Questions tagged [finite-element-method]
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.
618
questions
1
vote
0
answers
15
views
purpose of interpolants in Galerkin methods
I am learning about the finite element method in an abstract Banach and Hilbert space setting, especially for an application to differential forms, and I am a bit lost on the big picture about the ...
- 81
0
votes
0
answers
20
views
inhomogeneous Helmholtz equation does not obey superposition using FEM
I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM). The equation is;
$c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 p = f$
where ...
1
vote
0
answers
24
views
Is the weak form on this book about a level set FSI problem wrong?
I'm trying to repeat a level set FSI problem on the book :Level Set Methods for Fluid-Structure Interaction, on the page 89, the provided freefem code define a weak form of the discretized equation
...
- 301
0
votes
0
answers
12
views
Solving the weak form
$$
K \cdot w^e(x) \cdot \frac{d}{dx}T(x) \Bigg|_{x=0}^{x=L} - \int_0^L K \cdot w^e(x) \cdot \frac{d}{dx}T(x) \, dx + \int_0^L w^e(x) \cdot Q \, dx = 0
$$
I am learning Finite element analysis. In this ...
2
votes
1
answer
89
views
why $u \in H^2$ imply zero value of jump $[u] = u^{-}-u^{+}$
in page 13 of simple DG tutorial,
it gives following conclusion:
Define average $w=\frac{1}{2}\left(w^{-}+w^{+}\right)$ and recall jump $[w]=w^{-}*w^{+}$ then
$$
[a b]=[a]\{b\}+\{a\}[b]
$$
confusing ...
- 21
1
vote
0
answers
29
views
Discretize Burger's equation with upwind strategy
Let the 1D burgers equation be defined by $\frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = v \frac{\partial^2 u}{\partial x^2}$ where $v = \frac{1}{Re}$ where $Re$ is the number of ...
- 11
0
votes
1
answer
89
views
How to get shape functions for three node 2d triangular element
I am struggling with the shape function of finite element method. For sake of computation, most often we work with local coordinate. For 1d things are bit clear to me. but for 2d things getting hard ...
- 409
1
vote
0
answers
36
views
proving $ \int_{\Omega} (b \cdot\nabla u)\,v=-\int_{\Omega} (b \cdot \nabla v )\,u\,+\,\int_{\partial \Omega} v\,(n\cdot bu) $
Assume $b$ is a constant vector, $v \in V_h$ where $V_h$ is a space of functions and $\Omega$ is our domain of integration.
$$
\int_{\Omega} (b \cdot\nabla u)\,v=-\int_{\Omega} (b \cdot \nabla v )\,u\,...
- 2,511
1
vote
0
answers
25
views
Prove that the condition number of the stiffness matrix in Laplace equation is bounded by $h^{-2}$
I'm a little stuck trying to estimate a condition number in FEM context. I would like to prove the following:
Consider the stiffness matrix $K$ for piecewise linear functions on
a quasi-uniform mesh ...
- 31
0
votes
0
answers
54
views
Solving y'' - 4y = 0 Using Finite Element Analysis(using weak function)
What did I do wrong?please hint.
i solve this eq similar to bar eq $EA\frac{d^2U}{dx^2}+f=0$. I want to make stiffness matrix Kq=R
$y''-4y=0, BC1:y(0)=1, BC2:y(1)=2$
Try to the Finite Elements ...
- 1
0
votes
1
answer
39
views
Eigenmode problem of Maxwell's equations, verification.
According to https://www.sciencedirect.com/science/article/abs/pii/S004578250200539X?via%3Dihub
We are interested in finding solutions to Maxwell's equations which propagate along the source-free ...
0
votes
0
answers
33
views
How to use Bramble-Hilbert lemma to estimate the error of numerical intergration?
I am reading a paper that there a question confuses me. It says that ''...use $r+1$-point Gauss-Lobatto quadrature on the interval $I = [a,b](b-a = h)$, Since the Gauss-Lobatto quadrature on $[a,b]$ ...
- 41
0
votes
0
answers
21
views
Integrals for the the localized pyramid basis functions in Galerkin Method
I tried to show the following relations for the localized pyramid basis function $\phi_{i j}(x, y)=(1-|x| /$ $h)(1-|y| / h),|x|<h,|y|<h$, where $x$ and $y$ are measured from the site $(i, j)$. ...
0
votes
1
answer
27
views
Using Lipschitz continuity to prove a nonlinear operator is strongly monotone.
Suppose I have nonlinear operator $T: V \to V$ where $V$ is a finite dimensional Hilbert space with inner product $( \cdot, \cdot )$. I need to prove that $T$ is strongly monotone, i.e. there exists $...
- 2,359
0
votes
0
answers
14
views
The definition of compactly supported boundary conditions
In Chi-wang Shu's article Discontinuous Galerkin Methods: General Approach and Stability, "compactly supported boundary conditions" is mentioned in Proposition 3.2 and Proposition 3.3. I don'...
