Questions tagged [finite-element-method]

Question 1

Consider a classical BVP governed by linear elasticity $$ \begin{align*} -\nabla \cdot \boldsymbol{\sigma} = \boldsymbol{b} & \quad \textrm{in} \nobreakspace \nobreakspace \Omega \subset \...

Question 2

I’m sorry if this has already been answered; I couldn’t find a clear answer (maybe there is an indirect one and I failed to make the connection). In the context of numerical analysis / finite elements ...

Question 3

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 $𝑥_𝑖 = 𝑖ℎ, 𝑖 = 0,1,2, ... , 𝑁$ ...

Question 4

I am quite familiar with the numerical analysis of finite element methods. However, in the case of time-dependent PDEs I have a nagging doubt that I would like to finally be rid of. The short version ...

Question 5

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 ...

Question 6

I'm studying the scaled Dirichlet preconditioner for the FETI (Finite Element Tearing and Interconnecting) method and have a question about constraint regularity requirements. Use the same notations ...

Question 7

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 ...

Question 8

When showing the well-posedness of saddle point problems, one has to show the following condition, also called "inf-sup". Here, $b(\cdot,\cdot)$ is a continuous bilinear form defined on a ...

Question 9

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} \|...

Question 10

I would like to know whether there is a way to explicitly construct the Riesz representation for the following example. For an open domain $\Omega \subset \mathbb{R}^n$ consider the embedding $$ H_0^1(...

Question 11

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 ...

Question 12

I am working on improving my understanding of finite element methods for PDEs, and I was reading through Langtangen's book Introduction to Numerical Methods for Variational Problems. The book is quite ...

Question 13

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 ...

Question 14

I'm in a predicament where solving the Laplace equation in polar coordinates for the extension of a membrane yields a finite element solution that shows oscillatory behavior which I cannot explain, it ...

Question 15

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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Questions tagged [finite-element-method]

How do concentrated (nodal) forces manifest in the weak form of the linear elasticity pde?

$H^1$ functions and continuity

Finite Difference Discretization of $y'' + xy = 1$ with $N=4$: Forming $A$ and $b$

Bounding a term unique to time-dependent finite element analysis

Under what conditions (if any) is the application of a finite-element discretized operator squared the equivalent to discretizing the squared operator

Does the Scaled Dirichlet Preconditioner for FETI Require Additional Constraint Regularity?

How to implement a source term on the boundary of a finite volume (or discontinuous galerkin) method?

Inf-sup condition and equivalence of norms

Gateaux vs Fréchet differentiability in FEM

Explicitly computing Riesz representation of elements in $H^{-1}(\Omega)$

Predicted convergence rates in FEM

Finite elements: understanding the intuition behind the coefficients of the $A$ as inner products

weak formulation with Robin BC

Oscillatory non-converging behavior of finite element solutions

Finite Element Theory - Clement's Interpolation

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