Questions tagged [galerkin-methods]
98 questions
8
votes
1 answer
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 dimensional problem by introducing a $n$-dimensional subspace…
tom
- 4,737
- 1
- 23
- 43
8
votes
2 answers
Galerkin method for Poisson's equation
This is problem 3 from chapter 7 of Evans book:
Suppose $f\in L^2(U)$ and assume that $u_m=\sum_{k=1}^md_m^kw_k$ solves $$\int_UDu_m\cdot Dw_k=\int_Uf\cdot w_kdx$$
for $k=1,...,m$. Show that a subsequence of $\{u_m\}_{m=1}^\infty$ converges…
Aron
- 151
5
votes
2 answers
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 Collocation Revisited” which has a brief section on…
L. Young
- 71
4
votes
1 answer
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 trivial set satisfying the above boundary considitions…
4
votes
1 answer
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 differential equation $A(u)=f$, and an approximate…
Ohm
- 187
4
votes
1 answer
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:
$$
\begin{aligned}
1. & \; c = g_1, \quad (x, y) \in \Gamma_1 \\
2. & \;…
user16320
- 746
4
votes
1 answer
Why residual in Galerkin method is orthogonal to basis functions?
Let's consider equation of the form:
$L(f(x)) = g(x)$
In Galerkin method we substitute f(x) with it's approximation and we get residual of the form:
$$r(x) = \sum_{i=1}^N c_i \cdot \phi_i(x) - g(x)$$
Where the $c_i$ are coefficients and $\phi_i(x),…
maximus
- 177
4
votes
1 answer
Stiffness matrix for Galerkin method
I am working on a finite element approximation to solve the following ODE type:
$$-\dfrac{d}{dx}\bigg(a(x)\dfrac{du}{dx}\bigg)=f(x)$$
with $u(0) = u(1) = a(0)$, using a Galerkin method. We start by defining a mesh $x_j=jh$ over $N$ points. We take…
Wanderer
- 41
4
votes
1 answer
Existence of time derivative in the Galerkin equation of parabolic PDEs
Good day,
Let's take this initial-boundary value parabolic PDE
\begin{align} \partial_t u + Lu &=f \text{ in } \Omega_T=\Omega \times (0,T] \\
u&=0 \text{ on } \partial \Omega \times (0,T) \\
u(x,0)&=u_0(x) \text{ for } x \in \Omega
\end{align}
…
Cahn
- 4,623
4
votes
1 answer
Galerkin method and Schauder basis.
While studying the books: Monotone operators in Banach space and nonlinear partial differential equations from Showalter and Quelques méthodes de résolution des problèmes aux limites non linéaires from Lions, in the part of nonlinear evolution…
Tomás
- 22,985
3
votes
0 answers
Matrix formulation to the BVP using galerkin method
Derive a matrix formulation to the BVP using galerkin method:
\begin{align}
\frac{\partial^2f}{dx^2}+\frac{\partial^2f}{dy^2}&=0\text{ within }A\in(0,2)\times(0,2)\\
f=0\text{ on }C_1&:x=0,y=\pm2\\
\frac{\partial f}{\partial x}=K^\star\text{ on…
N00BMaster
- 721
3
votes
1 answer
PDEs: Derive source term of discontinuous advection equation (DGFEM)
We are looking at the equation
$$
\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 asked to derive $g(x,t)$ such that $u(x, t)=\sin (\pi(x-a(x)…
3
votes
0 answers
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 Finite Element Method.
I've been reading this book…
Quertie
- 31
3
votes
2 answers
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 denoted by $\mathbf{q}$, and let the source term…
WaltherArgyris
- 158
3
votes
1 answer
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!
Yidong Luo
- 451
- 2
- 12