take $ m > 1 $ an integer number, define the…
Math_learner
- 35
1
vote
1 answer
Show that bilinear form A(u , v) is continuous
I want to show that the following bilinear form is continuous:
$A:H^{1}(0,1)\times H^{1}(0,1)\rightarrow \mathbb{R}$, where $H^{1}(0,1)$ is the Sobolev space and $A$ is define by
$A(u , v) = \int_{0}^{1}\alpha(x)u'(x)v'(x)dx + \int_{0}^{1}u(x)v(x)dx…
VWasques
- 48
1
vote
1 answer
Finite Element Error - Why use Seminorm?
when using the Finite Element Method to solve the Poisson equation, one usually arrives at an (a priori) error estimate like
$$ \lVert u-u_h \rVert \leq c h |u|$$
where $\lVert \cdot \rVert$ and $| \cdot |$ are the Norm and Semi-Norm in $H^{1,2}$.…
g7349290
- 11
- 1
1
vote
2 answers
System of equations for vector valued functions problems
I am having trouble about determining the system to solve when using finite element method for vector fields.
Suppose we have the following problem:
$$-\Delta\mathbf{u}=\mathbf{f}\;\text{in}\;\Omega,$$
with some Dirichlet boundary conditions. The…
DOMiguel
- 494