Questions tagged [finite-element-method]

The finite element method is a popular method for approximating numerically on a computer the solution of partial differential equations. It is based on a variational (weak) formulation of the PDE, followed by discretization on a finite-dimensional ansatz space which reduces the problem to a sparse linear algebra one.

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Low order quadrature for low order terms in finite element method- convergence analysis

Consider the PDE $$\frac{\partial u}{\partial t} = \Delta u - \kappa u,$$ posed over a bounded domain $\Omega$ (say, $\Omega \subseteq \mathbb{R}^2$). Assume we have a family of triangulations $\...
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73 views

Typo in error a-priori estimate in a discontinuous Galerkin paper?

I'm looking at this famous paper which is available in the link below: Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
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272 views

Difference between variation and differential

Been trying to understand how the numerical formulation for structural elements used in FEM are derived. Came across this piece from "Fundamentals of FEM for Heat and Fluid Flow" by Roland ...
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47 views

Making rigorous a suggested Newton-Raphson like approach for finding root of a functional on $H^1_0$

I am trying to figure out the reasoning behind the proposed method in the following document. So we have the following non-linear equation on the unit square $\Omega=[0,1]^2$ given by $$-\mu \Delta u+\...
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38 views

Is it desirable to establish a CAD-like geometry medium for four dimensional space-time topologies and FEM? [closed]

Finite element and numerical methods over 4d space-time is a topic of interest for elastodynamics right now. Do you think it is desirable to establish a geometry/CAD system for space-time involving ...
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48 views

Continuity of weak solution of elliptic PDE

I am investigating the following standard elliptic PDE with mixed Dirichlet-Neumann boundary condition: $-\Delta u=f$ on $\Omega$; $u=0$ on $\Gamma_D$; $\left<n,\nabla u=g\right>$ on $\Gamma_N$ ...
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66 views

Boundary integration of weak form in FEM using DG elements

If we use CG elements (continuous Galerkin), the boundary integration in FEM can be easily converted to sum over quadrature points using node basis functions of the edges. However, in DG elements (...
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241 views

Generate a two-variable polynomial from its "roots [closed]

I know, from mathematics basis, that a polynomial with one variable can be factor in function of its roots, so I can generate any one-variable polynomial from its zeros. But I want know if is ...
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141 views

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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66 views

Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
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287 views

Galerkin finite element method for solving third order time dependent partial differential equation in the weak form

How to solve the third order time dependent partial differential equation $$u_t + 6u_x + u_xxx = 0$$ in weak form using galerkin finite element method?
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60 views

stability of parabolic problems where nonhomogeneous term in $L^2(0,T; H^{-1}(\Omega))$

There is a problem in this estimate. In Part 2, We can't derive the second inequality from the first inequality in Part 2. The main reason is the last term on the RHS in the first inequality is in $\|\...
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149 views

Solve 4th order ODE with variable coefficients

I am trying to solve a 4th order boundary value problem with variable coefficients, namely the problem of a rotating cantilever beam: $u'''' - \frac{((1-x^2)u')'}{2\eta} - \frac{\alpha}{\eta}((1-x)u')...
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76 views

Estimate a projection from a product space of $H^1(\mathbb{R}^3)$ to a finite dimensional space

Let $\mathcal{H}=(H^1(\mathbb{R}^3))^N$ be the product space with the associated norm $$ \Vert U\Vert_1=\left(\sum^N_{i=1}\Vert u_i\Vert_1^2\right)^{1/2} $$ where $U=(u_1,u_2,...,u_N)\in\mathcal{H}$. ...
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151 views

How to solve Poissons equation using FEM with integral BC?

I have a 3-dimensional domain D, with 3 types of BC which I am trying to solve the Poisson equation on. $$-\nabla \cdot (\sigma(x,y,z)\nabla u)=0$$ Insulator (Neumann BC) Electrode set at some ...
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252 views

Finite Element Method and Dirac eigenvalue problem

Consider Dirac equation in 2D with Hamiltonian given by (arb. units) \begin{equation} H=-i \begin{pmatrix} 0&\partial_x-i\partial_y\\ \partial_x+i\partial_y & 0\\ \end{pmatrix}. \end{equation} ...
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127 views

Time discretization of the variational formulation of the Navier-Stokes equation

Let $T>0$ $I:=(0,T]$ $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$$ and $$V:=\overline{...
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165 views

Bounding the determinant of the Jacobian between a set and its polyhedral approximation

My question is, essentially, suppose I have two simply connected subset of $R^n$, if I know that the boundary's of both are very close, how can I bound the determinant of the Jacobian between them. I ...
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56 views

Compatible Finite Elements [closed]

I have a basic question as a matter of definition. I am wondering what is meant by compatible finite elements? Does it has to do with the spaces over which the trial functions are defined?
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46 views

Approximation error estimate

I would like to find a good reference for the following or a similar, probably well-known, approximation error result: Let $\Omega\subset \mathbb{R}^d$ be bounded, $p\in [1,\infty]$, $l, m\in \mathbb{...
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263 views

Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation: $i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields: $i(u_t,v) + \beta (u_x,v_x) = 0$ ...
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281 views

Weak form of the Laplace-Beltrami operator on closed manifolds

Suppose we are dealing with diffusion over a boundaryless manifold $M$ (for simplicity let's say it's a surface). In that case, we have \begin{align} \int_M W \Delta U \mathrm{d} x & = -\int_M \...
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104 views

The classical two phase Stefan problems

What is the most commonly used treatment method of the moving interface in the classical two phase Stefan problems with the finite element method. Here I mean the water-ice two phase problem under ...
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64 views

Sobolev spaces, Finite Element Error Analysis

Let $\Omega\subset \mathbb{R}^{2}$ be a bounded, convex, polygonal domain and $H=\{(u_{1},u_{2})\in H^{\epsilon-\frac{1}{2}}(\Omega)\times H^{\epsilon-\frac{1}{2}}(\Omega),0<\epsilon<\frac{1}{2}:...
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604 views

Cea's lemma and norms

I would like your help understanding this article. Page 239 (3.2 A priori error estimates), I am quickly getting lost because of the type of norm that is always changing. Things I do not ...
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897 views

Comparison of cubic Hermite finite element and cubic B-spline finite element (regarding condition number of stiffness matrix)

Background: Consider the one-dimensional second-order elliptic PDE, $$ \left\{\!\! \begin{aligned} & -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\ & u(0)=u(1)=0 \end{aligned} \...
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Finite element method inverse estimate

$\DeclareMathOperator\diam{diam}$Looking for a proof in the literature of the following lemma: Let $K\subset\mathbb{R}^d$ be a bounded domain. Let $P_X$ be a finite dimensional subspace of $\mathcal{...