Numerical-PDE chapter README and notation crosswalk

Advanced results Master

This chapter opens the numerical-PDE and finite-element exterior calculus strand. Its role is to separate applied discretisation material from the smooth differential-geometry units while keeping explicit links to functional analysis and differential forms.

The chapter begins with Sobolev spaces, weak formulations, and saddle-point stability. These are the analytic prerequisites for classical finite elements, mixed finite elements, and FEEC. The later FEEC units then reinterpret many mixed methods as finite-dimensional subcomplexes of the de Rham complex with a bounded cochain projection.

The planned subchapters are:

• 24.01-sobolev-and-weak-pdes: Sobolev spaces, Sobolev differential forms, weak elliptic problems, and Babuška-Brezzi stability.
• 24.02-classical-fem: conforming Galerkin methods, Céa estimates, Bramble-Hilbert approximation, and mixed Poisson.
• 24.03-discrete-de-rham-and-feec: Whitney forms, polynomial differential-form spaces, discrete de Rham complexes, bounded cochain projections, and the FEEC convergence theorem.
• 24.04-applications: mixed Hodge Laplacians, Maxwell edge elements, elasticity complexes, smooth FEEC, isogeometric exterior calculus, and virtual element exterior calculus.

Notation follows the Arnold-Falk-Winther and Arnold CBMS conventions where possible ^{[Arnold-Falk-Winther]}. Smooth differential forms use the notation already established in 03.04.02 and 03.04.04.

The main crosswalk is:

• ${\Lambda }^k$ denotes smooth differential $k$-forms.
• $L^2{\Lambda }^k$ denotes square-integrable $k$-forms.
• $H{\Lambda }^k$ denotes $L^2$ $k$-forms whose exterior derivative is also $L^2$.
• $H^s(\Omega )$ and $W^{k,p}(\Omega )$ denote scalar Sobolev spaces on a domain $\Omega$.
• $d$ denotes the exterior derivative; $\delta$ denotes its formal adjoint when a Riemannian metric is present.
• $\Delta =d\delta +\delta d$ denotes the smooth Hodge Laplacian.
• ${\mathfrak{H}}^k$ denotes harmonic $k$-forms.
• ${\Lambda }_h^k$ denotes a finite-dimensional discrete space of $k$-forms.
• ${\pi }_h:{\Lambda }^k\to {\Lambda }_h^k$ denotes a projection, usually required to commute with $d$.
• ${\mathcal{P}}_r{\Lambda }^k$ and ${\mathcal{P}}_r^{-}{\Lambda }^k$ denote full and trimmed polynomial differential-form spaces.
• ${\lambda }_i$ denotes a barycentric coordinate on a simplex.
• ${\mathcal{W}}^{\sigma }$ denotes the Whitney form associated to a simplex $\sigma$.
• $a(\cdot ,\cdot )$ denotes a continuous bilinear form; $a_h(\cdot ,\cdot )$ denotes its discrete counterpart.
• $b(\cdot ,\cdot )$ denotes the off-diagonal bilinear form in a saddle-point problem.

The key sequencing principle is that continuous stability comes before discrete stability. The analytic units 24.01.01, 24.01.02, 24.01.03, and 24.01.04 state the norms, spaces, weak equations, and inf-sup conditions. The FEEC units later prove that specific finite element spaces preserve enough of this structure to inherit stability and convergence.

Synthesis. FEEC is not merely a list of special finite elements. Its foundational idea is that the topology of the de Rham complex controls the stability of numerical methods for differential-form PDEs. This chapter therefore puts Sobolev analysis, weak PDEs, finite elements, and cochain complexes in one applied-mathematics home while preserving links back to smooth geometry.

Connections Master

• Banach spaces 02.11.04. Sobolev spaces are Banach spaces, and many weak-PDE arguments are functional-analytic arguments in disguise.

• Hilbert spaces 02.11.08. The most common energy methods in elliptic PDE use Hilbert-space geometry.

• Differential forms 03.04.02. FEEC discretizes spaces of differential forms rather than only scalar functions or vector fields.

• Exterior derivative 03.04.04. The commuting relation ${\pi }_{h}d=d{\pi }_h$ is the central structural condition in FEEC.

• De Rham cohomology 03.04.06. Discrete de Rham complexes aim to preserve cohomological information at the finite-dimensional level.

Historical & philosophical context Master

Classical finite element analysis grew from variational methods for elliptic PDEs and became a general numerical framework for engineering and applied mathematics. Mixed finite elements added saddle-point structure and flux variables, while computational electromagnetism exposed the need for curl- and divergence-conforming spaces.

Arnold, Falk, and Winther unified these phenomena by placing finite element spaces inside the language of differential complexes ^{[Arnold-Falk-Winther]}. Arnold's later CBMS lectures presented this finite element exterior calculus viewpoint as a compact graduate path through the subject ^[Arnold].

The philosophical shift is from element-by-element invention to structure-preserving discretisation. Instead of asking only whether basis functions approximate well, FEEC asks whether the discrete spaces form the right complex, preserve the right projections, and reproduce the stability mechanisms of the continuous PDE.

Bibliography Master

@article{ArnoldFalkWinther2006FEEC,
  author = {Arnold, Douglas N. and Falk, Richard S. and Winther, Ragnar},
  title = {Finite element exterior calculus, homological techniques, and applications},
  journal = {Acta Numerica},
  volume = {15},
  pages = {1--155},
  year = {2006}
}

@book{Arnold2018FEEC,
  author = {Arnold, Douglas N.},
  title = {Finite Element Exterior Calculus},
  series = {CBMS-NSF Regional Conference Series in Applied Mathematics},
  volume = {93},
  publisher = {SIAM},
  year = {2018}
}

@book{BoffiBrezziFortin2013Mixed,
  author = {Boffi, Daniele and Brezzi, Franco and Fortin, Michel},
  title = {Mixed Finite Element Methods and Applications},
  series = {Springer Series in Computational Mathematics},
  volume = {44},
  publisher = {Springer},
  year = {2013}
}