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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.

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FEM: Testing basis functions in a subspace V_h

In the finite element method, at a certain point we arrive at the following Galerkian problem where it is desired to find the solution $u_h \space \in V_h$ that solves the following equation: $$ a(...
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First derivative approximation using function values in equidistant points

Given a function $f: [x_0,x_4] → \Bbb R$ and equidistant points $x_0, x_1, x_2, x_3, x_4$ so that $h=x_{i+1} - x_i > 0$. Normally I would do, $f'(x) \approx \dfrac{f(x+h)-f(x)}{h}$, but here I ...
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Determining a variational formulation of $u^{(4)} = f$ with $3$-rd order BC.

Below is a problem from a recent exam and I have some questions about it. Given the boundary value problem:$$\dfrac{\partial^4 u}{\partial x^4} = f,\quad f\in L^2(0,1)$$ $$u(0) = u''(0) = u'(1) = u''...
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variational formulation of second order differential equation

I am given the following differential equation. Let $\Omega = (a,b)\subset\mathbb{R},\ f:\Omega \rightarrow\mathbb{R},\ \alpha,\beta \in \mathbb{R}$ and $$ -u'' + u = f \\ u(a)= \alpha, u(b) = \beta ...
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1D Finite element method: Function contineously differentiable?

I m currently reading the book "The finite element method in electromagnetics" written by Jian-Ming Jin. On page 50 and 51 which I attach as a screenshot below they show for an example functional, ...
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Finite element method for nonlinear differential equation

I encounter this problem $$\frac{df(u(x))}{dx} = g(x)$$ with $$u(0) = u(1) = 0$$ I first convert it to weak form $f(u(x))v(x)]^1_0 - \int^{1}_0 \frac{dv(x)}{dx}f(u(x))dx =- \int^{1}_0 \frac{dv(x)}{dx}...
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Calculate surface normal and area for a non-planar quadrilateral

Given the four coordinates of the vertices, what is the best possible approximation to calculate surface area and outward normal for a quad? I currently join the midpoints of the sides, thus ...
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Choice of the family of the Basis Functions

As I have learned for now, there are several families of Polynomial-type Basis functions (Lagrange, Serendipity, Hermite, ...). My question is beside the order of the elements which affects the order ...
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Why are differential equations satisfied/defined only on open domains?

I am going through FEM lectures by University of Michigan, and in one of the lectures, the professor writes an ODE: $$\frac{d^2u}{dx^2}+f(x)= 0, \ \ \ x \in (0,L)$$ Why are these differential ...
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Iteration step of the Crank–Nicolson scheme

For many dynamic problems a masse matrix $M$ has to be taken into account too. Consider a discretized systems of the form $$ \textbf{M}\frac{d}{dt}\vec{u}(t) = -\textbf{A} \cdot \vec{u}(t) + \vec{f}(...
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For a bounded linear functional $l(v) := (f,v)_\Omega$, do we have $\|l\| \le \|f\|$?

Suppose we have a Poisson equation $-\Delta u = f$ on $\Omega$ and we want to derive its weak formulation, so we multiply it by an arbitrary test function $\forall v \in H^1_0$ and then take integral ...
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finite element non linear boundary value problem

I have the following small finite element non linear boundary value problem: $$ -u''(x) = 1 + u^2(x) \quad \text{for} \quad 0<x<1 \quad \text{with} \quad u(0) = u(1) = 0 $$ with a grid with ...
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Solving nonlinear equation resulting from finite element method

Using the finite element method (for a uniform mesh in the spatial domain) I have the system with initial conditions $u_j(0)=\cos(x_j)$ for $j=1,\dots, N$ $$\frac{d\vec{u}}{dt}=A\vec{u}+B\vec{c},$$ ...
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Does Picard iteration affect the convergence order of a numerical scheme?

I have a common nonlinear differential equation, for example, the one in Stokes problem: Find $u$ (velocity) and $p$ (pressure) such that $$\nabla\cdot(-\mu(u)\nabla u+p\,I)=f\qquad\textrm{ in }\...
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clarification on Galerkin method - in finite element method solution

I am unable to understand the marked step in the solution to the problem on finite element method (Galerkin Method). I shall be grateful for clarification. Mentioned Equations 24 and 32 have been ...
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FEA: Coordinate Transform of Point Rotation

In most finite element analysis' applications one can extract the Displacements [u1, u2, u3] and Rotations [r1, r2, r3] of a given node. They are given in the node's coordinate system CS1: 1, 2 and 3. ...
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Characteristic lengths of an hexahedron

