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Merge branch 'develop' into 263-move-data-contribution-guide
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lberti authored Mar 14, 2024
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:feelpp: Feel++
:stem: latexmath
:page-jupyter: true
:sectanchors: true
:sectnums: |,all|
:tocdepth: 3

== Introduction

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We suppose that stem:[\Omega] is a polyhedron of stem:[\mathbb{R}^3] and we consider a regular and conformal family of affine meshes of stem:[\Omega] that we note stem:[\left\{\mathcal{T}_h\right\}_{h>0}].
We choose as reference finite element stem:[\left\{\widehat{K}, \widehat{P}, \widehat{Sigma}\right\}] a Lagrangian finite element of degree stem:[k \geq 1].

Homogeneous Dirichlet problem::
=== Homogeneous Dirichlet problem

In order to construct a stem:[V_{\mathrm{D}}]-conformal approximation space, we pose
[stem]
++++
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The <<estimation-conv,estimation>> results from the lemma of Céa and the interpolation theorem which we apply component by component. The estimate results from the Aubin-Nitsche lemma.
====

Pure traction problem::
=== Pure traction problem

For the pure traction problem, one way to eliminate the arbitrary rigid displacement at the discrete level is to:
[lowerroman]
. impose that the displacement at one mesh node, stem:[a_0], is zero;
. choose three other mesh nodes, stem:[a_1, a_2, a_3], and three unit vectors, stem:[\tau_1, \tau_2, \tau_3], such that the set stem:[\left\{\left(a_i-a_0\right) \times \tau_i\right\}_{1 \leq i \leq 3}] forms a basis of stem:[\mathbb{R}^3];
. impose that the displacement at the node stem:[a_i] along the direction stem:[\boldsymbol{\tau}_i] is zero.

[.def#def:space]
.Pure Traction approximation space
[.def#def:space-pure-traction]
****
This leads to the approximation space
[stem]
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