Questions tagged [galerkin-methods]
The galerkin-methods tag has no usage guidance.
1
vote
1answer
26 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
21 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
35 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
48 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 $$ A = \int_\Omega \...
1
vote
1answer
59 views
Why the Galerkin Orthogonality Holds?
This is not homework. I'm going over my lecture notes to study for an exam.
For an Abstract Elliptic Problem such as the problem with V a Hilbert Space
$$\begin{cases}
\text{Find } u \in V \text{ ...
0
votes
0answers
16 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
68 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
129 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 “Orthogonal ...
0
votes
1answer
53 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:
$$ a(...
0
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0answers
68 views
Finite element method for nonlinear differential equation
I encounter this problem $$\frac{df(u(x))}{dx} = g(x)$$ with $$u(0) = u(1) = 0$$ I first convert it to weak form $f(u(x))v(x)]^1_0 - \int^{1}_0 \frac{dv(x)}{dx}f(u(x))dx =- \int^{1}_0 \frac{dv(x)}{dx}...
0
votes
0answers
43 views
clarification on Galerkin method - in finite element method solution
I am unable to understand the marked step in the solution to the problem on finite element method (Galerkin Method). I shall be grateful for clarification. Mentioned Equations 24 and 32 have been ...
1
vote
1answer
47 views
Why of this bound $||u_{m0}|| \leq ||u_0||$, where $u_{m0}$ is a projection of $u_0$?
I'm dealing with Navier-Stokes equations, using the book of Teman. Theres a bound of a projection of a function that i didn't understood, so i will introduce the main concepts with i am dealing with.
...
1
vote
0answers
31 views
Relationship between Fundamental Lemma of Calculus of Variations and the Weighted-Residual Statement
The Weighted-Residual Method states that the integral of the Residual R(x) times the weighting function w(x) is equal to zero which means that R(x) = 0
On the other hand, Fundamental Lemma of ...
0
votes
0answers
52 views
Advection-Diffusion across an interface using IPDG FEM
In the the Advection-Diffusion equation,
$$
\dfrac{\partial C}{\partial t} - \nabla \cdot (D \nabla C + \mathbf{u} C) = 0,
$$
do I have to add any terms to stabilize the flux in my finite element ...
0
votes
1answer
27 views
Flux Limiter for 2D Discontinuous Galerkin FEM
I want to learn about implementing convection-diffusion simulations using discontinuous Galerkin (DG) finite element methods to solve
$$
\dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J},
$$
...
1
vote
1answer
52 views
Regularity of coefficients in Galerkin method
Let $V\subset H\subset V^*$ be an evolution triple and suppose that $u\in W^{1,2}(0,T;V,H)$, where
$$W^{1,2}(0,T;V,H)=\{f\in L^2(0,T,V)\,|\,f'\in L^2(0,T,V^*)\}.$$
Now, let $\{w_1, w_2,...\}$ be a ...
3
votes
1answer
51 views
Basis functions for a Galerkin procedure
For a Galerkin procedure, I am trying to construct a set of linerly independent functions $\{\varphi_n\}_{n = 1}^N$ satisfying
$$
\varphi_n(0) = 0, ~~ \varphi_n'(1) = 0,
$$
for all $n \geq 1$.
A ...
0
votes
2answers
46 views
Weak closure of intersection in reflexive Banach space
Let $X$ be a reflexive Banach space. Let $\mathcal{S}$ be a family of finite-dimensional subspaces of $X$. Consider a bounded sequance $(x_Y)_{Y\in \mathcal{S}}\subset X$. Define
$$C_Y=\bigcup_{Y'\...
2
votes
0answers
75 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 $$ ...
1
vote
1answer
226 views
Understanding Galerkin method of weighted residuals
I have a puzzlement regarding the Galerkin method of weighted residuals. The following is taken from the book A Finite Element Primer for Beginners, from chapter 1.1.
If I have a one dimensional ...
1
vote
1answer
41 views
Galerkin $L^2$ Projections
I'm struggling with a question:
Consider the domain $Ω = (0, 1)$, and the space $V ⊆ L^2(Ω)$ spanned by the basis functions
$v_1 (x) = 1$
$v_2 (x) = 1 + x$
Find the $Ψ^δ ∈ V$ which is the Galerkin $L^...
1
vote
1answer
46 views
Why are boundary terms eliminated in the Galerkin method?
I'm trying to learn the Galerkin method for finite element.
I found this useful document with a 1D example about the stretching of a bar (pages 26 - 46).
This is the differential equation, boundary ...
2
votes
1answer
39 views
Help understanding subscript notation for Galerkin finite element
I'm following a few papers describing the Galerkin finite element method for a particular physical process. They all start with the same initial definition:
$$
h \approx \hat h(x,y,z,t) = \sum_{n=1}^...
1
vote
2answers
35 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 ...
0
votes
2answers
249 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
36 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
150 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
175 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:
$$
\begin{aligned}
1. &...
1
vote
0answers
71 views
Non-linear assumed form in Galerkin method
The wikipedia article on Method of mean weighted residuals has a section on choice of test functions, in particular it says about Galerkin method:
The Galerkin method, which uses the basis ...
2
votes
0answers
147 views
Why residual in Galerkin method is orthogonal to basis functions?
Let's consider equation of the form:
$L(f(x)) = g(x)$
In Galerkin method we substitute f(x) with it's approximation and we get residual of the form:
$$r(x) = \sum_{i=1}^N c_i \cdot \phi_i(x) - g(x)$$
...
1
vote
0answers
28 views
Galerkin Approximation for Finite Elements
so I've been trying my hand at this approximation but I can't quite seem to get a good answer. Would really appreciate if anyone can identify where I've gone wrong. Given a first order DE:
$$u''-\...
