Isogeometric exterior calculus pointer

Advanced results Master

Isogeometric exterior calculus combines FEEC with isogeometric analysis. Instead of building finite element spaces on piecewise affine meshes, it uses spline and NURBS spaces closely related to the functions used in computer-aided design.

The motivation is geometric as well as numerical. CAD models often represent domains with spline surfaces and volumes. Traditional finite element workflows approximate that geometry with a mesh, creating a gap between design geometry and analysis geometry. Isogeometric analysis tries to use the CAD representation directly.

Buffa, Rivas, Sangalli, and Vazquez developed three-dimensional isogeometric discrete differential forms. Their construction builds compatible spline spaces that form a de Rham complex under the relevant derivative maps. This imports the FEEC requirement of compatibility into the spline/NURBS setting.

The spline setting changes the design problem. Tensor-product B-spline spaces have high regularity and structured knot data. Differential-form spaces must be chosen with degree shifts and continuity shifts so that derivative maps send each space into the next. Piola-type transforms then move the reference complex to the physical CAD geometry.

This is especially useful for electromagnetism, incompressible flow, shells, and structural analysis, where geometry quality and compatible field spaces both matter. The FEEC lens ensures that one does not lose exactness, kernel control, or trace compatibility while taking advantage of spline smoothness.

For this curriculum, the unit is a pointer because the full theory requires spline technology beyond the main FEEC chain. The conceptual message is direct: FEEC's de Rham-complex principle is not tied to simplicial polynomial elements. It also governs spline spaces on CAD-derived geometries.

The main stability questions remain familiar. One needs compatible spaces, commuting interpolation or projection operators, approximation estimates, and careful handling of geometry maps. When those are available, the isogeometric complex inherits the same structural advantages as finite element de Rham complexes.

Synthesis. Isogeometric exterior calculus says that CAD-compatible splines and FEEC-compatible complexes can be built together. The resulting methods aim to preserve both exact geometry and exact differential structure.

Connections Master

• Polynomial differential form spaces 24.03.03. Isogeometric forms replace polynomial simplex spaces with spline and NURBS spaces.

• Discrete de Rham complex 24.03.04. The defining requirement is still a compatible de Rham sequence.

• Bounded cochain projection 24.03.05. Commuting interpolation remains the stability bridge between continuous and discrete complexes.

• Maxwell FEEC edge elements 24.04.02. Electromagnetic simulation is one of the natural application domains for compatible spline complexes.

• Smooth FEEC 24.04.04. Both directions exploit high-continuity spaces while preserving complex structure.

Historical & philosophical context Master

Isogeometric analysis was introduced by Hughes, Cottrell, and Bazilevs as a way to close the gap between CAD geometry and finite element analysis ^{[Hughes-Cottrell-Bazilevs]}.

Buffa, Rivas, Sangalli, and Vazquez brought the FEEC/de Rham viewpoint into this setting by constructing isogeometric discrete differential forms in three dimensions ^{[Buffa-Rivas-Sangalli-Vazquez]}. Their work showed that compatible discretization can be expressed in spline language rather than only in classical finite element language.

The philosophical lesson is that compatibility is representation-independent. Whether the discrete space is built from polynomial elements, smooth macroelements, virtual degrees of freedom, or NURBS splines, the same operator-complex constraints determine whether the method respects the PDE's structure.

Bibliography Master

@article{BuffaRivasSangalliVazquez2011,
  author = {Buffa, Annalisa and Rivas, J. and Sangalli, Giancarlo and Vazquez, Rafael},
  title = {Isogeometric discrete differential forms in three dimensions},
  journal = {Mathematical Models and Methods in Applied Sciences},
  volume = {21},
  pages = {1421--1448},
  year = {2011}
}

@article{HughesCottrellBazilevs2005IGA,
  author = {Hughes, Thomas J. R. and Cottrell, John A. and Bazilevs, Yuri},
  title = {Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement},
  journal = {Computer Methods in Applied Mechanics and Engineering},
  volume = {194},
  pages = {4135--4195},
  year = {2005}
}

@article{ArnoldFalkWinther2006IGAPointer,
  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}
}