- 1
0
votes
0
answers
20
views
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 ...
- 15
0
votes
1
answer
40
views
Derivative of a distribution, including an integral of abs(x)
I'm currently reading a book on finite element analysis and I lack knowledge on higher level of mathematics. My question is regarding the derivative of a distribution. I'm stuck on an example ...
- 1
0
votes
1
answer
74
views
PDEs: Derivating the weak form for the nonlinear poisson equation
I was reading a PDE tutorial on solving the nonlinear poisson equation. The author of the tutorial defines the problem and asserts the weak form, but does not provide the derivation. I tried my own ...
- 2,159
1
vote
0
answers
51
views
PDEs: Boundary term in the weak variational form of the 2D heat equation
I am looking at a tutorial on the Fenics software library for solving PDEs with finite elements. I have a question about the handling of the boundary term when we find the weak form for the 2D heat ...
- 2,159
1
vote
1
answer
77
views
Why do mathematicians use BDM or Hdiv finite elements to solve the Mixed Poisson partial differential equation
I am still new to finite element methods, and I was looking at some tutorials on a specific formulation of the Poisson equation that introduces an additional variable. Some of the tutorials call this ...
- 2,159
0
votes
0
answers
14
views
Choosing optimization procedure for solving non-linear PDE with finite-element method
I'm trying to write a code solving the Dirichlet boundary problem for $p$-Laplacian on an arbitrary planar domain $\Omega$:
$$\begin{cases}
-\Delta_p u = f \text{ on } \Omega, \\
u\big|_{\partial \...
- 11
0
votes
0
answers
29
views
Reasonable polynomial basis up to order k for finite-element method applied to ODEs and the error in the infinity norm of such a basis?
So for a first order polynomial basis I know that the following basis is standard:
$\phi_j(t) = \begin{cases} 0 \; t<t_{i-1}, \\ \frac{t-t_{i-1}}{t_i - t_{i-1}} \; t_{i-1} \leq t \leq t_i \\ \frac{...
- 653
1
vote
0
answers
54
views
Implementing gradient optimization methods for non-linear functionals
I'm trying to write a code solving the Dirichlet boundary problem for $p$-Laplacian on an arbitrary planar domain $\Omega$:
$$\begin{cases}
-\Delta_p (u) = f \text{ on } \Omega, \\
u\big|_{\partial \...
- 11
0
votes
1
answer
125
views
How was the weak form for this system of advection-diffusion-reaction equations derived?
I am looking at a tutorial create for the FENICs finite element method package. The tutorial shows a system of advection-diffusion-reaction equations occurring in a ...
- 2,159
1
vote
0
answers
37
views
Matrix of Finite element for −u′′′=f(x)
I am making progress in finite elements (I am studying on my own with my dog Toreto), and I got stuck with the following problem:
In the equation, equipped with the necessary conditions,
$$ - u(x)'' = ...
- 35
1
vote
0
answers
41
views
Help with an article on convex-splitting method
I trying to simulate gradient flow $$\frac{\partial u}{\partial t}=-\nabla_x F(u)$$ where F(u) is (in my case): $$F(u)=2u^4-u^2$$ for u $\in [-0.5,0.5]$. Since the equation is non-linear, I would like ...
- 11
1
vote
0
answers
35
views
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 ...
- 979
0
votes
0
answers
28
views
How to apply Runge Kutta method to this equation
I'm trying to solve three dimensional heat equation. I use the method of lines to discretize the spatial part,and choose five point stencil. I got the following equation:
$$ \frac{\mathrm{d} U}{\...
0
votes
0
answers
28
views
inverse laplace operator with matlab
I was implementing an algorithm mentioned in a paper(DOI:10.1109/TVT.2022.3229888). In the iterating process, inverse laplacian operator involved when updating phi(t,x).
ϕ^(k+1) (t,x)=ϕ^k (t,x)+σ(-Δ)^(...
- 1
0
votes
0
answers
25
views
PDEs: Is there a model to combine simulateous diffusive and congestive(concentrating) forces?
I am still pretty new to the field of PDEs, but I wanted to develop a novel economics model to study the spatial congestion of economic activity. For example, we see that cities tend to accumulate ...
- 2,159
1
vote
1
answer
76
views
Finite element method in polar coordinates
I need to solve the following problem
$$ -\dfrac{1}{r}\left( \dfrac{d}{dr} \left( r \dfrac{du}{dr} \right) \right) = f(r), ~~ r \in (0, 1)$$
$$ u'(0) = 0, ~~~ u(1) = 0 $$
with the finite element ...
