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
3
votes
1 answer
Stokes Problem, finite elements P1/P0
I'm looking for some algorithm of a finite elements to solve the problem
Find $u,p\in H_0^1(\Omega)\times L^2(\Omega)$ such that
$-\nu\Delta u+\nabla p=f$ in $\Omega\subset\mathbb{R}^2$
$\nabla\cdot u=0$ in $\Omega$
$u=0$ in $\partial\Omega$
where…
yemino
- 814
2
votes
0 answers
Where can I find the error estimates for using quadratures in FEM for parabolic equations?
guys.
I want to know where I can find something about variational crimes for using numerical integration in calculating integrals needed in FEM in case of a parabolic equation like Heat equation. All the articles and books about this matter are…
Nima
- 51
2
votes
1 answer
Why are identical test and basis functions preferred when using DG
In discontinuous Galerkin FEM (as in continuous FEM) you begin by deriving the weak form of the PDE you are attempting to solve. The weak formulation guarantees exact conservation of the function you are trying to solve multiplied by a set of…
2
votes
0 answers
Coordinate transformation of functions
I am given the triangle $\Delta$ in $\mathbb{R}^2$ in which the vertices are given as the points $e_1=(0,0), e_2=(1,0)$ and $e_3=(0,1)$. The space $\mathbb{P}$ is defined as the set of bivariate polynomials of degree $\leq 1$.
In the first step I am…
Andreas804
- 139
- 8
2
votes
2 answers
Element geometry in FEM for electrostatics
In a study of mine, I am using tetrahedral elements to solve for magnetic flux density from a given current density distribution. I extract the mesh structure from COMSOL. Since it is much easier to implement, I have chosen tetrahedral elements.…
strahd
- 111
- 1
- 7
2
votes
0 answers
How to Apply Gradient Operator to Nodal Basis Functions Expressed in Determinant
For a triangular element $T$ with nodes $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, I have the finite element basis functions $\eta_1$, $\eta_2$, and $\eta_3$ as:
$$\eta_j(x_k, y_k) = \delta_{jk}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$
which can be…
shashashamti2008
- 433
2
votes
2 answers
finite elements local vs global basisfunction
I always stumble across the term "local" and "global" basis functions for finite elements. But could not find an explanation what the difference is. What is the difference and where do they occur?
dba
- 999
2
votes
1 answer
Is my Finite Element Model for this problem correct?
The equation I'm trying to solve is
$$
\frac{d^2}{dx^2}u = - \frac{1}{1+x^2}
$$
on the domain $0 < x < 1$, with boundary conditions $u(0) = 0$ and $\frac{du}{dx}(1)=0$.
I got the weak form equation as:
$$\int_0^1 \frac{du}{dx} \frac{d\psi}{dx} \ dx…
Leir
- 23
1
vote
0 answers
What is a finite dimension subspace contained on an infinite dimension Hilbert space?
I am struggling to follow a book on finite elements method, and I believed I had understood what is a $u \in L²(a, b)$ . In my words, $u$ is any function $u: (a, b) \rightarrow \mathbb{R}$, where $(a,b)$ is a range in $\mathbb{R}$, that behaves…
lvella
- 715
1
vote
0 answers
Matrix of Finite element for −u′′′=f(x)
I am making progress in finite elements (I am studying on my own with my dog Toreto), and I got stuck with the following problem:
In the equation, equipped with the necessary conditions,
$$ - u(x)'' = f(x) $$
For the Stiffness Matrix $$ A_{i,j} =…
Math_learner
- 35
1
vote
0 answers
Reference for space-time finite element method
I want to study about solving time dependent problems using fem. I have references for solving such problem by first discretizing the space domain and then applying finite difference scheme in time variable. Another way to solve time dependent…
Shreya Chauhan
- 979
1
vote
0 answers
1d quadratic finite elements
I would like to show that in 1D the quadratic finite elements lead to the following
lumped mass matrices, obtained summing all line coefficients of the corresponding mass matrices onto the diagonal:
$$ A = \left( \begin{matrix} \frac{1}{6} & 0 & 0…
Andreas804
- 139
- 8
1
vote
1 answer
Rewrite function in a different coordinate system
Let's consider the triangle $\Delta$ in $\mathbb{R}^2$ with the vertices $e_1=(0,0), e_2=(1,0)$ and $e_3=(0,1)$. Let $\mathbb{P}$ denote the set of bivariate polynomials of degree $\leq 1$.
In the first step I am asked to find a basis $S_1,S_2,S_3$…
Andreas804
- 139
- 8
1
vote
0 answers
How the solution $u$ converges to the best approximation of the solution $u^{(k)}$ in the $H^1(\Omega)$-norm?
Let $\Omega$ be a polygonal domain in $\mathrm{R}^2$ and let $\mathcal{T}_0, \mathcal{T}_1,...$ be a sequence of triangulations of $\Omega$ formed by standard refinement. Let $\mathcal{P}_k^1$ be the space of continuous piecewise linear functions…
Ray Romero
- 11
1
vote
0 answers
Assembling the global finite element equation KU = F with gravitational body force
I am trying to model a basic solid cube that rests on a ground (z = 0) with gravity pointing in the negative z direction. To do this, I have a basic cube that is divided into 6 tetrahedrons. The entire solid is made up of these basic cubes so that…
Desiree Lin
- 11