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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Flux that can be represented by low and high resolution schemes.

In the wiki page of Flux limiter, it writes: If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the ...
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Adaptive step size for nonlinear static problem

Let's assume $F$ is an external load for a nonlinear static finite element problem. Normally, the problem will not converge if you apply $F$ fully. Instead, we multiply the load $F$ with a scaling ...
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What part of the Galerkin method ensures a solution with accurate nodal values?

I graduated over a year ago as a mechanical/industrial engineer and I've recently been re-studying my last year engineering courses that focused on numerical methods for simulation, including the ...
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Lipschitz-continuous boundary and boundedness issues

I have some questions about Ciarlet's book "Finite element method for elliptic problems" in page 12. The author defined Lipschitz-continuous boundary and wrote as follows: An open set $\...
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If a function is in $H^1(\Omega_i)$ then is $H^1(\Omega)$ in the whole domain?

This is my first question and english is not my first language, so sorry in advance. I have been recently started learning about finite elements in a graduate curse. I have a question about the ...
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Finite tetrahedron elements determine a funciton in $P_2(K)$

Let $K$ be a tetrahedron with vertices $v^i, i=1,...,4$. Denote $m^{ij}$ as the midpoint on the line between $v^i$ and $v^j$. (1)Show that a function $f \in P_2(K)$ is uniquely determined by the ...
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Equivalent functional integral of ODEs

I was going through the Ritz method in Finite Element analysis when I came across the statement here (see equation 1.25, its not same but similar. below I have taken a problem from my book.), ...
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Saddle Point Problem and Lagrangian

Let $X$, $M$ be Hilbert spaces. Consider two bilinear forms $a(\cdot,\cdot):X \times X\to \mathbb{R}$ and $b(\cdot,\cdot): X\times M\to \mathbb{R}$, and linear maps $f:X\to \mathbb{R}$, $g:M\to\mathbb{...
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The stiffness matrix for a triangle in the finite element mehod

In Wikipedia https://en.wikipedia.org/wiki/Stiffness_matrix under section "Practical assembly of the stiffness matrix" section, it says that: For example, for piecewise linear elements, ...
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Proving that $\{\varphi\in C^{\infty}(\Omega)\,: \varphi(b) = 0\}$ is dense in $\{u\in H^1(\Omega)\,:\, u(b) = 0\}$

Notation Given $\Omega :=(a,b)\subset\mathbb{R}$ a bounded open interval, let us define the Sobolev spaces $H^1(\Omega):=\{v\in L^2(\Omega)\,: \, v'\in L^2(\Omega)\}$, $H_0^1(\Omega):=\{v\in H^1(\...
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2D FEM — How to interpolate?

Let us consider a Poisson equation and a uniform mesh, as an example to demonstrate the piecewise linear basis functions and the finite element method: $$-(u_{xx} + u_{yy}) = f(x,y)$$ $$u(x,y)=0$$ on ...
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Finite Element model of Fourth order PDE in FreeFEM++

I am attempting to finite element model the/a solution to the Kuramoto-Sivashinsky equation $$u_{t}+\nu u_{xxxx}+u_{xx}+uu_{x}=0 $$ with 1-periodic boundary condition $$u(\cdot,0)=u_{0} $$ using the ...
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Calculating the contribution to nodes from the boundary vector in FEM-analysis - global vs local

The moisture content of a drying board of timber is being calculated with the finite element method and 9-node lagrangian elements, i.e. $$m = \alpha_1 + \alpha_2x+\alpha_3y+\alpha_4x^2+\alpha_5xy+\...
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Numerical solution of first-order linear PDEs

Consider a linear, first order partial differential operator $L$ with: $$ Lu = \sum^{N}_{i=1} a_{i}(x_1,\ldots,x_{N})\frac{\partial}{\partial x_{i}} u$$ For some---lets say Lipschitz continuous---...
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Examples of solutions of 2D Poisson Equation with homogeneous boundary conditions

I'm looking for analytic solutions of Poisson type equations $$ \Delta u(x,y) = f(x,y), \quad u_{| \Omega} = 0 $$ where $ \Omega = [0,1] \times [0,1] $ with different $f$. The purpose is to test the ...
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Legendre-Gauss-Lobatto Nodes via Newton-Raphson

I am trying to understand the code given here, which calculates the Legendre-Gauss-Lobatto nodes via the Newton-Raphson method. These nodes are given by the zeros of $(1 - x^2)\,P'_{N}(x)$ where $P_N(...
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Koszul differential and polynomial differential forms (finite element exterior calculus)

