Questions tagged [weak-solution]
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10
questions
0
votes
0answers
40 views
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.
...
1
vote
1answer
78 views
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,
\...
2
votes
1answer
155 views
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) = ...
0
votes
0answers
87 views
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 ...
0
votes
1answer
181 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 ...
0
votes
1answer
79 views
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 ...
1
vote
0answers
35 views
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 ...
0
votes
1answer
192 views
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", "...
3
votes
2answers
665 views
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 ...
14
votes
1answer
6k views
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?
