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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Construction of a polynomial with specified integral over several regions

This is appearing in the context of the finite element method. We want to find a $n$-variate polynomial, of order $\leq m-1$ in each variable, such that its integral on each of various subsets of the ...
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Finite Element Spring Analysis [closed]

Spring system enter image description here
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Is broken Sobolev space a Sobolev space?

The definition of a broken Sobolev space is as follows. Given infinite-dimensional (but mesh-dependent) spaces on an open bounded domain $\Omega \in R^3$ with Lipschitz boundary. The mesh, denoted by $...
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Confusing dot product and inner product in a weak formulation

I have been struggling with this for a while. Here, as you can see, they define the weak formulation of the Poisson equation as: $-\int_{\Omega}\nabla u\cdot\nabla v\,ds = \int_{\Omega}fv\,ds \equiv -\...
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Difference between the definition of the dot product in infinite and finite dimensions

I know that in $\mathbf{\mathbb{R}}^n$ the definition of the dot (or scalar) product is the following: $x.y=x^{\mathrm{T}}y$, with ''T" denoting the transpose of the vector x. How does this ...
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Weak formulation for coupled system

i am working on a problem with the FEniCS project for solving equations with finite elements. The problem is a coupled system of the biharmonic equation and a heat equation. The geometry on which i ...
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Understanding the Hypercircle Method

I'm trying to understand the hypercircle method from https://www.jstor.org/stable/43633616?seq=1 and https://homepage.ruhr-uni-bochum.de/dietrich.braess/nonconf.pdf but I don't see the main idea. So ...
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point of intersection of two lines in barycentric coordinate system

I am looking for an efficient way to determine the intersection point of two lines which go through a triangle (face) of a 3D triangular surface mesh. For both lines I know the two points at which ...
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Determine Jacobi Matrix of a unit simplex

Let $\Phi$ be a affine linear mapping with $\Phi(K)=\hat K$, where $\hat K$ is the unit simplex and $\Phi$ is of the form $$\Phi(\hat x)=A\hat x+b$$ with $A\in\mathbb R^{d\times d}$ and $b\in\mathbb R^...
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Operator form of the pure Neumann problem in Finite element analysis

Let the elliptic problem of form $-\Delta u + u = f$ on $\Omega\subset\mathbb{R}^2$, where $\Omega$ is a sufficiently smooth bounded domain. Let $u = 0$ at $\partial \Omega.$ The finite element ...
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Reference request: $L_2$ projection over the boundary of triangles of the triangulation

Let $\Omega\subset\mathbb{R}^2$ a bounded domain. Define triangulations $\mathcal{T}_h=\{K\}$ of $\Omega,$ with $h = \max_{\mathcal{T}_h}diam(K)$ and $\kappa:=\sup_{\mathcal{T}_h}\kappa_K,$ where $\...
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Second order shape funtions on first order element

In a Finite Element context, say I have a 1D finite element with 2 nodes, $x=0$ and $x=1$. The typically used shape funcions are $N_1(x) = 1 - x$ $N_2(x) = x$ Would it make sense to use second order ...
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$L_\infty$-stability for forward heat equation $u_t=\frac12u_{xx}$

I am considering the forward heat equation $$\frac{\partial u}{\partial t}=\frac12\frac{\partial^2u}{\partial x^2}$$ with $0<x<1$, $t>0$ under the explicit Euler scheme such that $$U_j^{(n+1)}...
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Galerkin method approximation limit $\|u_n(f)-u(f)\| \to 0$

Let $H$ a Hilbert space and let $\{H_n\}_n$ be a sequence of finite-dimensional subspaces of $H$, such that $H_{n}\supset H_{n-1} \quad \forall n \in \mathbb{N}$ and $\cup_n H_n$ is dense on $H$. Let $...
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Finite elements convergence issue with 2D elliptic equation

I deal with a 2D Helmholtz equation solved via the P1 FEM on a given geometry. The shape of my solution is as expected, but I encounter an issue with its amplitude which depends linearly on the size ...
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Gauss-Lobatto quadrature and nodal points for FEM

By using the Legendre-Gauss-Lobatto (LGL) quadrature formula (QF) and LGL nodal points one achives a diagonal mass-matrix for finite element problems. (More specifically, the spectral element method.) ...
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Proving an inequality involving $L_2(\Omega)$ norms of $L_2$ and $C^{\infty}$ functions

Say $v\in L_2(\Omega)$, then need to show that there's a $w\in C_0^{\infty}(\Omega)$ such that $$\|v-w\|_{L_2(\Omega)}\leq\frac{1}{2}\|v\|_{L_2(\Omega)}$$. So far I've done this: \begin{align*}\|v-w\|...
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What is the error of FEM-like spline discretizations?

