Questions tagged [galerkin]
The galerkin tag has no usage guidance.
23
questions
3
votes
0
answers
143
views
Galerkin Method - Why does integration-by-parts eliminate need to enforce Neumann boundaries?
I've posted this to MathStackexchange but I figured I'd also post here as well as I have yet to receive an answer on my original post, and that I would be more likely to encounter users of the ...
2
votes
1
answer
181
views
Galerkin projection in AMG
In the context of Classical AMG for elliptic problems discretised with finite elements (DG or CG), one has the (fine) matrix of the problem, say $A_0$, and the coarser operators of the hierarchy $\{...
- 405
1
vote
2
answers
116
views
How to handle non bilinear weak form?
I solved the 2D heat equation using the finite element method. It all went well first with the adiabatic case, however problems occured when I introduced cooling with the enviroment.
I modeled the ...
- 63
2
votes
2
answers
35
views
How do we implement the balance of stress on interface in ALE FSI method?
""we aim for a consistent variational-monolithic coupling scheme in which we need all equations defined in the same domain; therefore, $\mathrm{ALE}_{f x}$ was introduced. In variational ...
- 193
0
votes
1
answer
52
views
How to implement the interface extension of fluid "displacement" in ALE?
In ALE, we first set a referenced space for fluid, then we extend the boundary fluid displacement to the whole fluid region, take harmonic extension as an example, we need $$\Delta \left ( \hat{u} \...
- 193
5
votes
1
answer
298
views
Does the weighted residual method not use energy minimization in any form?
I've come across several texts/papers utilizing the concept of a minimum potential energy state corresponding to an equilibrium state, and I know that it is used in FEM formulations that are based on ...
- 73
0
votes
0
answers
90
views
A Finite Element Method for a first order PDE?
I want to develop a finite element method to solve for $u(x,t)$ the PDE:
$$u_t+c u_x= \frac{-c}{x}u$$
where $c$ is a constant.
so I am trying the following ( as Rothe's method? ) :
Letting $k= t_n- ...
- 151
0
votes
0
answers
54
views
Can you describe the Galerkin numerical method to solve the wave equation?
How would you describe the Galerkin method to solving the 3D wave equation
$$u_{tt}= c^2\Delta u$$
to someone who wants to implement it immediately?
More precisely, we want to solve the Cauchy problem
...
- 151
1
vote
1
answer
84
views
Roothaan equations and Galerkin method
When we do Hartree-Fock computations by solving the Roothaan equations $FC = SC \varepsilon$ is it a Galerkin method?
- 331
3
votes
1
answer
208
views
How to measure the error of Finite Element approximation in satisfying the PDE?
In Galerkin methods, we seldom can measure the accuracy of an approximation by tracking the value of the residual. For example, take the wave equation:
$ u_{tt} = u_{xx}$, $(x,t) \in (0,L) \times (0,T)...
- 131
1
vote
1
answer
224
views
3d schrodinger equation weak form
SCHRODINGER’S EQUATION
$$-ih u_{t}(x,y,z,t) = \frac{h^2}{2m} u_{xx}(x,y,z,t)+ \frac{e^2}{r}u(x,y,z,t)$$
The potential $\frac{e^2}{r}$ is a variable coefficient.
So, let’s take the free Schrodinger ...
user38211
5
votes
0
answers
357
views
Galerkin Least-Squares stabilization for FEM solution advection (hyperbolic) equations
I am playing with Galerkin Least-Squares stabilization to solve advection diffusion problem in the context of the finite element method. This works very well for steady-state advection-diffusion ...
- 1,157
0
votes
1
answer
210
views
Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?
I am using the Galerkin method (Discontinuous to be precise) to discretize in space the scalar linear wave equation and the explicit second order centered finite difference scheme to discretize in ...
- 155
0
votes
0
answers
54
views
Unsteady Stokes equations in ALE framework
I'm trying to solve Unsteady Stokes equations on a moving domain, using an ALE formulation, that is
$$\frac{\partial \mathbf{u}}{\partial t} - \mathbf{w}\cdot \nabla\mathbf{u} = \nu\Delta\mathbf{u} - \...
- 11
0
votes
0
answers
115
views
How to solve odd-order differential equations in FEM? Petrov-Galerkin?
I've recently learned about using weighted residuals with the Galerkin method to numerically approximate even-order differential equations (for linear elements, I'm still a beginner). It seems for odd-...
- 9
3
votes
1
answer
364
views
What is the difference between Methods of Weighted Residuals and Spectral Methods?
Methods of Weighted Residuals (MWR) [1] usually include Galerkin, collocation, method of moments, least-squares and subdomain methods.
Spectral methods [2] usually include Galerkin, tau and ...
- 188
5
votes
1
answer
1k
views
Is a symmetric bilinear form necessary to ensure a weak formulation has a solution?
Problem
I want to convert the general second order linear PDE problem
\begin{align}
\begin{cases}
a(x,y)\frac{\partial^2 u}{\partial x^2}+b(x,y) \frac{\partial^2 u}{\partial y^2} +c(x,y)\frac{\...
- 165
0
votes
1
answer
105
views
Interpolation of function onto mesh gives different results, depending on mesh density
I wanted to test the numerical accuracy of my program. For that I wanted to interpolate the function $$f=I_0\exp\left(-100x^2\right)\exp(-100y^2)$$ onto a grid, defined on $$\Omega=[0,1]^2$$ by using ...
- 563
3
votes
3
answers
631
views
$H^1$-convergence rate of finite element method for Poisson equation, depending on element order
I wanted to verify my FEM-program by applying the method of manufactured solutions, while solving the Poisson equation in two dimensions using the continuous Galerkin method
$$-\nabla^2u=f$$
with
$$u=...
- 563
2
votes
0
answers
44
views
Solving Compressible Euler in Primitive Variables with Galerkin P1 FEM
I have implemented a small compressible Euler solver, discretizing in primitive variables (rho, u, v, p) with standard Galerkin FEM P1 triangular elements, and mixed isotropic and anisotropic/...
- 21
2
votes
1
answer
163
views
Prevent single node spikes in a FEM-simulation (using continuous Galerkin)
I am trying to solve a non-linear time-dependent heat equation
$$\partial_tT=\nabla \left(k_T(T)\nabla T\right) + f$$ (similar to question Solving a non-linear heat equation with the galerkin method ...
- 563
4
votes
4
answers
912
views
Role of weight function in Galerkin methods
I have difficulties in understanding the role of the weight function $w(x)$ that occurs in the solution of PDEs via the Galerkin approach. Consider a linear differential equation of the form
$$
\...
- 3,197
5
votes
2
answers
353
views
Motivation behind Collocation Method
In the previous question "Motivation behind Galerkin method", Paul gives a good and easy-to-understand explanation indicating that the Galerkin method is a kind of projection method. Can anyone ...
- 53