1
vote
0
answers
37
views
Gear scheme for heat equation
I have the following finite difference approximation to the standard 1D heat equation.
$$
\dfrac{3u_j^{n+1} - 4u_j^n + u_j^{n-1}}{2\Delta t} - \alpha\dfrac{u_{j-1}^{n+1} - 2u_j^{n+1} + u_{j+1}^{n+1}}{...
0
votes
0
answers
50
views
Poisson-Boltzmann with finite element method
I'm trying to solve the Poisson-Boltzmann equation with the Finite element approach:
$\begin{matrix}
\frac{\partial^2 y}{\partial x^2}=\sinh{y}, & \\
\frac{\partial y}{\partial x} = a,b & at ...
- 1
1
vote
0
answers
47
views
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:
$$...
- 139
0
votes
0
answers
39
views
Confusion with implementation of PDE constrained Bayesian Inverse Problem
Consider a PDE,
$$\partial_t u -a \nabla u - ru (1-u) = 0$$
at a given snapshot in time. The inverse problem is to find the diffusion coefficient $a \in L^{\infty}$ from a noisy measurement $$Y = \Phi(...
- 101
1
vote
1
answer
42
views
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 ...
- 139
2
votes
0
answers
33
views
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 ...
- 139
0
votes
0
answers
45
views
use finite element method to solve the PDE
Hi so I want to use the finite element method to solve this question
∂/∂x (y^2 ∂u/∂x (x,y))+∂/∂y (y^2 ∂u/∂y (x,y))-yu(x,y)=-x (x,y)∈D
where the boundary conditions are:
u(0,y)=0 for 0.5 ≤y ≤1 ...
1
vote
0
answers
20
views
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 ...
- 11
1
vote
0
answers
63
views
How to perform integration by parts on a term containing the gradient of a gradient?
I have $u\in\mathbb{R}^3$ and the term $\epsilon = (\nabla u)^\top + \nabla u$.
Since $\nabla u = \begin{pmatrix}
\frac{\partial u_1}{\partial x} & \frac{\partial u_1}{\partial y} &...
- 111
2
votes
2
answers
58
views
How to prove that $a(u,v)=\int_0^1 u'v' dx$ is coercive
Let $u,v \in H^1_0 (0,1)$
$a(u, v)=\int_0^1 u^{\prime} v^{\prime} dx$
I did not understand how they established the inequality between the 2nd and 3rd line. Can someone explain it to me in detail?
$$
\...
0
votes
1
answer
58
views
Representation of the analytical solution to the Poisson equation?
I have the following 1-dimensional Poisson's problem with the corresponding boundary conditions:
$$
u_{xx} = -f(x) ; \quad u(1) = g, \quad -u_x(0) = h
$$
on a open domain
$$
\Omega = (0,1)
$$
Is it ...
- 17
1
vote
2
answers
73
views
Divergence of gradient of temperature in heat method for geodesic distance approximation
The is a paper from some years ago describing how to approximate the geodesic distance on a surface.
The method solve the unsteady heat equation for a certain amount of $t$, then computes the ...
- 143
0
votes
0
answers
22
views
Mass matrix of a Tetra10 element
How or where can I find the mass matrix of a 10 nodes tetraedrhal element for FEM computation? Every nodes has 3 dof (x,y,z) so that the element has 30 dof. The Stiffness Matrix is a 30x30 matrix and ...
- 73
0
votes
1
answer
51
views
Change of variables in iso parametric shape function for FEM
I am having some struggle understanding how the change of variable works when using the iso-parametric elements in FEM.
From the weak form of the steady heat equation, I get two integrals
$$
\int_\...
- 143
2
votes
1
answer
170
views
FEM with no homogeneous Dirichlet boundary conditions
I am mechanical engineer and I have been using the FEM regularly for several years. However, I feel that I have never really understand the basis of the principle. I am using the heat equation to try ...
- 143
1
vote
0
answers
44
views
trying to solve a 2D finite element analysis FEA problem by hand
I am trying to solve this problem:
image of the problem
so far I could do some math to reach here:
Multiply equation (1) by a test function $v$ and integrate over the domain $\Omega$. Then, we apply ...
- 13
0
votes
0
answers
16
views
Bell's quantum nonlocality and finite elements
Nonlocality theorem concerns writing a covariance as $C(a,b)=\int A(a,x)A(b,x)dx$ with A in {1,-1}.
This is not without remembering the weak formulation of FEM with a continuous basis index a or b.
...
- 286
8
votes
1
answer
273
views
Symmetric formulation for the heat equation
Consider the heat equation: $$\partial_t u-div(A\nabla u)=f$$
with $u(0)=0, u=0$ on the boundary of the domain of definition, call it $U$. Consider a test function $v=v(x,t)$, and perform the ...
- 179
0
votes
0
answers
40
views
In finite element analysis, which boundary conditions affect the left hand side?
I'm currently working on a problem related to heat transfer, and thus I encountered robin boundary condtions for the first time while working with the finite element method. Unlike neumann boundary ...
- 1