Let's have $f(x)$ a nice*, scalar valued function. It is approximated by $$ f(x)\approx g(x)=\sum_i f(x_i) N_i(x), $$ where $x_i = x_0+i \Delta x$ are uniformly spaced points, $N_i(x) = N(x-x_i)$, ...
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How can I solve this problem with Finite element method?

consider the problem: $$-(a(x)u'(x))'=f(x)$$ $$x \in(a,b)$$ $$u(a)=u(b)=0,$$ and $$0<a_{0}\leq a(x)\leq a^{0}$$ $$\forall x \in [a,b]$$. Solve this problem with finite element method and proceed to ...
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Basis funcions Discontinuous Galerkin

I am learning about Disconinuous Galerkin methods. I fail to understand how he basis funcions are constructed. I understand that typically Legendre polynomial are used, but I can't see how they relate ...
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FEM computations in the case of an axisymmetric problem

I have been struggling with this question for a while now... When running an FEM simulation to compute a value, say a capacitance between two conductors, and when the problem is simplified to an ...
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What reference would you recommend for someone willing to implement a FEA generalized shell element?

I'm trying to find works describing the composition of (FEM) stiffness and mass matrices for shell elements which can predict the out-of-plane displacement component, in the 3D space if possible. The ...
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Proving a version of Green's formula in functional analysis

$\textbf{b}:\Omega\to\mathbb{R}^2$ is a $C^1$ vector field. Need to prove the following version of the Green's formula: $$(\nabla u,\textbf{b}v)=-(u\textbf{b},\nabla v)-(u,v\nabla\cdot \textbf{b})$$ ...
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Proving a Sobolev norm inequality

I need to show that $\|u\|_{H^1}^2\leq \|u\|_{L_2}\|u\|_{H^2},\forall u\in H^2\cap H^1_0$ using the Helmholtz equation: \begin{align*}-\Delta u+u&=f,\text{ in }\Omega\\u&=0,\text{ on }\partial\...
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Axisymmetric Finite elements problem

I would like to know please how does an axisymmetric geometry simplifies the FEM problem to be solved. I am really new to the FEM world so I would really appreciate if you go really easy on me. Thanks ...
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Requirements for Korn's inequality on piecewise $H_1$ vector field

I am looking at the Korn's inequality on $H^1$ vector fields, as described in this paper by Brenner. In particular, I am looking at how the seminorms defined in examples 2.3 - 2.5 satisfy the ...
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Change of variable of surfaces integrals

Let $\mathbb{S}^{2}$ be the unit sphere, $\Delta_{s}$ the Laplace-Beltrami operator, $U_h$ the finite element space in $\mathbb{S}^{2}$ and $V_{h}$ the space of constant function associated to $U_{h}$....
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Proving an Inequality from Finite Element Analysis

There is a uniform triangulation $\tau_h$. $v_h(x)$ is a piecewise linear function such that $$v_h(x)=\sum_{i=1}^nv_i\phi_i(x)$$ where the $v_i$'s are the nodal values, $\phi_i(x)$'s are the hat ...
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Is there a measure of “indefiniteness” of a matrix?

Question Is there a measure of the "indefiniteness" of a matrix? And what constitutes a "highly indefinite" matrix? Explanation I have to solve a linear system computationally, ...
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Finite Element Code for Stokes Problem using P2P1 Quad Elements

I am trying to write some MATLAB code to solve fluid flow problems using the finite element method. I am starting with the relatively-simple steady Stokes problem. I am using the quadrilateral P2P1 ...
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Infinite shell transformation in FEM [closed]

I would like to know please, what would be the expression of the transformation matrix for an infinite shell transformation in FEM. It's been really 4 years since I've had real math courses so I would ...
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$H_1$ seminorm of an error in 2-D

I have a finite element scheme with triangulation. Say $u_h$ is the approximations found by the scheme and $u$ is the exact values at the nodes$-$which are here the vertices of the triangles. I am ...
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Consequence of trace theorem

Trace theorem says that there exists unique continous linear mapping $\gamma : W^{k,p}(\varOmega) \rightarrow L^q(\varOmega)$ that $\gamma(u) = u|_{\partial \varOmega}$. It is correct to write $||u||_{...
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Weak derivative of a multivariable function and Sobolev spaces

I have to check if the function $f(x,y)=\sqrt{x+y}$ is in the Sobolev space $H^1(\Omega)$. I also have to check if it has a restriction to the boundary that is $L^2(\Gamma)$. Here $\Omega=(0,1)\times(...
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Convert differential equation to variational statement

