Questions tagged [galerkin-methods]
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79
questions
1
vote
0answers
26 views
Variational formulation using galerkin scheme
Consider the boundary problem
$$
\dfrac{\partial u}{\partial t}+ A^2 u+Af(u)+\lambda g(u)=0\ \ \ \ \ \mbox{ in }\ \ \ \Omega.
$$
$$
\dfrac{\partial u}{\partial \nu}=\dfrac{\partial \Delta u}{\partial \...
2
votes
0answers
13 views
References for discontinuous Galerkin method with one dimensional case?
Could anyone recommend some references on discontinuous Galerkin method for beginners? It is nice to contain the one-dimensional case. Thanks in advance!
1
vote
1answer
32 views
Chebyshev differentiation matrices and Galerkin method
I've been reading the "Spectral Methods in Matlab" by Lloyd N. Trefethen and I'm interested in solving PDE's with spectral methods. From what I understand this book gives us a method (at ...
1
vote
0answers
29 views
Finite-dimensional truncations of unbounded operators and resolvent convergence
Let $A$ be an unbounded self-adjoint operator in a Hilbert space $\mathcal{H}$, with domain $\mathrm{dom}\,A$, and consider a growing sequence $(P_n)_{n\in\mathbb{N}}$ of orthogonal projectors in $\...
3
votes
1answer
122 views
Requirements for Korn's inequality on piecewise $H_1$ vector field
I am looking at the Korn's inequality on $H^1$ vector fields, as described in this paper by Brenner. In particular, I am looking at how the seminorms defined in examples 2.3 - 2.5 satisfy the ...
0
votes
0answers
26 views
Inequality between inner product of functions in dual space
I'm reading Brezzi's paper on DG method and is currently puzzled on how equations (5.7) is used to derive equation (5.8).
Further searches lead me to learning that the space defined $\mathbf{V}'$ is ...
0
votes
0answers
28 views
Help deriving a Fourier-Galerkin approximation for a variable coefficient partial differential equation
I've been trying to derive a Fourier-Galerkin approximation of
\begin{equation}
\frac{\partial u}{\partial t} + \sin(x) \frac{\partial u}{\partial x} = 0
\end{equation}
But I'm not sure that my steps ...
2
votes
1answer
71 views
Galerkin method for system of ode's
I have the following system of ode's
$$
\begin{cases}
\frac{du_1}{dx}=u_1+u_2,\\
\frac{du_2}{dx}=u_1u_2
\end{cases}
$$
BCS
$$\frac{du_1}{dx}|_{x=0}=1,\frac{du_2}{dx}|_{x=1}=2 $$
$$x \in\Omega=[0,1]$$
...
3
votes
2answers
112 views
Galerkin method for nonlinear ode
I'm trying to solve the following differential equation:
$$\frac{d^2u}{dx^2}=\frac{du}{dx}u+u^2+x$$
$$x \in \Omega=[0,1]$$
$$BCS:u|_{x=0}=1;\frac{du}{dx}|_{x=1}=1$$
You can see that the right side ...
2
votes
1answer
66 views
Solving 1D Poisson equation using finite element method and understanding the Galerkin orthogonality
Let's consider the following test problem
$$
u'' = 12x^2 - 36x + 18 \qquad u(0) = u(3) = 0
$$
Analytical solution is
$$
u(x) = (x-3)^2 x^2
$$
I'm solving this using the finite element method, ...
0
votes
0answers
12 views
Hat Function for Galerking Method
I'm currently looking into Finite Element Method. I've started with the simple 1D case. I'm wondering how the entries in the stiffness matrix are calculated. I found the following interesing site.
...
0
votes
0answers
21 views
Compute the $L_2(0, 1)$ projection of the exact solution u into $P^3(0, 1)$
I'm attempting to self-teach myself the discontinuous Galerkin method, and this seems to be some of the basics to know beforehand based off this introduction found online. Here's the question (2.3 in ...
