Basis functions

13. Basis functions#

  • Hierarchical basis using lowest order, plus bubbles

  • Basis for higher order tensor-fields

  • Here: 3D case

13.1. Basis functions for \(H^1\):#

continuous functions

  • vertex basis: barycentric coordinatges \(\lambda_i\)

  • edge, face, cell bubbles: scalar-valued polynomial functions depending on barycentric coordinates

    \[u_i(\xi), \quad u_i(\xi) v_j(\eta), \quad u_i(\xi) v_j(\eta) w_k(\zeta)\]

13.2. Basis functions for \(H(\operatorname{curl})\):#

tangential-continuous functions

  • lowest order Whitney forms: \(\lambda_i d\lambda_j - \lambda_i d \lambda_j\)

  • high order: \(d u_i, u_i d v_j - v_j du_i\)

forms can be canonically restricted to facets (see course by Ralf)

Implemented in Euklidean space: \(u_i dv_j - v_i du_j = u_i \nabla v_j - v_j \nabla u_i\)

See thesis Sabine Zaglmayr, 2006 for various element shapes (trigs, quads, tets, prisms, hexes)

13.3. Basis functions for \(H(\operatorname{div})\):#

normal-continuous functions

  • lowest order Whitney forms: \(\lambda_i d\lambda_j \wedge d \lambda_k + \lambda_j d\lambda_k \wedge d\lambda_i + \lambda_k d\lambda_i \wedge d\lambda_j\)

  • high order: \(u_i dv_j \wedge dw_k\)

two-forms, identified with vectors \(\nabla \lambda_i \times \nabla \lambda_j\) in \({\mathbb R}^3\)

13.4. Matrix-valued spaces (high order tensors with symmetries)#

13.4.1. Regge space \(H(\operatorname{curl curl})\):#

\(tt\)-continuous matrix fields

  • lowest order: \(d \lambda_i \odot d \lambda_j\) (symmetric product)

  • high order: \(u_i v_j w_k d \lambda_i \odot d \lambda_j\)

Snorre Christiansen 2011, Lizao Li 2018, Michael Neunteufel 2021

13.4.2. MCS space \(H(\operatorname{curl div})\)#

\(nt\)-continuous matrix fields:

  • \((d u_i \wedge d v_j) \otimes d w_k\)

3-tensors with skew-symmetry are identified with 2-tensors \(A\) (matrices):

\[ A(\nabla u_i \times \nabla v_j, \nabla w_k) = (d u_i \wedge d v_j) \otimes d w_k \]

Bianchi identity leads to trace-free matrices

Theses Philip Lederer, 2019

13.4.3. TDNNS space \(H(\operatorname{div div})\)#

\(nn\)-continuous matrix fields:

  • \((d u_i \wedge d v_j) \odot (d w_k \wedge d u_i)\)

4-forms with skew-symmetries are identified with symmetric 2-tensors (matrices):

\[ A(\nabla u_i \times \nabla v_j, \nabla w_k \times \nabla u_i) = (d u_i \wedge d v_j) \odot (d w_k \wedge d u_i) \]

Theses Astrid Sinwel (aka Pechstein), 2009

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