Questions tagged [galerkin-methods]
The galerkin-methods tag has no usage guidance.
96
questions
2
votes
1
answer
89
views
Evans - existence of parabolic PDE, why does $B(u_m,v)\to B(u,v)$?
In Evans book, chapter 7.1, he establishes existence of weak solutions of
$$\partial_t u + Lu = f$$
where $L$ is a uniformly elliptic differential operator.
He first shows that for any $m$, the ...
- 120
2
votes
1
answer
56
views
Weak form FEM discretisation of Non-Linear System
I am trying to derive the FEM solution for the equation $u''(x) + \left( u'(x) \right)^2=0$ with $u(1)=0, u'(0)=1$ over the interval $[0, 1]$.
Constrain the trial function, $v$, to also be zero at $x=...
- 73
1
vote
0
answers
59
views
Galerkin Method - Why does integration-by-parts eliminate need to enforce Neumann boundaries?
Disclaimer: I have not taken any class in functional analysis and the only class I have ever taken in differential equations is at the barest-bones introductory level.
From what I know of the Galerkin ...
0
votes
0
answers
27
views
Seeking Insights: Partial Differential Equation and Galerkin Method
I'm trying to solve the evolution problem below using the standard Galerkin method:
...
- 111
0
votes
0
answers
35
views
Which subspaces are considered in the Bubnov-Galerkin method?
A PDE problem in the form
$$−∇⋅(k(x)∇u)=f$$
can be expressed in the form
$$Au=f$$
where the linear operator $A$ is defined by the expression
$$Au=−∇⋅(k(x)∇u)$$
In the Bubnov-Galerkin method, $u$ is ...
- 315
2
votes
0
answers
83
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 ...
- 101
1
vote
1
answer
157
views
Essential Boundary Condition and Natural Boundary Conditions in Weak Form Galerkin
Some days ago my teacher gave me a question about using weak form Galerkin to solve an ODE. I'm not so good at solving Differential Equations so I'm here asking for some help(By the way my English is ...
- 11
3
votes
0
answers
94
views
Matrix formulation to the BVP using galerkin method
Derive a matrix formulation to the BVP using galerkin method:
\begin{align}
\frac{\partial^2f}{dx^2}+\frac{\partial^2f}{dy^2}&=0\text{ within }A\in(0,2)\times(0,2)\\
f=0\text{ on }C_1&:x=0,y=\...
- 701
0
votes
0
answers
27
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
2
answers
93
views
Why does my implementation of the galerkin method yield the correct values, but with inverted sign
I tried to implement the galerkin-method for the one dimensional poisson equation. I chose for the boundary conditions
$$
u(0) = u(1) = 0
$$
and for the "source" term $f(x)$
$$
\frac{\...
3
votes
1
answer
113
views
PDEs: Derive source term of discontinuous advection equation (DGFEM)
We are looking at the equation
$$
\partial_t u+a(x) \partial_x u=g(x, t), \quad x \in[-2,2]
$$
with
$$
a(x)= \begin{cases}1.5 & |x| \leq 0.5 \\ 1 & \text { otherwise }\end{cases}
$$
We are ...
0
votes
1
answer
75
views
Basis functions for Galerkin approximation of BVP
Given the following boundary-value problem (BVP) for $\lambda$:
$$y^{(4)}+ay''+\lambda b y'+\lambda^2 y=0,\quad y=y(x),\quad 0\le x\le 1,\\ y(0)=y''(0)=y''(1)=y^{(3)}+ay'(1)=0$$ how can we ...
- 147
0
votes
1
answer
58
views
Numerical methods for $u_t+c u_x= \frac{-c}{x}u$?
I am looking for possible numerical methods to solve the PDE
$$u_t+c u_x= \frac{-c}{x}u$$
for $u(x,t):\mathbb{R}\times \mathbb{R} \to \mathbb{R}$ and where $c>0$ is a constant.
I am particularly ...
- 940
0
votes
0
answers
110
views
Can you describe the Galerkin numerical method to solve the wave equation?