Let $r$ and $k$ denote, respectively, a polynomial degree and the degree of differential forms over $\mathbb R^n$. For $r,k \ge 0$ with $r+k > 0$, denoting by $\kappa$ the Koszul differential, by $\...
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Finite element method stencil

I'm a bit new in the field, As I understand one of the way to solve PDE is by converting the problem to system of linear equations. Finite element method is a common way to do so. In case of 2D ...
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Quadrature rule for a triangle with quadratic Lagrange elements

I am trying to code up some finite element approximation in 2D. Suppose each element is a triangle. You can use linear Lagrange basis $P_1$, where each node coincides with a vertex. I want however to ...
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Finite Elements or Finite Difference for the Heat Equation, Dissipates the Entropy?

Given an initial density $\rho_0$ on $\Omega\subset \mathbb{R}^d$, it is well known that the Heat Equation $$ \partial_t \rho(t)=\Delta\rho(t),~~~~~~~\rho(0)=\rho_0, ~~~~~(1)$$ dissipates the entropy $...
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Finite elements method: why test fuction vanishes on boundaries

I am trying to understand why (and exactly when) the test functions must vanish at the boundaries when Dirichles conditions are applied to a PDE. The context is the learning of the finite element ...
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Does small interpolation error imply regularity of function?

There's plenty of literature on results that say that interpolation errors are small for smooth functions. But, I can't find much information about the converse, that is - does convergence of ...
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Strong vs Weak solution to one-dimensional elliptic PDE

Consider the elliptic PDE \begin{align} &-\frac{\mathrm{d}}{\mathrm{d} x} \left(a(x) \frac{\mathrm{d}}{\mathrm{d} x}u(x)\right) = 1, \qquad 0 < x < 1,\\ &u(0) = u(1) = 0. \end{align} ...
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How to handle non-linear terms on FEM for PDEs

When using FEM for solving a PDE you first have to do a discretization. when we have a linear PDE is quite straightforward. You find the week form of the PDE and then make the discretization. But, ...
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Help understanding the role of barycentric coordinates in affine mapping of reference element to actual element

I am slogging through FEM self study leaning heavily on the 1st Zienkiewicz text. I think the book is great but I cannot seem wrap my head around switching between the element of interest and the ...
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Preferable finite element for thin-plate simulation

I want to simulate a vibration of a thin plate (the Kirchhoff-Love model) on triangular meshes. Can you advise me on an introductory-level review of different elements for thin-plate FEM simulation? I ...
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Assembling the global finite element equation KU = F with gravitational body force

I am trying to model a basic solid cube that rests on a ground (z = 0) with gravity pointing in the negative z direction. To do this, I have a basic cube that is divided into 6 tetrahedrons. The ...
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Advantage of Helmholtz-Hodge Decomposition in detecting singularities of a vector field

I'm trying to understand the advantages of the Helmholtz-Hodge Decomposition (HHD) in detecting singularities (sources, sinks, centers of rotation) of a vector field in a discrete setting. Since I am ...
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What is the relation between the finite difference method's solution and the weak solution?

I never understood the justification of some authors using the Finite Difference Method to solve a PDE numerically while theoretically they only proved the existence of weak solutions. Is the use of ...
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Integration by parts of vector fields (divergence, gradients)

I have the following system of equations in 3D: $\boldsymbol{\Delta}\mathbf{u}(\mathbf{x})-\boldsymbol{\nabla}div\mathbf{u}(\mathbf{x})=\mathbf{F}(\mathbf{x})$ I'm solving for $\mathbf{u}(\mathbf{x})$ ...
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FEM for Elliptic equations with gradient term

I have a question about the following problem. Let $-\Delta u = \|\nabla u\|^p+f$ in $B(0,1)$, $u=0$ on $\partial B(0,1)$, where $B(0,1)\subset\mathbb{R}^2$ is a bounded regular domain, $f$ is ...
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Consistency finite differences vs finite elements

Whether a discretisation is consistent (and the order of consistency) in the FDM setting is defined by the the truncation term, e.g.: $$\partial_{xx} = \frac{1}{h^2}\begin{bmatrix}1 & -2 & 1\...
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Reproducing function from locally-integral moments

I am looking for a consistent theory (that is, something that is quotable, and that provides all relevant theorems) on the fact that a polynomial function can be reproduced by knowing its integration ...
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Why solving a system of linear equations by minimizing residual is not working?

