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‎dolfinx/main/cpp/demos/demo_biharmonic.html

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\forall \ K \in \mathcal{T} \right\}
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\]</div>
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<p>and considering the boundary conditions</p>
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<div class="amsmath math notranslate nohighlight" id="equation-46dc1b42-6e2f-4e54-9a6c-e8cdbc7e69ee">
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<span class="eqno">(1)<a class="headerlink" href="#equation-46dc1b42-6e2f-4e54-9a6c-e8cdbc7e69ee" title="Permalink to this equation"></a></span>\[\begin{align}
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<div class="amsmath math notranslate nohighlight" id="equation-4645999e-9631-4a07-a981-facbeb708596">
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<span class="eqno">(1)<a class="headerlink" href="#equation-4645999e-9631-4a07-a981-facbeb708596" title="Permalink to this equation"></a></span>\[\begin{align}
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u &amp;= 0 \quad {\rm on} \ \partial\Omega, \\
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\nabla^{2} u &amp;= 0 \quad {\rm on} \ \partial\Omega,
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\end{align}\]</div>

‎dolfinx/main/python/_downloads/4108f87eb9b678931168ffba26e2c302/demo_mixed-poisson.py

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reason = ksp.getConvergedReason()
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assert reason > 0, f"Krylov solver has not converged {reason}."
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ksp.view()
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sigma.x.scatter_forward()
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u.x.scatter_forward()
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# -
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# We save the solution `u` in VTX format:

‎dolfinx/main/python/_downloads/9561cf6cd1a70079b986f4eb31580cbd/demo_poisson_matrix_free.ipynb

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"cells": [
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{
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"cell_type": "markdown",
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"id": "6d0de063",
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"id": "09e1e263",
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"metadata": {},
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"source": [
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"# Matrix-free conjugate gradient solver for the Poisson equation\n",
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "633f21a3",
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"id": "2f8640b8",
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"metadata": {},
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"outputs": [],
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"source": [
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "77d44a81",
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"id": "13e718b0",
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"metadata": {},
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"outputs": [],
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"source": [
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "c350213f",
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"id": "615215ca",
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"metadata": {},
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"outputs": [],
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"source": [
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},
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{
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"cell_type": "markdown",
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"id": "d2b90a39",
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"id": "29c41679",
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"metadata": {},
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"source": [
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"We begin by using {py:func}`create_rectangle\n",
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "ef89e9a2",
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"id": "59d9fcb4",
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"metadata": {},
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"outputs": [],
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"source": [
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "05ab025a",
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"id": "23d7d3cf",
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"metadata": {},
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"outputs": [],
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"source": [
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},
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{
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"cell_type": "markdown",
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"id": "4973aa5c",
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"id": "2cb5311d",
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"metadata": {},
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"source": [
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"The second argument to {py:class}`functionspace\n",
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We now find the degrees of freedom that are associated with the boundary\n",
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"source": [
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"and use {py:func}`dirichletbc <dolfinx.fem.dirichletbc>` to define the\n",
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{
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"cell_type": "code",
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"execution_count": null,
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},
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{
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"cell_type": "markdown",
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"Next, we express the variational problem using UFL."
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{
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"cell_type": "code",
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"execution_count": null,
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"For the matrix-free solvers we also define a second linear form `M` as\n",
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{
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"cell_type": "code",
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"execution_count": null,
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},
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{
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"cell_type": "markdown",
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"### Matrix-free conjugate gradient solver\n",
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{
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{
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},
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"metadata": {
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"lines_to_next_cell": 2
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{
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"cell_type": "code",
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},
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{
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},
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"This matrix-free solver is now used to compute the finite element\n",
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{
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