How would you describe the Galerkin method to solving the wave equation
$$u_{tt}= c^2\Delta u$$
to someone who wants to implement it immediately?
More precisely, we want to solve the Cauchy problem
$$...
- 940
3
votes
0
answers
207
views
What part of the Galerkin method ensures a solution with accurate nodal values?
I graduated over a year ago as a mechanical/industrial engineer and I've recently been re-studying my last year engineering courses that focused on numerical methods for simulation, including the ...
- 31
1
vote
1
answer
617
views
Finite elements method: why test fuction vanishes on boundaries
I am trying to understand why (and exactly when) the test functions must vanish at the boundaries when Dirichlet conditions are applied to a PDE. The context is the learning of the finite element ...
- 125
2
votes
0
answers
68
views
Radial basis functions for spectral solution of PDE in spherical coordinates
I want to solve the following PDE defined in 3D-space and time:
$(z \partial_t-F(t)\partial_z)f(t,\vec{x})+C[f]=S(t,z,r),$
where $r=\sqrt{x^2+y^2+z^2}$ and $C[f]$ is a linear integral operator. The ...
- 21
1
vote
1
answer
206
views
General non-FEM Galerkin, boundary conditions
Consider the weak form of our all-time favorite Poisson-equation,
$$- \int_{\Omega} \nabla u \cdot \nabla v\, \mathrm{d} x = \int_{\Gamma}v \nabla u \cdot \mathrm{d}x +\int_{\Omega} fv\, \mathrm{d} ...
- 283
3
votes
2
answers
768
views
Conservation laws in the finite element method for domains with curved boundaries
Consider, for concreteness, the finite element (FE) method applied to stationary heat conduction in a domain $\Omega \subset \mathbb{R}^3$. Let the heat flux (thermal energy per unit area and time) be ...
- 158
0
votes
1
answer
310
views
Test function for Weak formulation
Can anybody give an elaborate explanation as to why we multiply a test function on both sides of the boundary value problem and then integrate it into the procedure of weak formulation?
e.g A BVP
$$ -\...
1
vote
0
answers
53
views
Variational formulation using galerkin scheme
Consider the boundary problem
$$
\dfrac{\partial u}{\partial t}+ A^2 u+Af(u)+\lambda g(u)=0\ \ \ \ \ \mbox{ in }\ \ \ \Omega.
$$
$$
\dfrac{\partial u}{\partial \nu}=\dfrac{\partial \Delta u}{\partial \...
- 11
3
votes
1
answer
216
views
References for discontinuous Galerkin method with one dimensional case?
Could anyone recommend some references on discontinuous Galerkin method for beginners? It is nice to contain the one-dimensional case. Thanks in advance!
- 439
1
vote
1
answer
207
views
Chebyshev differentiation matrices and Galerkin method
I've been reading the "Spectral Methods in Matlab" by Lloyd N. Trefethen and I'm interested in solving PDE's with spectral methods. From what I understand this book gives us a method (at ...
- 403
1
vote
0
answers
102
views
Finite-dimensional truncations of unbounded operators and resolvent convergence
Let $A$ be an unbounded self-adjoint operator in a Hilbert space $\mathcal{H}$, with domain $\mathrm{dom}\,A$, and consider a growing sequence $(P_n)_{n\in\mathbb{N}}$ of orthogonal projectors in $\...
- 393
3
votes
1
answer
167
views
Requirements for Korn's inequality on piecewise $H_1$ vector field
I am looking at the Korn's inequality on $H^1$ vector fields, as described in this paper by Brenner. In particular, I am looking at how the seminorms defined in examples 2.3 - 2.5 satisfy the ...
- 115
0
votes
1
answer
153
views
Inequality between inner product of functions in dual space
I'm reading Brezzi's paper on DG method and is currently puzzled on how equations (5.7) is used to derive equation (5.8).
Further searches lead me to learning that the space defined $\mathbf{V}'$ is ...