I've faced a totally unexpected situation while I'm studying on FEA and optimization. I have the following system of linear equations (according to FEA) with constraints. $$ \begin{aligned} \mathbf K \...
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General non-FEM Galerkin, boundary conditions

Consider the weak form of our all-time favorite Poisson-equation, $$- \int_{\Omega} \nabla u \cdot \nabla v\, \mathrm{d} x = \int_{\Gamma}v \nabla u \cdot \mathrm{d}x +\int_{\Omega} fv\, \mathrm{d} ...
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Why the normal trace operator for H(div;) is surjective? I think proof is wrong..

prerequisite: For $\underline{q} \in H(\operatorname{div}, \Omega)$,we can define $\left.\underline{q} \cdot \underline{n}\right|_{\Gamma} \in H^{-\frac{1}{2}}(\Gamma)$ and $$ \int_{\Gamma} \...
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How does the compactness work?

T is a isosceles right - angled triangle in 2-dim and F is any edge of T. v $\in P^{k}(T)$ (polynomial space with degree at most k). In the book --mathematical aspects of discontinuous galerkin method,...
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Weak form of cahn hilliard equation

I am trying to write the weak formulation of Cahn-hilliard equation as under. $\frac{\partial \phi}{\partial t} = \nabla .\{\phi(1-\phi) [\phi(1-\phi)\nabla \mu - \nabla (A(\phi)q)]\}$ $\frac{\partial ...
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Hexahedral mesh generation around two cylinders junction

I need to design a mesh for Finite Element Method around the junction of two hollow cylinders. Here is an example of mesh geometry: https://i.stack.imgur.com/V0hVX.png. I've found how to generate a ...
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Why the error using backward Euler is less than using Crank--Nicolson?

I'm reading the following paper https://www.wias-berlin.de/people/john/ELECTRONIC_PAPERS/JMR06.CMAME.pdf where calculates the solution of the time-dependent Navier--Stokes equations using different $\...
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Proving the equality in the attached Image

I am trying to learn the mathematics behind Finite elements as an engineer. can anyone help me with the proof below ? I have almost figured out the solution. The problem is I get a multiplication of ...
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Question related the pressure plot in time-dependent Navier--Stokes

I'm simulating the time-dependent Navier--Stokes equations using a Taylor-Hood finite element scheme: $$\dfrac{\partial u}{\partial t}-\nu\Delta u+(u\cdot\nabla)u+\nabla p=f$$ $$\nabla \cdot u=0$$ The ...
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How to discretize the trilinear form $m(w;,u,v) := \int_\Omega w^{2p} u \ v$ to get a matrix for FEM methods

I need to solve the following elliptic non-linear problem using the finite elements method with polynomials of degree $1$ and $2$: \begin{cases} -\Delta u + (u)^{2p} u = f \qquad & \text{in} \ \...
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Maximum norm stability for elliptic finite elements

I'm interested in an estimate of the type $$||u_h||_\infty \leq C ||f||_2$$ for $u_h$ solving $-\Delta u_h+c_0u_h = f$ ($u_h$ belongs to standard piecewise affine finite element space that are zero on ...
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Finite Element Method: discretization for off-diagonal elments of the mass matrix

I am reading a really excellent book on numerical methods for PDE called Computational Seismology, but Igel. He has a really nice treatment of finite element methods, but I was a bit confused about ...
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Why the inlet and outlet fluxes are different when I solve the Poisson equation by FEM over a trapezoidal domain?

I solved the Poission equation which is given by \begin{equation} \Delta h = 0, \end{equation} where $h$, in my case, is the hydraulic pressure. Because I want to solve steady-state flow, the $\nabla ...
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A posteriori estimates for Ricci-flat metrics

One formulation of the Calabi conjecture, proved by Yau, is the following (p.100 in Joyce: Compact Manifolds with Special Holonomy): Let $M$ be a compact, complex manifold of complex dimension $m$, $...
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Finite elements in $\mathbb R^d$ with obstable in $\mathbb R^{d-1}$

Is it possible to use some version of finite element method in a fluid problem (for example Stokes or Navier--Stokes problem) in a bounded domain $\Omega\subset\mathbb{R}^d$ with an obstacle (velocity ...
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MATLAB - Linear System With Symbolic And Numerical Constants

How do I solve the following system on MATLAB Click here to see the system The answer should be F1x = -1 and F5x = 1. Also, u2 = 0.005, u3 = 0.01, and u4 = 0.015
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Why are weather prediction models--such as hurricane forecasts--so accurate?

I have been watching the hurricane forecasting for Hurricane Henri, and it made me think about the accuracy of the forecasting for hurricane tracks. While I don't know much about meteorology, it seems ...
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