I know how to convert a differential equation with constant coefficients to variational form. But the question in the picture has non-constant coefficients. How do people get around that? Any ...
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Deriving shape function for 2d finite element analysis

I have recently learned about finite element analysis. I mainly focus on structural mechanics. A common element for 2d is the triangle with 3 nodes. I will first make an example of how I would compute ...
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Trouble in using finite difference method to solve a boundary value problem (attempts and pictures included)

I want to numerically solve (using FDM)$$-y''(t)+2y'(t)=1, t\in (0,1)\\y(0)=1, y(1)=3$$ First, I check with symbolab that the analytic solution would be $1-\frac{3}{2\left(-1+e^2\right)}+\frac{3}{2\...
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Advection diffusion equation: weak formulations and Lax-Milgram Lemma

This exercise is taken from Numerical Models for Differential Problems Authors: Quarteroni, Alfio. There's no solutio, so I need a check. Consider the one-dimensional advection diffusion problem $$\...
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Find norm of functional and coercivity constant for a bilinear form (Cèa Lemma)

Let $V=H_0^1(0,1)$ and consider the bilinear form acting on $V$: $$a(u,v) = \int_0^1 \frac{1}{1+x} u'(x)v'(x) dx $$ and let $F:V \rightarrow \mathbb{R}$ the functional defined as $$F(v)=\int_0^1 \Bigl(...
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Weak formuation $1-D$ advection diffusion equation: test function and solution in the same space $H_0^1$

I need a check on the following problem: Consider the system $$ \begin{cases} -(\mu u')' + b u' = 0 \\ u(0)=0 \\ u(1)=1 \end{cases} $$ with $b$ and $\mu$ functions in $L^2$. I want to find the weak ...
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Principle of virtual work and weak form

In finite element method, one wants to derive a so-called weak form of the differential equation to solve. This latter is obtained by multiplying both sides of the equation by a "test function&...
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How I can determine coefficients of shiftness matrix of the below boundary value problem?

I want to give an approximate solution $u$ of the following problem $(P)$ : $$(P)\begin{cases} -u''(x)+6u(x)=(-4x^2-6)\exp(x^2),x\in{\Omega}=]-1,1[\\u(-1)=u(1)=0 \end{cases}$$, I have multiplied by a ...
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Inequality between inner product of functions in dual space

I'm reading Brezzi's paper on DG method and is currently puzzled on how equations (5.7) is used to derive equation (5.8). Further searches lead me to learning that the space defined $\mathbf{V}'$ is ...
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$L_p$ discrete norm and norm $L_p$

I want to prove this inequality and I don't have idea: $ \exists c>0, s.t. \forall 1\leq p < +\infty \text{ and } \phi_h\in V_h:$ $$ \frac{1}{c} \Vert \phi_h \Vert^p_{L^p(\Omega)} \leq h^d ...
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Is what I have done to Show that each element of $(E_h)$ is written as a linear combination of the functions $(\phi)_ i$?

Let $\phi_{i}$ are continious over mesh every integer, and define $E_h=\text{vect}\{\phi_i, i=1,\cdots N\}$ I want to Show that each element of $(E_h)$ is written as a linear combination of the ...
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What is this “1” in this stiffness matrix?

What is this "1" in this stiffness matrix? http://www.annualreviews.org/article/suppl/10.1146/annurev.earth.35.031306.140104?file=ea.35.rayfield.pdf $$K_e= \frac{E^e A^e 1}{L^e} \begin{...
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derivation of finite element analysis in elasto static mechanics

I am currently studying finite element methods to solve mechanical problems. I was only given a few papers which I am working through right now. The problem is that one of the first few steps is ...
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derivation of finite element analysis in elasto static mechanics

I am currently studying finite element methods to solve mechanical problems. I was only given a few papers which I am working through right now. The problem is that one of the first few steps is ...
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12 views

Understand how the cubic Hermite element is a valid finite element

I'm having trouble understanding how the cubic Hermite element is a valid finite element. The element is defined as follows: $K$ is a non-degenerate triangle with vertices $z_{1}, z_{2}, z_{3}$ $P_{3}...
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51 views

$H^1$-conforming approximation for elliptic PDE with discontinuous coefficient?

everyone Suppose I want to solve the diffusion equation $$-\nabla\cdot a \nabla u=f, \\u=0 \text{ on } \partial \Omega,$$ $f \in L^2(\Omega)$, $\partial \Omega$ is smooth. I use the standard node-...

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