Questions tagged [galerkin-methods]

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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 \...
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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!
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 $$\frac{du_1}{dx}|_{x=0}=1,\frac{du_2}{dx}|_{x=1}=2 $$ $$x \in\Omega=[0,1]$$ ...
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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 ...
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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, ...
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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. ...
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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 ...
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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 ...
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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 ~~~~...
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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;...
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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-- $$ -\Delta u + cu = f\ \ \mathrm{in}\ Ω $$ $$ u = g\ \ \mathrm{on}\ ...
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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 - ...
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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 ...
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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 ...
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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 {\...
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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 $k=1,...,m$...
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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 ...
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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 ...
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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 ...
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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 \...
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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{ ...
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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 ...
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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 |...
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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 “Orthogonal ...
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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(...
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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}...
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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. ...
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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 $$ \dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J}, $$ ...
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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 ...
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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 ...
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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'\...
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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 $$ ...
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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 ...
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1answer
73 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^...
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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 ...
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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}^...
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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 ...
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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 ...
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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 ...
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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 ...
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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. &...
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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 ...
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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: $$r(x) = \sum_{i=1}^N c_i \cdot \phi_i(x) - g(x)$$ ...
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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''-\...
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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 ...