Questions tagged [galerkin-methods]
97 questions
8
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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
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8
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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 weakly…
Aron
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6
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
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4
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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
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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: $$ \begin{aligned} 1. & \; c = g_1, \quad (x, y) \in \Gamma_1 \\ 2. & \;…
user16320
- 746
4
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1 answer
Why residual in Galerkin method is orthogonal to basis functions?
Let's consider equation of the form: $L(f(x)) = g(x)$ In Galerkin method we substitute f(x) with it's approximation and we get residual of the form: $$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)$$ with $u(0) = u(1) = a(0)$ , using a Galerkin method. We start by defining a mesh $x_j=jh$ over $N$ points. We take…
Wanderer
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4
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Existence of time derivative in the Galerkin equation of parabolic PDEs
Good day, Let's take this initial-boundary value parabolic PDE \begin{align} \partial_t u + Lu &=f \text{ in } \Omega_T=\Omega \times (0,T] \\ u&=0 \text{ on } \partial \Omega \times (0,T) \\ u(x,0)&=u_0(x) \text{ for } x \in \Omega \end{align}…
Fritz
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4
votes
1 answer
Galerkin method and Schauder basis.
While studying the books: Monotone operators in Banach space and nonlinear partial differential equations from Showalter and Quelques méthodes de résolution des problèmes aux limites non linéaires from Lions , in the part of nonlinear evolution…
Tomás
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3
votes
0 answers
Matrix formulation to the BVP using galerkin method
Derive a matrix formulation to the BVP using galerkin method: \begin{align} \frac{\partial^2f}{dx^2}+\frac{\partial^2f}{dy^2}&=0\text{ within }A\in(0,2)\times(0,2)\\ f=0\text{ on }C_1&:x=0,y=\pm2\\ \frac{\partial f}{\partial x}=K^\star\text{ on…
N00BMaster
- 721
3
votes
1 answer
PDEs: Derive source term of discontinuous advection equation (DGFEM)
We are looking at the equation $$ \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
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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
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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…
WaltherArgyris
- 158
3
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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
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