I am working on an error estimation strategy for a finite element code working on unstructured anisotropic meshes. In order to compute this error estimator I need the characteristic lengths of any ...
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Finite Element formulation of mixed BVP of Variational Problem

Suppose we are given the followin where $f$,$u$, $g$ are given functions: $-\Delta u = f$ in $\Omega$ $u=u_o$ on $\Gamma_1$ $\frac{du}{dn}=g$ on $\Gamma_2$ So in order for me to form the ...
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Relationship between Fundamental Lemma of Calculus of Variations and the Weighted-Residual Statement

The Weighted-Residual Method states that the integral of the Residual R(x) times the weighting function w(x) is equal to zero which means that R(x) = 0 On the other hand, Fundamental Lemma of ...
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How to refine NURBS mesh in isogeometric analysis?

Let we have a coarse description of 2D domain using NURBS. That is, we have two sets of knot vectors $\{\xi_1, \dots, \xi_n\}$, $\{\eta_1, \dots, \eta_m\}$, the set of control vectors $\{B_{ij}\}$, ...
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The mass matrix and the stiffness matrix in finite element method for heat equation

On the page 99 in the Chapter 5.2 of Prof. Endre Süli's lecture notes on FEM for PDEs (see: https://people.maths.ox.ac.uk/suli/fem.pdf), he derived the mass matrix for forward Euler scheme and the ...
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Biholomorphic functions and delaunay triangulation

Lets have a look at the two simply connected domains $D,G \subset \mathbb{C}$ and a biholomorphic function $f:D \rightarrow G$ which maps $D$ conformal onto $G$. For some $n \in \mathbb{N}$ there ...
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Numerical integration in Finite Element Method (and implementation in Matlab)?

i'm trying to solve the p-Laplace Equation: \begin{align} \begin{cases} \text{div} (\sigma |\nabla u|^{p-2} \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \text{in } \partial\Omega \...
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Computation of piece-wise linear hat functions

I have a discretized 3D surface for which I want to compute piece-wise linear hat functions. I assumed these functions are of the following form: $$\phi = ax + by + cz + d$$ with the property of $\...
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Use trace theorem to define $H^2_0$ space and the requirement of the boundary?

For homogeneous biharmonic problems, the solution space is in general defined as $$H^2_0(\Omega):=\{u\in H^2(\Omega): u=\frac{\partial u}{\partial \mathbf{n}}=0\text{ on }\partial \Omega\}.$$ With ...
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$\underset{v \in S_h}{\Vert u - v_h \Vert_{1,\Omega}} \leq ch\Vert u \Vert_{2,\Omega} $ implies $H^2(\Omega) \hookrightarrow H^1(\Omega)$

We have the following that is derived from the Bramble-Hilbert Lemma $\underset{v \in S_h}{\Vert u - v_h \Vert_{1,\Omega}} \leq ch\Vert u \Vert_{2,\Omega} $ implies $H^2(\Omega) \hookrightarrow H^1(\...
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Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions

Consider Poisson’s equation $$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$ with following mixed boundary cconditions $$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$ $$\frac{{\...
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Minimizing a nonlinear functional over finite elements

I am looking for a reference (an url, a book, or a paper) that could help me in discretization and minimization of the following cost function $J\left(\omega\right)$, over 3D tetrahedrons (finite ...
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Solution and Test space Euler bernoulli Beam

Let $\Omega := (0,1)$ be the domain, with boundary $\partial \Omega = \lbrace 0, 1 \rbrace$, such that $\bar{\Omega} = \Omega \cup \partial \Omega$. Let $\alpha \in \mathbb{R}$ be a constant, we ...
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Finding an analytical solution to a simple 2D Finite Element Method problem

diagram Is it possible to find the value of scalar function u(x, y) anywhere in the region $\Omega$, given the following: $\nabla \cdot$ ($\nabla$u) = f, u(x,y) = g when y = 0, L$_2$ ($\nabla$u)$...
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Are finite element mesh considered as planar graph?

Planar graphs have the intersection of edges only at nodes. So the finite element mesh also satisfies the definition of planar graph. Should we consider a FE mesh as a planar graph? I am trying to ...
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Advection-Diffusion across an interface using IPDG FEM

In the the Advection-Diffusion equation, $$ \dfrac{\partial C}{\partial t} - \nabla \cdot (D \nabla C + \mathbf{u} C) = 0, $$ do I have to add any terms to stabilize the flux in my finite element ...
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Can we mesh a rectangle using only heptagons?

