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.

20 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
5 votes
1 answer
2k views

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{...
alext87's user avatar
  • 3,167
3 votes
0 answers
116 views

Is Stokes equation a saddle point problem or a minimum problem?

Consider $\Omega \subset \mathbb{R}^2 $ (or $\mathbb{R}^3$). The well known stationary Stokes equations in the incompressible case are \begin{equation} \begin{cases} - \Delta u + \nabla p = f \text{ ...
tommy1996q's user avatar
3 votes
0 answers
69 views

Confusion with implementation of PDE constraint Bayesiain inverse problem

Consider a PDE, $$\partial_t u -a \nabla u - ru (1-u) = 0$$ at a given snapshot in time. The inverse problem is to find the diffusion coefficient $a \in L^{\infty}$ from a noisy measurement $$Y = \Phi(...
Jarwin's user avatar
  • 81
3 votes
0 answers
385 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 \...
Yamamoto's user avatar
2 votes
0 answers
47 views

finding weak form of nonlinear differential equation for FEM simulation

The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
Hari Sam's user avatar
2 votes
0 answers
237 views

A general question about spectral methods vs finite element methods

According to this Wikipedia article: Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple ...
UserA's user avatar
  • 587
2 votes
0 answers
64 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 $\Gamma_N$ ...
Zherong Pan's user avatar
2 votes
0 answers
221 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{{\...
David's user avatar
  • 21
2 votes
0 answers
162 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')...
resalmon's user avatar
2 votes
0 answers
75 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}:...
s.chowdhury's user avatar
2 votes
0 answers
919 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} \...
Roun's user avatar
  • 163
1 vote
0 answers
49 views

Nitsche's method for p-Laplace equation

My question is about how you impose Dirichlet boundary conditions for the p-Laplace equation. The minimization form of this problem is to find the function $u$ in $W_1^p(\Omega)$ that minimizes the ...
Daniel Shapero's user avatar
1 vote
0 answers
43 views

Holomorphic "quasi-interpolation" of a function sequence

I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...
Sébastien Loisel's user avatar
1 vote
0 answers
75 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 ...
mueller_seb's user avatar
1 vote
0 answers
61 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 $\|\...
Ariel So's user avatar
  • 111
1 vote
0 answers
79 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 $U=(u_1,u_2,...,u_N)\in\mathcal{H}$. ...
Q-Y's user avatar
  • 163
1 vote
0 answers
50 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{...
user35593's user avatar
  • 2,286
0 votes
0 answers
49 views

How to solve with FEM a semilinear elliptic equation?

I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
Bogdan's user avatar
  • 1,330
0 votes
0 answers
32 views

Quadrature error estimates for $n$-rectangular finite elements in the context of elliptic second order problems

In Ciarlet's book "Finite Element Methods for Elliptic Problems" from 1978, in Chapter 4.1 "The Effect of Numerical Integration", the following Theorem is stated and proved: #######...
Phasefieldlover's user avatar
0 votes
0 answers
138 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{...
0xbadf00d's user avatar
  • 161