3
votes
0answers
90 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 ...
2
votes
0answers
60 views
Convergence rate of finite element method?
Suppose I have a convex polygon $\Omega$ with $n$ sides and I want to find eigenvalues of the Laplacian with mixed Robin boundary conditions:
$$
-\Delta u=k^2u\text{ in }\Omega
$$
$$
a_iu+b_i\frac{\...
2
votes
0answers
381 views
Stiffness Matrix for Galerkin Method (Finite Element Approx)
I'm working on a finite element approximation to solve the following ODE type:
$-\dfrac{d}{dx}\bigg(a(x)\dfrac{du}{dx}\bigg)=f(x)$
$u(0) = u(1) = a(0)$ using a Galerkin method.
We start by defining ...
0
votes
1answer
57 views
Galerkin Method
Can someone please explain this to me, the part where we take the integrals, why is it not $ \frac{1}{ \Delta x^{2}} $ for K11 how do they get 2\deltax >?
2
votes
0answers
267 views
Numerical Integration of a Fredholm Integral Equation of the First Kind
I'm trying to test out a numerical integration scheme for a Fredholm integral equation of the first type with an integral equation that has an exact solution. My test equation is:
$$
\int_{-1}^{1}\...
1
vote
1answer
68 views
Difference between Galerkin approximations and Yosida approximation in Open Problem Navier Stokes Equations
To approximate a solutions of Navier-Stokes equation, we use sometime Galerkin approximation and some time we use Yosida approximations. Why we use two different approximations for approximate the ...
1
vote
1answer
321 views
Relation between global and local basis functions in finite element methods
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be a bounded domain
$\mathcal T$ be a finite set of compact subsets of $\mathbb R^d$ with $$\stackrel{\circ}K_1\cap\stackrel{\circ}K_2=\emptyset\;\;\;...
2
votes
0answers
193 views
Evaluating a lift operator
In a lot of papers discussing discontinuous Galerkin methods for solving elliptic/parabolic PDE's, they mention a "lift" operator.
For example, consider a simple Poisson system:
\begin{gather}
\nabla ...
1
vote
1answer
50 views
Computing error towards reference solution
In a term paper about Finite Elements, I need to compute empirical convergence rates without knowing the exact solution of an elliptic PDE.
Given a family of grids $\mathcal{T}_h$ with $h \in \{h_1, \...
0
votes
1answer
104 views
Choice of the global basis functions in the Galerkin finite element method
Let
$a,b\in\mathbb R$ with $a<b$
$\Lambda:=(a,b)$
$n\in\mathbb N$
$h:=(b-a)/n$
$z_k:=a+kh$ for $k\in\left\{0,\ldots,n\right\}$
$e_k:=[z_{k-1},z_k]$
$r\in\mathbb N$
Moreover, let $$\tilde V:=\left\...
3
votes
1answer
179 views
Existence of time derivative in the Galerkin equation of parabolic PDEs
Good day,
Let's take this initial-boundary value parabolic PDE
\begin{align} \partial_t u + Lu &=f \text{ in } \Omega_T=\Omega \times (0,T] \\
u&=0 \text{ on } \partial \Omega \times (0,...
0
votes
0answers
297 views
hierarchical basis functions finite element
I am using the hierarchical basis functions on the reference element $[-1,1]$ which are
$$\theta_1(\xi)=1/2(1-\xi) $$
$$\theta_2(\xi)=1-\xi^2$$
$$\theta_3(\xi)=1/2(1+\xi) $$
By using the Galerkin ...
0
votes
1answer
466 views
solution of poisson problem with finite element method in 1D
I am going to solve
$$-u''=-1-10\delta (x-0.2)$$
$$u(0)=u(1)=0$$
by finite element method in 1D.
The exact solution that I got is
$$u(x) = \left\{ {\begin{array}{*{20}{c}}
{\frac{{{x^2}}}{2} - \...
0
votes
1answer
74 views
How exact do you need to approximate integrals in the finite element method
Consider the finite element method for the approximate solution of the equation
$$
\nabla\cdot(a(x)\nabla u(x))=f(x) \text{ (in some domain)}
$$
or, in weak form (assuming the right boundary ...
7
votes
0answers
163 views
Manifold Galerkin method
Standard Galerkin method reduces the problem
Find $u\in V$ such that $a(u,v) = f(v)$ for all $v \in V$,
where $V$ is Hilbert space, $a$ is bilinear form and $f\in V^*$.
to a finite ...
0
votes
1answer
156 views
Galerkin method for initial-boundary value problem
Consider to
$$u_{t}(x,t)=u_{xx}(x,t)$$
$$u(0,t)=u(1,t)=0$$
$$u(x,0)=f(x)$$
I want to solve this problem by the Galerkin method based on finite dimensional space $X_{N}$, please help me. ...
2
votes
0answers
87 views
The 3rd term of the energy estimates in chapter 7 Evans PDE
Hi I wonder someone could help me check my understanding of getting an inequality of the estimate for the 3rd term
$\|u'_m\|_{L^2(0,T;H^{-1}(U))}$ correctly.
This inequality need to be checked is ...
1
vote
1answer
304 views
energy estimates Evans PDE chapter 7
Hi I am looking at the proof of theorem 2 energy estimates in Evans PDE. I have some difficulties regarding the estimate for each term.
First for the first term.
Q1 I am a little vague how (23) is ...
1
vote
1answer
218 views
Intuition behind steps in formulating Finite Element Method
Let's consider the classic elastostatics case where the strong form of the PDE is:
$\sigma _{ij,j}+b_i =0$ on V
By multiplying through by weighting functions and integrating we can create an ...
