# 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.

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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 ...
187 views

### 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, ...
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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 $\... 1 vote 1 answer 115 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 ... 1 vote 0 answers 123 views ### 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$... 166 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. ...
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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, \...
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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) = ...
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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 ...
187 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 ...
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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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### 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 ...
215 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", "...
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?