Computational Science

Questions tagged [weak-solution]

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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 ...
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1answer
120 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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1answer
67 views

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 \ ...
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1answer
137 views

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{...
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1answer
169 views

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 $\...
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1answer
104 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 ...
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0answers
120 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 $...
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2answers
159 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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1answer
86 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, \...
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1answer
289 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) = ...
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94 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 ...
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1answer
186 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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1answer
81 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 ...
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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 ...
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1answer
201 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", "...
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2answers
799 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 ...
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1answer
7k 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?

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