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chapter1/complex_mode.py

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# extension: .py
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# jupytext_version: 1.18.1
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# display_name: Python 3 (DOLFINx complex)
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chapter1/fundamentals.md

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FEniCSx is based on the finite element method, which is a general and
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efficient mathematical technique for the numerical solution of
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PDEs. The starting point for finite element methods is a PDE
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expressed in _variational form_. For readers not familiar with variational problems, we suggest reading a proper treatment on the finite element method, as this tutorial is meant as a brief introduction to the subject. See the original tutorial {cite}`FenicsTutorial` (Chapter 1.6.2).
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expressed in _variational form_. For readers not familiar with variational problems, we suggest reading a proper treatment on the finite element method, as this tutorial is meant as a brief introduction to the subject. See the original tutorial {cite}`fd-FenicsTutorial` (Chapter 1.6.2).
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The basic recipe for turning a PDE into a variational problem is:
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where $\frac{\partial u}{\partial n}=\nabla u \cdot \vec{n}$ is the derivative of $u$ in the outward normal direction $\vec{n}$ on the boundary.
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Another feature of variational formulations is that the test function $v$ is required to vanish on the parts of the boundary where the solution $u$ is known. See for instance {cite}`Langtangen_Mardal_FEM_2019`.
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Another feature of variational formulations is that the test function $v$ is required to vanish on the parts of the boundary where the solution $u$ is known. See for instance {cite}`fd-Langtangen_Mardal_FEM_2019`.
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In the present problem, this means that $v$ is $0$ on the whole boundary $\partial\Omega$. Thus, the second term in the integration by parts formula vanishes, and we have that
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## References
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```{bibliography}
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:filter: cited and ({"chapter1/fundamentals"} >= docnames)
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:filter: cited
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:labelprefix:
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:keyprefix: fd-
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```

chapter1/fundamentals_code.ipynb

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"\n",
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"Author: Jørgen Schartum Dokken\n",
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"\n",
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"This implementation is an adaptation of the work in {cite}`FenicsTutorial` to DOLFINx.\n",
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"This implementation is an adaptation of the work in {cite}`fundamentals-FenicsTutorial` to DOLFINx.\n",
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"\n",
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"In this section, you will learn:\n",
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"- How to use the built-in meshes in DOLFINx\n",
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"so it is sufficient to use a common space for the trial and test function.\n",
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"\n",
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"We use the {py:mod}`Unified Form Language<ufl>` (UFL) to specify the varitional formulations.\n",
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"See {cite}`ufl2014` for more details."
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"See {cite}`fundamentals-ufl2014` for more details."
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]
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},
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{
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"metadata": {},
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"source": [
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"```{bibliography}\n",
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" :filter: cited and ({\"chapter1/fundamentals_code\"} >= docnames)\n",
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" :filter: cited\n",
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" :labelprefix:\n",
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" :keyprefix: fundamentals-\n",
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"```"
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]
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}

chapter1/fundamentals_code.py

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#
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# Author: Jørgen Schartum Dokken
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#
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# This implementation is an adaptation of the work in {cite}`FenicsTutorial` to DOLFINx.
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# This implementation is an adaptation of the work in {cite}`fundamentals-FenicsTutorial` to DOLFINx.
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#
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# In this section, you will learn:
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# - How to use the built-in meshes in DOLFINx
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# so it is sufficient to use a common space for the trial and test function.
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#
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# We use the {py:mod}`Unified Form Language<ufl>` (UFL) to specify the varitional formulations.
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# See {cite}`ufl2014` for more details.
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# See {cite}`fundamentals-ufl2014` for more details.
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# -
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# ```{bibliography}
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# :filter: cited and ({"chapter1/fundamentals_code"} >= docnames)
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# :filter: cited
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# :labelprefix:
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# :keyprefix: fundamentals-
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# ```

chapter1/membrane_code.py

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chapter1/nitsche.py

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chapter2/amr.py

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# ---
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# # Adaptive mesh refinement with NetGen and DOLFINx

chapter2/diffusion_code.py

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chapter2/heat_code.py

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chapter2/helmholtz_code.py

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