0
votes
0answers
13 views
Convergence of finite element approximation to ODE solution
For a particular application, I wish to represent the solution to the ODE
$$
\mathcal{L}u = \frac{\mathrm{d}^2u}{\mathrm{d}x^2} = \mathrm{f}(x), \quad x\in[0,1], \qquad (1)
$$
with boundary ...
0
votes
1answer
94 views
FEM: Steady-State heat diffusion and convection
So the strong form of the heat diffusion and convection PDE is given as
$\rho c_m \mathbf{v}\cdot\nabla T - \nabla \cdot \nabla T = \dot{q}\\
T(\mathbf{x},t) = T_e(\mathbf{x},t)~~~ on ~~~\Gamma_e ~~~~...
1
vote
1answer
132 views
How to solve this simple nonlinear ODE using the Galerkin's Method
I'm trying to solve a more complicated differential equation using the Galerkin's Method, but before that, I'm trying to understand how I would solve this simpler one:
$$ \cfrac{d^2u}{dx^2} + u^2 = 1;...
0
votes
1answer
63 views
Galerkin approximation for an elliptic BVP
What is the process for deducing Galerkin approximation for an elliptic boundary value problem?
For example, for a problem like--
$$
-\Delta u + cu = f\ \ \mathrm{in}\ Ī©
$$
$$
u = g\ \ \mathrm{on}\ ...
1
vote
1answer
50 views
RK4 gives nan for finite element galerkin method with 8+ basis
Exact Solution = $e^t \sin(\pi(x)) $
$f(x,t) = e^t(1-\beta \pi^2)\sin(\pi(x))$
$H = 1/N$
$ \phi_0 = (H - x)/H \hspace{5mm} in \hspace{5mm} [0, H] \hspace{5mm} else \hspace{5mm} 0 $
$ \phi_N = (x - ...
0
votes
0answers
40 views
Using finite element galerkin to solve the heat equation (homogeneous bcs)
I 'm trying to solve the 4 types of heat equations homogeneous linear, non-homogeneous and non linear. This is the code that I have written for linear homogeneous case for beta = -1 and N of your ...
0
votes
0answers
74 views
Singular matrix for solving Helmholtz PDE with Neumann boundary condition
I am trying to solve the Helmholtz PDE $-\nabla^2u + u = f \quad \text{on} \; \Omega$ with homogeneous Neumann boundary condition $\partial_n u = 0 \; \text{on} \; \partial \Omega$.
I noted that an ...
1
vote
1answer
88 views
Orthogonal projection in $L^2(\Omega)$ and $W_{0}^{1,2}(\Omega)$
While stydying the proof of the existence theorem for weak solutions for parabolic equations using the Galerkin approximation I encountered the following problem:
Assume that $\Omega \subseteq {\...
1
vote
0answers
38 views
Existence of Galerkin Approximation for Navier-Stokes equations
In some existence proof for the stationary incompressible Navier-Stokes equations I came across the following Lemma:
For $\Omega \subset \mathbb{R}^n$, $\psi _k \in C_c^\infty(\Omega)$ for $k=1,...,m$...
1
vote
1answer
95 views
Scope of (Petrov-)Galerkin methods
(I acknowledge that this question is somewhat conceptual, as I'm not an expert in the subject.)
I would like to understand what the general consensus is in the numerical analysis community on when to ...
1
vote
0answers
52 views
Galerkin Method - Beam Natural Modes
I try to solve Euler-Bernoulli Beam with Galerkin method, being clamped at one side and being free at the other to find the natural frequencies of vibration.
I assume the solution field as follows ...
-1
votes
1answer
805 views
What is the difference between trial and test functions in the context of numerical integration?
I know that in, for example, Galerkin's method we try to approximate the solution via a sum
$$
\sum_{j=1}^{n} u_{j} a\left(e_{j}, e_{i}\right)
$$
with $a\langle \cdot ,\cdot \rangle$ a bilinear form ...