- 115
0
votes
0
answers
47
views
Help deriving a Fourier-Galerkin approximation for a variable coefficient partial differential equation
I've been trying to derive a Fourier-Galerkin approximation of
\begin{equation}
\frac{\partial u}{\partial t} + \sin(x) \frac{\partial u}{\partial x} = 0
\end{equation}
But I'm not sure that my steps ...
- 155
2
votes
1
answer
364
views
Galerkin method for system of ode's
I have the following system of ode's
$$
\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]$$
...
- 125
3
votes
2
answers
928
views
Galerkin method for nonlinear ode
I'm trying to solve the following differential equation:
$$\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$$
You can see that the right side ...
- 125
2
votes
1
answer
488
views
Solving 1D Poisson equation using finite element method and understanding the Galerkin orthogonality
Let's consider the following test problem
$$
u'' = 12x^2 - 36x + 18 \qquad u(0) = u(3) = 0
$$
Analytical solution is
$$
u(x) = (x-3)^2 x^2
$$
I'm solving this using the finite element method, ...
- 21
0
votes
1
answer
309
views
FEM: Steady-State heat diffusion and convection
So the strong form of the heat diffusion and convection PDE is given as
$\rho c_m \mathbf{v}\cdot\nabla T - \nabla \cdot \nabla T = \dot{q}\\
T(\mathbf{x},t) = T_e(\mathbf{x},t)~~~ on ~~~\Gamma_e ~~~~...
- 163
2
votes
1
answer
547
views
How to solve this simple nonlinear ODE using the Galerkin's Method
I'm trying to solve a more complicated differential equation using the Galerkin's Method, but before that, I'm trying to understand how I would solve this simpler one:
$$ \cfrac{d^2u}{dx^2} + u^2 = 1;...
- 245
0
votes
1
answer
114
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--
$$
-\Delta u + cu = f\ \ \mathrm{in}\ Ω
$$
$$
u = g\ \ \mathrm{on}\ ...
1
vote
1
answer
175
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
0
answers
213
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
0
answers
165
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 ...
- 19
1
vote
1
answer
354
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 {\...
- 511
1
vote
0
answers
60
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$...
- 451
1
vote
1
answer
160
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 ...
- 163
1
vote
0
answers
127
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 ...
- 103
1
vote
1
answer
4k
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)
$$
with $a\langle \cdot ,\cdot \rangle$ a bilinear form ...
- 367
0
votes
1
answer
332
views
Stiffness Matrix Formation for PDE with Neumann Boundary
Given the problem $$-\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 \...
- 242
1
vote
1
answer
2k
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
$$\begin{cases}
\text{Find } u \in V \text{ ...
- 53
0
votes
1
answer
161
views
About Finite Elements Method and stabilization using Upwind Scheme
I'm asking here cause I have a doubt about approximate solution of this problem:
\begin{equation}
\begin{cases}-\epsilon u''+bu'=0 \\ u(0) = 0, u(1)=1 \end{cases}
\end{equation}
which is a diffusion ...
- 1,963
0
votes
1
answer
102
views
How Young inequality was applied?
Let $|\cdot|$ a norm in $L^2$ and $\|\cdot\|$ a norm in $H_0^1$. Then
$\begin{align}
&|u_h^n|^2 + 2 \theta \Delta t h^{-1} |u_h^n|\|u_h^{n+\theta}\| |u_h^{n+ \theta} - u_h^n| +2 \theta \Delta t |...
5
votes
2
answers
1k
views
Courant (1943) and History of Finite Element Method
I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called “Orthogonal ...
- 71
0
votes
1
answer
160
views
FEM: Testing basis functions in a subspace V_h
In the finite element method, at a certain point we arrive at the following Galerkian problem where it is desired to find the solution $u_h \space \in V_h$ that solves the following equation:
$$ a(...
0
votes
0
answers
217
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}...
- 851
1
vote
1
answer
64
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
1
answer
190
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
$$
\dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J},
$$
...
