Questions tagged [weak-solution]
For questions about weak solutions to differential equations. These typically involve finding a function which satisfies some transformed (typically integrated) version of the differential equation for a class of smooth test functions.
21 questions
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Query about FE approximation of a Poisson equation with non-constant coefficients
Consider the standard weak form of the Poisson equation with coefficient $\alpha$:
\begin{equation}
\int_\Omega \dfrac{\partial v}{\partial x} \dfrac{ \partial \left( \alpha u \right) }{\partial x} = ...
- 506
0
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0
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74
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How to set Neumann BC for coupled transport problem in weak form?
Consider
$$\begin{aligned}
\partial_t v + b\cdot \nabla \phi &=0 \\
\partial_t \phi + b\cdot \nabla v &= 0
\end{aligned}$$
for $v:(x,y)\mapsto \mathbb{R}$, unknown and time-dependent ($...
- 90
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0
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159
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Interface condition for 1D Helmholtz equation using finite element method
I want to implement a 1D Helmholtz equation with jump condition. The domain is $x=[0,1]$ and both ends have Dirichlet boundaries($p$=0). The 1D strong formulation is;
$$c^2\nabla^2p + w^2p=0 \qquad \...
- 11
1
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0
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84
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Finding the weak form of a PDE with a tensor argument
I am trying to solve for the order parameter ($A$) in the Ginzburg Landau equations. I had asked on the math SE site but was recommended to ask here.
We are trying to solve the following equation, (...
- 21
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votes
2
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509
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Nitsche's method for imposition of Dirichlet boundary conditions: implementation standpoint
I'm trying to understand how Nitsche's method works in practice. I understood the theoretical principle behind it, but what I can't understand is its implementation. More precisely, I'd like to solve ...
- 435
2
votes
1
answer
374
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SIPG method for $-\nabla \cdot (\nu \nabla u)=f$
Consider the diffusion equation with a coefficient $\nu$: $$-\nabla \cdot (\nu \nabla u)=f$$ with Dirichlet boundary conditions $u = g_D$ in $\partial \Omega$.
If the coefficient would be constant, ...
- 435
1
vote
1
answer
298
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Finite element method with two different Dirichlet boundary conditions
I have the problem like this
$$
-\triangle u = f \ \ on\ \Omega \\
u = g_1 \ \ on \ \partial \Omega_1 \\
u = g_2 \ \ on \ \partial \Omega_2
$$
If we choose
$$
V_1 = \{ \nu_1 \in H^1 : \nu_1 = 0 \ ...
- 111
4
votes
1
answer
289
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Discontinuous Galerkin: confusion about the weak formulation for linear advection equation
In an introduction to Discontinuous galerkin methods, I have some problems in checking the weak formulation. I'm looking at page 16 here
The context is the advection reaction equation: $$\operatorname{...
- 435
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votes
1
answer
685
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Weak Formulation of Poisson's Equation for Electrostatics with Surface Charge
I am currently attempting to use FEnICS to solve an electrostatic problem with two materials of different permittivity $\varepsilon_1$ and $\varepsilon_2$ forming an interface:
Consider a domain $\...
- 43
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3d schrodinger equation weak form
SCHRODINGER’S EQUATION
$$-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)$$
The potential $\frac{e^2}{r}$ is a variable coefficient.
So, let’s take the free Schrodinger ...
user38211
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0
answers
138
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Weak form of Elliptic problem with mixed Dirichlet & Neumann conditions
Let $\Omega \subset \mathbb{R}$ in a bounded polygon domain and $f:\Omega \to \mathbb{R}$ known function.We split the boundary into two parts $\partial \Omega_{1}$ and $\partial \Omega_{2}$ such that $...
user38211
2
votes
2
answers
192
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Manufactured solution for $-\operatorname{div}(a(x) \nabla{u}) = f$ when $\alpha(x)$ is discontinuous
I'm studying the dealii tutorial number 4,5 and I understand the workflow. I've also been able to find the EOC by using manufactured solution where $f$ is a smooth r.h.s. and $\alpha(x)$ smooth too.
...
- 435
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vote
1
answer
110
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Discretization of a nonlinear boundary value problem
I am trying to use finite element method to discretize the following problem
\begin{align}
\min_{u \in H^1_0(\Omega)} \int \| \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3 \|^2_2 \ d\Omega,
\...
- 13
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answer
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Weak formulation for advection diffusion reaction
I need a check on the following exercise about weak formulations and finite elements.
Consider the advection diffusion system
$$
\begin{cases}
-(\mu u')' + \beta u' + \gamma u = f \\
u(a)=0 \\
u(b) = ...
- 153
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130
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Constraining the total volume in Finite Element Methods
I have a diffusion problem which can be broken down to be:
$-\Delta u = f(u) $ on $\Omega ~/~ \Omega_{int}$
$u = 1$ on $\Omega_{int}$
Note that this is an internal Dirichlet constraint to the ...
- 3,065
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votes
1
answer
237
views
Unphysical Behaviour Characteristic-Wise WENO5-Z
I am currently working on a scheme that uses finite differences WENO5-Z with 3rd Order Runge-Kutta time integration for solving the Euler equations. The code projects the conserved variables and ...
- 41
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votes
1
answer
110
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Deriving weak form of a set of scalar equations
I have the equilibrium equation in elasticity for a static case.i.e
Div T=0.
For certain implementation, I have to get the x and y component equations and then derive the weak form separately. How is ...
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0
answers
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How do 'virtual kinematics/functions' play a role during deriving weak form formulations for physical problems?
I wanna ask a question that confuses me quite a long time. I saw many guys, in the context of computational mechanics, they seemed to choose the virtual functions or kinematics in a way that some ...
- 321
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Ritz, Galerkin, Weak Form, FEM: How to catch up the basics? [duplicate]
I have to deal with FEM and the numerical solution of PDEs a lot. While I'm doing ok when just applying or implementing it, I observe a lack of understanding when authors begin to argue with "Ritz", "...
- 1,463
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FEM: Obtaining the Weak Form
In the the Finite Element Method (FEM), we attempt to obtain the Weak Form of the described equation. I understand that this is an attempt to reduce the order regularity of the equation, but what are ...
- 319
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Strong vs. weak solutions of PDEs
The strong form of a PDE requires that the unknown solution belongs in $H^2$. But the weak form requires only that the unknown solution belongs in $H^1$.
How do you reconcile this?
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