0
votes
1answer
89 views
Stiffness Matrix Formation for PDE with Neumann Boundary
Given the problem $$-\nabla u + u = f$$ $$ n\cdot\nabla u = g \quad\text{on} \quad \Gamma$$ I can show the discretization given through the Galerkin formulation is $Au=b$ where $$ A = \int_\Omega \...
1
vote
1answer
193 views
Why the Galerkin Orthogonality Holds?
This is not homework. I'm going over my lecture notes to study for an exam.
For an Abstract Elliptic Problem such as the problem with V a Hilbert Space
$$\begin{cases}
\text{Find } u \in V \text{ ...
0
votes
1answer
91 views
About Finite Elements Method and stabilization using Upwind Scheme
I'm asking here cause I have a doubt about approximate solution of this problem:
\begin{equation}
\begin{cases}-\epsilon u''+bu'=0 \\ u(0) = 0, u(1)=1 \end{cases}
\end{equation}
which is a diffusion ...
0
votes
1answer
78 views
How Young inequality was applied?
Let $|\cdot|$ a norm in $L^2$ and $\|\cdot\|$ a norm in $H_0^1$. Then
$\begin{align}
&|u_h^n|^2 + 2 \theta \Delta t h^{-1} |u_h^n|\|u_h^{n+\theta}\| |u_h^{n+ \theta} - u_h^n| +2 \theta \Delta t |...
4
votes
2answers
408 views
Courant (1943) and History of Finite Element Method
I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called āOrthogonal ...
0
votes
1answer
73 views
FEM: Testing basis functions in a subspace V_h
In the finite element method, at a certain point we arrive at the following Galerkian problem where it is desired to find the solution $u_h \space \in V_h$ that solves the following equation:
$$ a(...
0
votes
0answers
155 views
Finite element method for nonlinear differential equation
I encounter this problem $$\frac{df(u(x))}{dx} = g(x)$$ with $$u(0) = u(1) = 0$$ I first convert it to weak form $f(u(x))v(x)]^1_0 - \int^{1}_0 \frac{dv(x)}{dx}f(u(x))dx =- \int^{1}_0 \frac{dv(x)}{dx}...
1
vote
1answer
59 views
Why of this bound $||u_{m0}|| \leq ||u_0||$, where $u_{m0}$ is a projection of $u_0$?
I'm dealing with Navier-Stokes equations, using the book of Teman. Theres a bound of a projection of a function that i didn't understood, so i will introduce the main concepts with i am dealing with.
...
0
votes
1answer
76 views
Flux Limiter for 2D Discontinuous Galerkin FEM
I want to learn about implementing convection-diffusion simulations using discontinuous Galerkin (DG) finite element methods to solve
$$
\dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J},
$$
...
1
vote
1answer
59 views
Regularity of coefficients in Galerkin method
Let $V\subset H\subset V^*$ be an evolution triple and suppose that $u\in W^{1,2}(0,T;V,H)$, where
$$W^{1,2}(0,T;V,H)=\{f\in L^2(0,T,V)\,|\,f'\in L^2(0,T,V^*)\}.$$
Now, let $\{w_1, w_2,...\}$ be a ...
3
votes
1answer
60 views
Basis functions for a Galerkin procedure
For a Galerkin procedure, I am trying to construct a set of linerly independent functions $\{\varphi_n\}_{n = 1}^N$ satisfying
$$
\varphi_n(0) = 0, ~~ \varphi_n'(1) = 0,
$$
for all $n \geq 1$.
A ...
0
votes
2answers
56 views
Weak closure of intersection in reflexive Banach space
Let $X$ be a reflexive Banach space. Let $\mathcal{S}$ be a family of finite-dimensional subspaces of $X$. Consider a bounded sequance $(x_Y)_{Y\in \mathcal{S}}\subset X$. Define
$$C_Y=\bigcup_{Y'\...