I know that according to Euler's formula in graph theory the average number of edges or vertices in a planar connected graph cannot exceed six. So it seems according to Euler's formula its not ...
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Flux Limiter for 2D Discontinuous Galerkin FEM

I want to learn about implementing convection-diffusion simulations using discontinuous Galerkin (DG) finite element methods to solve $$ \dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J}, $$ ...
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Scaled trace theorem for $H(div)$

The scaled trace theorem is well known, i.e. there holds \begin{align*} \|v\|_{L^2(\partial K)} \leq C \left(h_k^{1/2} \|\nabla v\|_{L^2(K)} + h_k^{-1/2}\|v\|_{L^2(K)}\right) \quad \forall v\in H^1(K)...
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Calculate the surface area of each face of a hexahedron and it's outward normal,given coordinates

Basically, I have a hexahedral finite element mesh. I know the coordinates of elements, I used the coordinate transformation into an isoparametric structure and shape(basis) functions to calculate the ...
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Raviart-Thomas elements for mixed u-p formulation

I'm trying to solve a mixed u-p problem given by the following equations using the finite-element method: $\int_{\Omega} \delta\epsilon_{ij}2Ge_{ij} \, d\Omega - \int_{\Omega} \delta\epsilon_{ij}\...
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Shape function in Finite Element Method

Why is it that the choice of polynomial for 6-nodes rectangular element(linear in sides 1 and 3, quadratic in sides 2 and 4) in FEM does not follow normal pascal triangle regular arrangement? i.e $u=...
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Volume and barycentric coordinates of $k$-simplex in $\Bbb{R}^{n}$

How can the volume and barycentric coordinates (aka area/triangular coordinates) of a $k$-simplex in $\Bbb{R}^{n}$ be calculated given the vertices? In general $k \le n$ but any special cases for $k=n$...
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Modal Analysis for Transverse vibrations in Euler Bernoulli Beam

I have a beam clamped at one end, with a linear spring at the other. I was able to simulate the vibrations using finite element analysis, and I also found the first mode using separation of variables. ...
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Show that if $w$ is continuous on $[0,1]$ and $\int_0^1wv\, dx=0$ then $w=0$ everywhere in $[0,1]$

I was hoping someone could check my proof of the following. The problem comes from section 1.1 of Finite Element Method by Claes Johnson. Problem statement: Let $$V:=\{v: v \text{ is a continuous ...
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Sobolev inequality on a strip of size $h$

Let $\Omega$ be a sufficiently regular domain of $\mathbb{R}^2$ and let $h>0$ be the size of a triangulation of $\Omega$. Define $$ U_h:=\{x\in\Omega: \operatorname{dist}(x,\partial\Omega)\le h \}. ...
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Mixed formulation with Volterra integrals?

I'm going straight to the point. When gathering Volterra equations of the second kind with mixed formulations, one can arrive to the following problem: Let $X,M$ Hilbert spaces and $\mathcal{J}=[0,T]$,...
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Why minimize $ \Vert u_1 + u \Vert_1 $ in this Finite Element Analysis variational problem?

$\textbf{The problem reads:}$ Let $\Omega$ be bounded with $\Gamma: = \partial \Omega$ and let $g:\Gamma \rightarrow \mathbb{R} $ be a given function. Find the function $u\in H^1(\Omega)$ with ...
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Thoroughly understanding the LBB condition

I'm a mechanical engineer who's just gotten into FE analysis. The more I read about FE methods for Navier-Stokes, the more I run into the "LBB condition". I understand that it talks about the order of ...
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$L^\infty$ error estimate for optimal $L^2$ approximants

I have a function $u\in H^m(\Omega)$ for which I find $u_h \in \mathbb{P}^p(\Omega)$, the best $p$ polynomial approximant in an $L^2$ sense $$ u_h = \arg \inf_{v\in\mathbb{P}^p(\Omega)} \| u - u_h \|...
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Existence of closed solution of a certain type of Poisson equations

I want to check whether my numerical FEM solver is correct or not. So I seek a solution of \begin{equation} -\Delta u(x,y)=f(x,y) \text{ on } [-1,1]\times[-1,1]\\ u(-1,y)=1\\ u(x,y)=0\text{ on other ...
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Finite element approximation of weighted p-laplacian - error estimation?

i'm currently working on the following dirichlet problmen: \begin{cases} \text{div} (\sigma(x) |\nabla u|^{p-2} \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \text{in } \partial\...
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39 views

Right way to label the vertices in local bases using finite elements

I'm having a problem using local coordinates basis in my finite elements code, depending how i label the vertices, my results goes completly wrong, results of integral doesn't match and functions are ...
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Gradient operator tangent to surface

enter image description here What is gradient operator tangent to the surface of bubble, when a gas bubble moving in fluid?. How can be solved if we using finite element method.? Hi all. I am ...