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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How to project a function represented by a neural network into finite element spaces
Suppose we trained a neural network to fit a solution of a PDE, but we want to do something in a Finite Element Space, so we need transform our neural network to the latter. What is the way to do this ...
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Regularity of eigenfunctions of elliptic operator with piecewise-constant coefficients
Consider the generalized eigenvalue problem:
$$ [- \nabla \cdot (D(\mathbf{x}) \nabla) + \Sigma_a(\mathbf{x})] \phi(\mathbf x) = \lambda \Sigma_f(\mathbf x) \phi(\mathbf x)$$
Some specifications:
The ...
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Approximation of $H_\text{imp}$-functions by Nédélec-functions
All references refer to "Finite Element Methods for Maxwell's Equations" by Monk.
Preliminaries: Let $\Omega\subset \mathbb{R}^3$ be a bounded lipschitz domain. The space $H(\text{curl})$ is ...
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What is the standard notation for bilinear, biquadratic, etc... spaces?
A typical notation for the polynomials of degree $k$ is $P_k$. The space $P_k$ is considered well-suited for interpolation on simplices, although that is hard to put into practice in full generality.
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Lumped mass matrices and bubble functions for tetrahedral elements
For whatever reason, I stubbornly decided to use tetrahedral elements and find myself needing to use P3 elements with bubble functions ("P3b3d" in FREEFEM-style denomination).
The 2d case is ...
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Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)
Intro
Suppose we have the following static linear equations (e.g. of an elastostatic problem):
$$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$
We want a multipoint constraint of the type
$$\boldsymbol{\...
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Nitsche's method for p-Laplace equation
My question is about how you impose Dirichlet boundary conditions for the p-Laplace equation.
The minimization form of this problem is to find the function $u$ in $W_1^p(\Omega)$ that minimizes the ...
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Is Stokes equation a saddle point problem or a minimum problem?
Consider $\Omega \subset \mathbb{R}^2 $ (or $\mathbb{R}^3$). The well known stationary Stokes equations in the incompressible case are
\begin{equation}
\begin{cases}
- \Delta u + \nabla p = f \text{ ...
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Confusion with implementation of PDE constraint Bayesiain inverse problem
Consider a PDE,
$$\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(...
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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 ...
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Pressure integrated by parts in finite element method
Most FEM texts or tutorials apply FEMs on diffusion equations where the 2nd spatial derivative is integrated by parts during weak formulation. For convection diffusion equations, there is a first ...
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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 ...
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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 ...
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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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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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