2
votes
0answers
160 views
Galerkin method, formulate the weak form, finite difference method for PDE
Consider the differential equation
$$ \frac{\text d^2 u}{\text d x^2} + \lambda_1 \frac{\text d u}{\text d x} + \lambda_2 u = -f(x),\quad\text{for}\quad x\in [a,b], $$
with boundary conditions $$ ...
1
vote
1answer
842 views
Understanding Galerkin method of weighted residuals
I have a puzzlement regarding the Galerkin method of weighted residuals. The following is taken from the book A Finite Element Primer for Beginners, from chapter 1.1.
If I have a one dimensional ...
1
vote
1answer
73 views
Galerkin $L^2$ Projections
I'm struggling with a question:
Consider the domain $ā¦ = (0, 1)$, and the space $V ā L^2(ā¦)$ spanned by the basis functions
$v_1 (x) = 1$
$v_2 (x) = 1 + x$
Find the $ĪØ^Ī“ ā V$ which is the Galerkin $L^...
1
vote
1answer
81 views
Why are boundary terms eliminated in the Galerkin method?
I'm trying to learn the Galerkin method for finite element.
I found this useful document with a 1D example about the stretching of a bar (pages 26 - 46).
This is the differential equation, boundary ...
2
votes
1answer
71 views
Help understanding subscript notation for Galerkin finite element
I'm following a few papers describing the Galerkin finite element method for a particular physical process. They all start with the same initial definition:
$$
h \approx \hat h(x,y,z,t) = \sum_{n=1}^...
1
vote
2answers
51 views
Does operator have to be linear for weak formulation?
Wikipedia says, that we find a weak formulation of the equation
$$
Au = f
$$
by defining a bilinear form $a(u,v)$. And in the examples, the operator $A$ is always linear, however it doesn't specify ...
1
vote
2answers
644 views
Heat equation energy estimates
We consider the following 3D and periodic Heat equation $$\partial_t u-\Delta u=0,~~~u(0)=u_0\in L^2(\mathbb{T}^3)~~\mbox{for } (t,x)\in \mathbb{R}_+\times \mathbb{T}^3$$
Where $u(t,x)$ stands for ...
0
votes
1answer
47 views
Inner product (functions) tending to infinity
I have a quite simple question, which I'm not really able to answer. Assume that you have to functions $f,g$ on an infinite dimensional function (normed) space. Define the usual inner product on a ...
2
votes
1answer
309 views
What is the difference between Orthogonal collocation and Weighted Residual Methods
I know that a lot of topics within FEM have already been dealt within in here.
However, I myself still need the big picture of FEM, and I were not able to retrieve it from the questions already being ...
3
votes
0answers
316 views
Galerkin method + FEM - clarification for Poisson equation with mixed boundary conditions
I will be refering to this link, but I am interested in slightly easier equation:
$$
-\Delta c = f, \quad (x, y) \in \Omega
$$
with the following (mixed) boundary condition:
$$
\begin{aligned}
1. &...
1
vote
0answers
89 views
Non-linear assumed form in Galerkin method
The wikipedia article on Method of mean weighted residuals has a section on choice of test functions, in particular it says about Galerkin method:
The Galerkin method, which uses the basis ...
4
votes
1answer
337 views
Why residual in Galerkin method is orthogonal to basis functions?
Let's consider equation of the form:
$L(f(x)) = g(x)$
In Galerkin method we substitute f(x) with it's approximation and we get residual of the form:
$$r(x) = \sum_{i=1}^N c_i \cdot \phi_i(x) - g(x)$$
...
1
vote
0answers
43 views
Galerkin Approximation for Finite Elements
so I've been trying my hand at this approximation but I can't quite seem to get a good answer. Would really appreciate if anyone can identify where I've gone wrong. Given a first order DE:
$$u''-\...
3
votes
0answers
161 views
What is a test function and why do we need them?
I'm taking a class right now (graduate level), an Introduction to Nodal Discontinuous Galerkin methods. I've got a little numerical analysis background, meaning I've covered a lot of topics over the ...
