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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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Direct sum factorization of polynomials

I have been recently reading the paper "Mixed finite elements for second order elliptic problems in three variables" by Brezzi et. al. I noticed the claim in the proof of Lemma $2.1$, which basically ...
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FDM variable coefficients

Usually authors of the books demonstrate the usage of FDM on the following equation $\frac{\partial f}{\partial t} - a\frac{\partial^2 f}{\partial x^2} = f(x,t)$ where $a$ is some constant. Is it ...
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Finite elements for biharmonic equation

I have a question regarding the conforming finite elements for biharmonic equation: if we want to discretize the weak formulation for this problem using a conforming finite element space $V^h$, is it ...
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Why the Galerkin Orthogonality Holds?

This is not homework. I'm going over my lecture notes to study for an exam. For an Abstract Elliptic Problem such as the problem with V a Hilbert Space $$\begin{cases} \text{Find } u \in V \text{ ...
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How do theoretical convergence speeds translate into real life when using FEM?

I'm currently solving the wave equation in a 1d and a 2d domain using the finite element method in space and the leapfrog or crank-nicolson method in time. Theoretically, I expect a convergence of $O(...
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67 views

How Young inequality was applied?

Let $|\cdot|$ a norm in $L^2$ and $\|\cdot\|$ a norm in $H_0^1$. Then $\begin{align} &|u_h^n|^2 + 2 \theta \Delta t h^{-1} |u_h^n|\|u_h^{n+\theta}\| |u_h^{n+ \theta} - u_h^n| +2 \theta \Delta t |...
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Finite Element Method for vector valued functions

I need help with the finite element method for the following problem I present in weak formulation. Certain details are left out since they are not important for the essence of this question. This is ...
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How to implement the Kutta condition for potential flow and FEM?

I have a potential flow FEM solver (basic Laplace equation) which works well and is validated for potential flow around cylinder and symmetrical airfoils (such as NACA0012). However, when the angle of ...
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FEM approach to obtain the projection of a function onto the mesh?

I need to explain the background of the problem I'm working on, and the questions themselves are a couple of paragraphs down. I am reading this book and implementing the algorithms along (so far in ...
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Courant (1943) and History of Finite Element Method

I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called “Orthogonal ...
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Show that the variational formulation has at most one solution

We have the problem: $$ -u''(x) + u(x) = f(x) ,\quad \quad x \in [0,L] $$ $$u(0) = 0 $$ $$u'(L) + u(L) = 4 $$ I then put it into variational form (hopefully correctly done) with introduction of $v \...
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Property of a quasi-uniform triangulation

I have some type of proof for the inverse inequality: $|\nabla v |_{H^1} \le C |v|_{H^1}$ This proof uses the following property for quasi-uniform triangulations: $ \frac{\int_{{K}^\wedge}{|\nabla v^\...
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Why does my newton-Raphson iteration fail?

Suppose that I have the following energy equation that is a function of $\varepsilon$, the strain, and $\eta$, the hardening parameter. $\phi=\frac{1}{2}E \varepsilon_e^2+\frac{1}{2}H \eta^2$, $\...
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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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44 views

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}\...