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# Questions tagged [finite-element]

A means of solving ordinary and partial differential equations. The domain of the problem is broken up into elements, and the solution in each element is expanded in a basis of functions. The Finite Element Method lends itself well to adaptive refinement, irregular geometry, and good error estimates.

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### Could you please recommend some books or lectures about immersed boundary FEM methods?

I want to learn IB for FEM and do some FSI problems, could you please recommend some great books or lecture notes on it?thank you
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### What do diagonal (DOF-to-self) terms of stiffness matrix physically mean?

I am used to interpreting each entry of a solid mechanic system's stiffness matrix as a 1D (linear or angular) spring joining one DOF (column index) to another (row index). But this interpretation ...
1 vote
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### How to constraint the tangential gradient on a boundary in FEniCS?

The problem I'm considering is a 2D scalar PDE. The domain $\Omega$ is a disk with two holes $\partial\Omega_1$ and $\partial\Omega_2$ and an external boundary $\partial\Omega_0$. The PDE and boundary ...
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### Can finite element exterior calculus be used for the proof of discrete stability?

I've heard about Finite Element Exterior Calculus (FEEC) and its applications in numerical simulations, but can FEEC be utilised to prove discrete stability in computational methods? If so, can the ...
1 vote
656 views

### If FEM is exact at the nodes, why do first and second-order elements give very different results?

I'm looking at the solution to a structural mechanics problem that is modeled with first-order elements and then as a comparison with second-order elements. It is clear that the first-order elements ...
1 vote
132 views

### Can domain decomposition methods also be applied to linear systems resulting from finite difference discretizations?

Question In general, do domain decomposition methods (DDMs) require a linear system of equations $Au = f$ to be formed by finite element discretization/method (FEM) of a PDE? Or could one simply use a ...
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### Gradient of field computed using FEM (L2 Projection)

From my understanding, in the finite element context, an L2 projection can be formulated as the following linear system: $$M\ \phi = \beta$$ where $M$ is the mass matrix, $\phi$ the resulting ...
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### Computational efficiency of Galerkin projection in AMG

I have been using recently AMG as preconditioner for CG with several meshes for simple elliptic problems discretised with linear elements on "complicated" three dimensional geometries and I ...
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1 vote
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### Computing Tangential Derivative using the Dirichlet value

Let $\Gamma$ be a smooth boundary of a domain $\Omega$. Let $u = g$ on $\Gamma$. How can I compute the tangential derivative of the function $u$ using the information that $u = g$ on $\Gamma$? Please ...
130 views

### "Cleanest, most professional" approach to implement a finite differences scheme in dimension greater than two

Coding a finite difference algorithm in 1D does not require a complex mesh. In higher dimensions, you would need a mesh and its connectivity graph to compute the differential operators. Finite ...
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### Interpolation constant on triangles

There are quite a few references regarding the estimation for the interpolation error for the piece-wise affine finite elements. I find one particular estimate interesting (and useful in my case), ...
51 views

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### Quintic Hermite shape functions

I am trying to use quintic Hermite basis functions for FEM applications, could someone please direct me to the general formula that would help me generate quintic Hermite shape functions? In natural ...
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### Any code on ALE method for N-S equations or advection-diffusion equations?

I'm currently reading the book computational fluid-structure interaction, but the book doesn't provide any code, and I can't understand the ALE method, could you please provide some codes written in ...
1 vote
92 views

### Reference request: graph Laplacian approximation for domains/manifolds

Is there any reference on solving the heat equation on irregular geometries via creating a mesh and using the graph Laplacian instead of FEM techniques? (Convergence to real solution). That is to say, ...
1 vote