# 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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### Change of variables via Jacobian with Voigt notation for differential operator

I'm reading Non-linear Finite Element Analysis of Solids and Structures (De Borst, R., Crisfield, M. A., Remmers, J. J., & Verhoosel, C. V). As part of the weak form of the governing equations of ...
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### How to prove this inverse inequality about boundary flux when using FEM?

I'm currently studying the article by Pehlivanov et al.: Pehlivanov, A. I.; Lazarov, R. D.; Carey, G. F.; Chow, S. S., Superconvergence analysis of approximate boundary-flux calculations, Numer. Math. ...
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### How can the Finite Element Method be used to solve a Multi-equation System?

Every explanation I've seen of FEM show it solving an equation of the form $\mathcal{L}\phi=f$, with $\mathcal{L}$ being a differential operator, and every solution (whether it be Ritz or Galerkin) ...
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### What does this text mean in the context of the approximating solution for method of weighted residuals?

I am going through some introductory material on spectral/hp element methods and the book has a section on method of weighted residuals. I am paraphrasing the contents of the book below. If we ...
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### Variational formulation of the vector Laplace equation in cylindrical coordinates

I want to solve the vector Laplace equation $\nabla^2 \mathbf{v}=\mathbf{f}$ in arbitrary coordinate systems using finite-elements. The usual way to derive the variational form necessary for the ...
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### How to solve a boundary value problem with Neumann Boundary conditions using the Finite Element Method

I have been given the question: Consider the boundary value problem: $$-u'' + u = f(x) \forall x \in (0,1), u(0) = 0, u'(1) = 0.$$ Using continuous piecewise linear basis functions on a uniform mesh ...
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### Seeking name of "trick" involving operators like $A + \tau B$, where $B$ is Lipschitz.

Theorem. On Hilbert space $V$, suppose $T: V \to V$ is nonlinear and that $T = A + \tau B$ where $A$ is linear and strongly monotone, $B$ is nonlinear and Lipschitz, and $\tau > 0$ can be made ...
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### Mixed Laplace equation and missing one step in the integration by parts

I am looking at the Deal.ii finite element package docs, and was reading the section about vector valued problems. These are simply systems of PDEs that are solved together. I am trying to derive the ...
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### Finite element method: what are bilinear and linear forms?

Sample problem Consider this ODE: $$u : \Omega \to \Re$$ $$\nabla u-u=0 \qquad \mathrm{in} \qquad \Omega$$ $$\Omega\subset\Re$$ $$u = 1 \qquad \mathrm{on} \qquad \Gamma$$ $\Omega\subset\Re$ is the ...
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