#Choosing an operator

Seven classes project a field onto finite element forms. They differ in what they guarantee, what input they accept and what they cost. Every statement below is one that the validation suite checks, with the evidence named there; where something does not hold, that is said here too.

#The short answer

#At a glance

"Commutes" means d Π u = Π d u. "Keeps traces" means the degrees of freedom on the boundary of Π u equal those of u. "Bounded" means the L² norm of Π u on a cell is bounded by the norm of u on the cells around it, with a constant that does not depend on the mesh size.

"On its class" means: commuting is proved and tested for fields whose boundary trace is a lowest-order trace, and measured to fail by about 5·10⁻³ on a 4×4×4 mesh for other fields. The papers' statement is wider than its proof; the reason is explained in Validation.

#Projector

The canonical lowest-order projection: the coefficient of each vertex, edge and face is the point value, the edge integral or the face flux of the input, taken directly. It is a projection, keeps boundary traces and commutes (Stokes' theorem makes this exact). It is not bounded: a field that is unbounded at a vertex has no point value there, and Projector raises ProjectError instead of returning Infinity. Use it when your data are smooth and you want the cheapest operator. It also provides projectAtPoint, sampler (project many fields at fixed points) and refine.

#CanonicalProjector

The same idea in any polynomial degree (degree: p gives Lagrange, Nédélec, Raviart–Thomas and discontinuous spaces of order p + 1). Checked for p = 0, 1, 2 in all four form degrees: it is a projection, the traces are conforming, it commutes with the exterior derivative, and it converges at the expected orders. Like Projector it is not bounded and needs regular input.

#ZeroTraceProjector

The interior building block of the papers: a projection onto the forms with zero trace, locally L²-bounded and commuting, computed from weight functions instead of point values, so it is finite for rough input. By itself it is useful for problems with homogeneous boundary conditions. StableProjector and TracePreservingProjector are built on it.

#StableProjector

Π u = Π_∂ u + Π̊ (u − Π_∂ u) with a canonical boundary part. The interior is stable for rough input (a field like |x − c|^(−0.4), unbounded at an interior vertex, is projected with a finite result); the boundary part uses point values and fluxes, so the boundary data must be finite. Commuting holds to roundoff on its class.

#BoundedCommutingProjector

The operator of Theorem 5.1 of Ern, Guzmán, Potu and Vohralík with no boundary condition: a weight for every entity, boundary included. It is the only lowest-order operator here that is both bounded and commuting for every L² input with an L² derivative. It does not keep boundary traces. Choose it when you need commuting and stability and can live without exact boundary values (for instance when the boundary condition is imposed afterwards).

#TracePreservingProjector

The full operator of the trace-preserving paper, Π = mΠ (I − Π_∂) + Π_∂, where Π_∂ is built from the paper's boundary weights on Worsey–Farin splits of the boundary and mΠ is the zero-trace bounded projection. It reproduces the spaces, keeps the discrete trace on the boundary and commutes with the exterior derivative for fields whose trace is a discrete trace. You supply the derivative of your field as well as the field: the gradient for a 0-form, the curl for a 1-form, the divergence for a 2-form.

At degree: 0 (the default) it has coefficients on the Whitney forms and, at the boundary, is stable (ratio about 1.01 for 0-forms and 1.006 for 1-forms, flat under refinement). With degree: p for p ≥ 1 it works field by field: projector.local(u, du, l) gives the value and the derivative of the projection at a point of a cell, computed lazily from the weights near that cell. Validated at p = 1 for all four form degrees (a projection to 5·10⁻¹³; commuting for fields with a discrete trace to 10⁻¹¹ at quadrature order 8) and at p = 2 for 0-forms. Its stability constant above the lowest order has not been measured.

#HigherOrderProjector

The locally L²-bounded commuting projection of Arnold and Guzmán for any degree p. It reproduces its own space, commutes with d for every L² field with an L² derivative (measured to 3·10⁻¹² for 0→1, 1→2 and 2→3 at p = 1), and its norm on narrow bumps did not grow under refinement (ratio between 0.995 and 1.007 at p = 1 for all four form degrees). With boundaryConditions: true it is the zero-trace version. It does not keep boundary traces. It needs a mesh fine enough that the extended stars are topological balls; on a coarse mesh near the boundary the stars grow and the local problems become very large (see Performance).

#What to do about boundary values

If the boundary values of your field must be reproduced exactly and your field is smooth, use Projector or CanonicalProjector. If you need both exact boundary values and stability for rough interior data, use StableProjector (finite boundary values, lowest order) or TracePreservingProjector (any degree). If you need a degree above the lowest, stability and commuting without preserving the boundary trace, HigherOrderProjector is the cheaper choice. See Limits for what is validated at each degree.