Questions tagged [finite-element-method]
A method of obtaining (numerically) approximate solutions to (usually) differential equations. It consists of a method of discretization splitting the domain into disjoint subdomains over each of which the problem has a simpler (approximate) solution, and a method of reassembling those pieces to obtain a solution over the whole domain. It is closely tied to the calculus of variations.
665 questions
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How do concentrated (nodal) forces manifest in the weak form of the linear elasticity pde?
Consider a classical BVP governed by linear elasticity
$$
\begin{align*}
-\nabla \cdot \boldsymbol{\sigma} = \boldsymbol{b} & \quad \textrm{in} \nobreakspace \nobreakspace \Omega \subset \...
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Finite Difference Discretization of $y'' + xy = 1$ with $N=4$: Forming $A$ and $b$
Problem:
Given the ODE-problem
$\frac{𝑑^2𝑦}{𝑑𝑥^2}$ + 𝑥𝑦 = 1, 𝑦(0) = 1, 𝑦(1) = 0
Discretize with the finite difference method (FDM) the problem on a grid $𝑥_𝑖 =
𝑖ℎ, 𝑖 = 0,1,2, ... , 𝑁$ ...
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Under what conditions (if any) is the application of a finite-element discretized operator squared the equivalent to discretizing the squared operator
The following is very loose, but as an example, consider the derivative operator $D = \frac{d}{dx}$, and some set of finite element basis functions $\varphi_i$ (perhaps Lagrange elements).
Take a ...
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How to implement a source term on the boundary of a finite volume (or discontinuous galerkin) method?
I've written a 3D linear acoustic discontinuous galerkin method that works on an unstructured tetrahedral mesh. This models pressure, $p$ and velocity, $\mathbf{u}=(u,v,w)$ and has fully reflective ...
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Gateaux vs Fréchet differentiability in FEM
I am trying to figure out the practical relevance of having only Gateaux differentiability vs Fréchet differentiability. As an example consider the Dirichlet energy
$$E(u) = \frac{1}{2}\int_{\Omega} \|...
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Predicted convergence rates in FEM
When computing the $L_2$ error in Finite Element Analysis with a theoretical predicted convergence rate $\beta$ in $||u-u_h||_{L_2} \leq C h^{\beta}$, how common is it to achieve the predicted ...
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weak formulation with Robin BC
I’m working with a weak formulation of a problem involving the equation:
$$
\sigma = \nabla u \quad \text{in} \quad \Omega, \quad \mathrm{div} \, \sigma = -f \quad \text{in} \quad \Omega, \quad \sigma ...
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Finite Element Theory - Clement's Interpolation
I'm having some trouble trying to understand the construction of Clement's Interpolation. The first part of the theorem states that if $T_h$ is a shape-regular triangulation of the domain $\Omega$, ...
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prescribe discontinuity in hybridized discontinuous Galerkin
I am working on a Poisson problem in two subdomains where I want to couple the interface between two subdomains such that we have (1) flux conservation, (2) flux = function of jump of variable between ...
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Numerically solve $\nabla\cdot(u\nabla f)=0$ for $u$, with $f$ known
I am interested in learning about numerical methods (ideally finite-element/Galerkin-type discretizations on simplicial meshes) to approximate solutions to the steady-state reaction-advection equation
...
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The relation between surface integral and volume integral in the weak form of the advection term
I have a question about a derivation based on weak form of the advection term. The original weak form is give as:
$$
\int_\Omega \phi\boldsymbol{c}\cdot\nabla f d\Omega,
$$
where $\boldsymbol{c}$ is a ...
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Showing the solution to the variational formulation satisfies a pure Nuemann condition
I am having a problem with showing the solution to the variational formulation satisfies the Nuemann boundary condition.
Condsider the PDE
$$
\begin{aligned}
-\Delta u &= f \quad \text{in} \space \...
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Density argument in $C^0$ interior penalty method for HJB equation (in the linear case). Proving stability(?)
This is my first question here. Please let me know if I am doing anything wrong
I am following an article from Susanne C. Brenner and Ellya L. Kawecki (ADAPTIVE $C^0$ INTERIOR PENALTY METHODS FOR ...
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Raviart--Thomas space in tensor context, does it exist? (mixed finite element method)
The Raviart--Thomas space is defined as
$RT_m(T)=\mathbb{P}_m(T)^n+x\,\mathbb{P}_m(T)$
where $\mathbb{P}_m(T)$ is the polynomial space with degree less than $m$, and $n=2$ or $n=3$. Here $T$ is a ...
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Jacobian of a variational formulation
Background
I am solving a nonlinear problem that is mixed domain and also mixed finite element. I have the variational formulation and want to take the Jacobian so that I can solve the problem using ...
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Change of variables via Jacobian with Voigt notation for differential operator
I'm reading Non-linear Finite Element Analysis of Solids and Structures (De Borst, R., Crisfield, M. A., Remmers, J. J., & Verhoosel, C. V).
As part of the weak form of the governing equations of ...
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How can the Finite Element Method be used to solve a Multi-equation System?
Every explanation I've seen of FEM show it solving an equation of the form $\mathcal{L}\phi=f$, with $\mathcal{L}$ being a differential operator, and every solution (whether it be Ritz or Galerkin) ...
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How to transform an integro-differential equation into weak form for FEM
I have the following 2D boundary-value problem which I would like to solve numerically using FEM software:
\begin{equation}
a(x,y)\nabla^2 u(x,y) + \int\int K(x,y,x',y')u(x',y') dx' dy' = f(x,y),
\end{...
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Error in the energy norm for inhomogeneous Dirichlet boundary conditions
Short description
When conducting FEM analysis with inhomogeneous Dirichlet boundary conditions, I compute the error in the energy norm with an expression that should only work for problems with ...
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Weak formulation of the biharmonic equation
The biharmonic equation is
$\Delta^2 u = 0$.
Tutorials show that the stiffness term in the weak formulation is:
$\int \langle\Delta \varphi_i, \Delta \varphi_j\rangle$
However, FEM implementation uses
...
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Weak formulation for $-u''+u=1$
I am trying to solve the following problem.
Now, based on other examples I have seen, it seems the weak formulation should be something in terms of an inner product. In particular, we see
$$(-u+1,v)=(...
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Uniform equivalence of grid functions
I am studying the finite element method and have found a problem:
For a regular and quasi-uniform family of simplicial partitions, prove uniform equivalence
$H^1$ norms of grid functions to the norm:
$...
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L2−conforming (discontinuous) vs integration points
Finite element discretization spaces
Full de Rham complex
The picture below taken from here, displays from left to right:
H1−conforming (continuous)
H(curl)−conforming (continuous tangential ...
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Weak form FEM discretisation of Non-Linear System
I am trying to derive the FEM solution for the equation $u''(x) + \left( u'(x) \right)^2=0$ with $u(1)=0, u'(0)=1$ over the interval $[0, 1]$.
Constrain the trial function, $v$, to also be zero at $x=...
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In finite element method for second order elliptic problem with neumann boundary value, is the solution weakly satisfies the boundary conditions?
For example, let's consider the problem
\begin{equation}
-\Delta u+u = 0
\end{equation}
For $g\in H^{\frac{1}{2}}((D)$, where $D$ is the domain, assume that $u$ is the solution of
\begin{equation}
(u,...
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